A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS arXiv:2110.01649v1 [math.KT] 4 Oct 2021 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Abstract. Let Γ be a finite dimensional Lie group and consider the smooth double loop group, i.e. the Fréchet Lie group of smooth maps from the 2-torus to Γ. For a finite dimensional Hilbert space V , let H denote the Hilbert space of vector valued L2 -functions on the 2-torus. The purpose of this paper is to construct a higher central extension of the smooth double loop group from the representation of the smooth double loop group on H induced by a smooth action of Γ on V . This higher central extension comes from an action of the smooth double loop group on a 2-category and yields a group cohomology class of degree 3 on the smooth double loop group. We show by a concrete computation that this group cohomology class is non-trivial in general. We relate our higher central extension to the Kac-Moody extension of the smooth single loop group as a higher dimensional analogue of the latter. More generally, given a group G acting on a bipolarised Hilbert space, we apply higher category theory to construct a group cohomology class of degree 3 on G. As a second motivating example, we use these ideas to introduce a higher central extension of the group of invertible smooth functions on the noncommutative 2-torus. Contents 1. Introduction 1.1. Higher central extensions 1.2. Groups acting on higher categories and group cocycles 1.3. Relation to the Tate tame symbol and the multiplicative character 1.4. The structure of the paper 1.5. Acknowledgements 2. Determinants of Z/2Z-graded vector spaces 3. The torsion isomorphism of Fredholm operators 4. The perturbation isomorphism 4.1. Perturbation commutes with torsion 5. The stabilisation isomorphism 6. The group 2-cocycle associated to a group action on a category 6.1. The group 2-cocycle on the restricted general linear group 7. Comparison with the multiplicative character 8. The group 3-cocycle associated to a group action on a coproduct category 8.1. Coproduct categories 8.2. 3-cocycles 9. Categories associated to representations of a ring 9.1. The category Lp and its dual L†p 9.2. The composition of morphisms 10. The coproduct category associated to representations of a ring 11. Change of base point 12. Group actions and main result 13. Coproduct categories and group cocycles from bipolarised representations 13.1. Group 3-cocycles on double loop groups 2 2 4 8 10 11 11 14 15 17 19 20 21 22 25 25 27 32 34 35 38 41 43 45 48 2020 Mathematics Subject Classification. 18N10, 20J06, 22E67, 47A53; 18F25, 19K56, 46L80, 58B34. Key words and phrases. Higher category theory, Group cocycles, Picard categories, Loop groups, Algebraic K-theory, Multiplicative character, Fredholm determinants, Perturbation isomorphisms, Torsion isomorphisms. The first author gratefully acknowledge the financial support from the Independent Research Fund Denmark through grant no. 9040-00107B and 7014-00145B. The second author would like to acknowledge the support of the Danish National Research Foundation through the Center for symmetry and Deformations (SYM). The third author is partially supported by NSF grants DMS-1400349, DMS-1811846 and DMS-1944862. 1 2 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON 14. Non-triviality of group 3-cocycles on double loop groups 14.1. The coproduct 14.2. The group action 14.3. The composition 14.4. Non-triviality 15. Proofs of properties of the composition in Lp and its dual L†p 15.1. Associativity 15.2. Unitality 15.3. Duality 16. Proofs of properties of the change of base point 16.1. Cofunctoriality 16.2. The dual change of base point 16.3. The symmetrised change of base point 16.4. Binary and ternary versions of the change of base point 16.5. Functoriality References 50 53 54 56 60 61 62 68 68 69 70 71 74 77 82 85 1. Introduction 1.1. Higher central extensions. Let Γ be a finite dimensional Lie group. The smooth double loop group of Γ is the Fréchet Lie group C ∞ (T2 , Γ) of smooth maps from the 2-torus to Γ. The purpose of this paper is to construct higher central extensions of the smooth double loop group by the group C∗ of invertible complex numbers. These higher central extensions yield classes in the third group cohomology of C ∞ (T2 , Γ) (viewed as a discrete group) and we show that these classes are non-trivial in general. The motivation for the constructions in this paper comes from the central extension of a group G acting on /H / GL(H) and an orthogonal projection P : H a polarised Hilbert space H. For a representation ρ : G we define the bounded idempotent Pg := ρ(g)P ρ(g −1 ) for all g ∈ G . / GL(H) when the difference Pg −P : H / H is a Hilbert-Schmidt We recall that P is a polarisation of ρ : G operator for all g ∈ G. The data of a polarisation of a representation determines a central extension of the group in question by the group C∗ of invertible complex numbers: / C∗ /E /G /1. 1 Suppose for example that the finite dimensional Lie group Γ acts smoothly on a finite dimensional Hilbert space V and let H := L2 (T) ⊗ V denote the Hilbert space of vector valued L2 -functions on the 1-torus. The smooth single loop group G := C ∞ (T, Γ) is then represented on H and the orthogonal projection P onto the vector valued Hardy space H 2 (T) ⊗ V ⊆ L2 (T) ⊗ V gives a polarisation of this representation. The resulting central extension is the Kac-Moody extension of C ∞ (T, Γ), and plays a central role in conformal field theory and quantum field theory. The two-dimensional analogue of a polarisation is given by the following notion of a bipolarisation: / GL(H) be a representation of a group on a separable Hilbert space. A bipoDefinition 1.1. Let ρ : G larisation of ρ consists of a pair of orthogonal projections P, Q ∈ L (H) such that the two operators /H [Pg , Qu ] and (Pg − Ph )(Qu − Qv ) : H are of trace class for all u, v, g, h ∈ G. The main application of our constructions is the following result (see Theorem 13.8): / GL(H) be a representation of a group on a separable Hilbert space. A bipolariTheorem 1.2. Let ρ : G sation of ρ determines a higher central extension of G by the group C∗ , i.e. a crossed sequence /N /E /G / 1. / C∗ 1 This extension yields a group cohomology class in H 3 (G, C∗ ) which is non-trivial in general. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 3 Bipolarisations arise frequently in nature: Example 1.3. Let Γ be a Lie group acting smoothly on a finite dimensional Hilbert space V . The smooth double loop group G := C ∞ (T2 , Γ) can then be represented on the Hilbert space H := L2 (T2 ) ⊗ V of vector b 2 (T) we obtain two valued L2 -functions on the 2-torus. Using the unitary isomorphism L2 (T2 ) ∼ = L2 (T)⊗L 2 2 2 2 b (T)) ⊗ V ⊆ H and (L (T)⊗H b (T)) ⊗ V ⊆ H and the corresponding orthogonal closed subspaces (H (T)⊗L projections P and Q yield a bipolarisation of our representation of the smooth double loop group. In partic
ular, we obtain an associated class in group cohomology [c] ∈ H 3 C ∞ (T2 , Γ), C∗ corresponding to a higher central extension of the smooth double loop group. We show by an explicit computation that this class is non-trivial in general (see Section 14). One of the advantages of our framework is that it easily adapts to a noncommutative setting: Example 1.4. Let θ ∈ R be an irrational number, and let G := C ∞ (T2θ )∗ denote the group of smooth units of the noncommutative 2-torus. Using the crossed product description of the noncommutative 2-torus we b 2 (Z). The Hilbert space H contains the closed may represent the group G on the Hilbert space H := L2 (T)⊗ℓ 2 2 2 2 b (Z) and L (T)⊗ℓ b (N0 ) and the corresponding orthogonal projections P and Q yield a subspaces H (T)⊗ℓ bipolarisation of our representation of the smooth units of the noncommutative 2-torus. In particular, we
obtain an associated class in group cohomology [c] ∈ H 3 C ∞ (T2θ )∗ , C∗ corresponding to a higher central extension of the smooth units of the noncommutative 2-torus. The group cohomology classes constructed in the present text can detect information about the singular homology (with integer coefficients) of the classifying space BGδ where the superscript “δ” signifies that the group G is equipped with the discrete topology. Indeed, one may identify the singular homology of BGδ with the group homology of the group G. In the above examples, the group G can as well be considered as a topological group in a natural way (e.g. using that C ∞ (T2 ) is a Fréchet space) and we therefore also have the classifying space BGtop , where we are taking this topology into account. Explicit computations show that the origin of our group cohomology classes is “not topological” in the sense that they do not in general arise from singular cohomology classes in H ∗ (BGtop , C∗ ) via pullback along the comparison map / BGtop . BGδ In the case where the group G = GL(R) agrees with the general linear group over a unital ring R we obtain a class in group cohomology [c] ∈ H 3 (GL(R), C∗ ) from a bipolarisation of a representation of GL(R) on a separable Hilbert space. We may use this class to obtain numerical information about the third algebraic / H3 (GL(R), Z) and the pairing between K-group of R by means of the Hurewicz homomorphism K3alg (R) / C∗ . The above examples fit group cohomology and group homology H 3 (GL(R), C∗ ) × H3 (GL(R), Z) in this context with the unital ring R being either the smooth functions on the 2-torus C ∞ (T2 ) or the smooth functions on the noncommutative 2-torus C ∞ (T2θ ). In particular, we obtain numerical invariants / C∗ and K alg (C ∞ (T2 )) / C∗ and in Section 14 we present concrete computations showing K3alg (C ∞ (T2 )) 3 θ that the first of these invariants is in fact non-trivial. From the algebraic point of view, one may consider the formal double loops into C, namely the field C((t))((s)) (which is also a 2-Tate space). The automorphism group of C((t))((s)) is an interesting object and in the papers [ArKr10, FrZh12], the construction of a group 3-cocycle on this automorphism group is carried out. This group 3-cocycle arises from the explicit description of 2-category theoretic data and the corresponding character on the algebraic K-theory of the field C((t))((s)) is a particular case of the characters described in [BGW21, OsZh16, GoOs15]. The difference between the present work and the above constructions of group 3-cocycles can (to some extent) be clarified by considering the difference between the formal double loops into C, C((t))((s)), and the smooth double loops into C, C ∞ (T2 ). This passage from the formal setting to the smooth setting is the cause of a whole range of analytic problems which we take up in this paper. At the level of linear operators we are no longer working with infinite matrices subject to vanishing conditions on the entries but with bounded operators on Hilbert spaces. Similarly, we are systematically replacing finite rank operators with trace class operators on Hilbert spaces and their corresponding analytically defined Fredholm determinants. As a consequence, there seems to be no correct notion of lattices in our context: finite dimensionality conditions on (quotient) subspaces are not adequate for dealing with the presence of infinitely many eigenvalues subject to decay conditions. Lattices in 2-Tate spaces form a core ingredient in the approach developed in [ArKr10, FrZh12]. 4 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We introduce here an entirely new machinery which allows us to build group actions on 2-categories from representations of groups on Hilbert spaces. Our ideas are related to the program of A. Connes in noncommutative geometry and in particular to the notion of finitely summable Fredholm modules and their link to analytic K-homology, Kasparov’s KK-theory and index pairings, [Con85, Con94, HiRo00, Kas80]. The two idempotents appearing in our definition of a bipolarisation are thought of as yielding two classes in KK-theory such that their Kasparov product yield a 3-summable Fredholm module. The associated index pairings with the K-theory classes represented by elements in the group G then yield actual Fredholm operators and their determinants provide us with the correct definition of the 2-morphisms in our 2-category. One may thus view the present paper as an attempt to align certain index theoretic constructions pertaining to noncommutative geometry with the framework of higher category theory. We now explain the basic category theoretic ideas involved in the present approach to higher central extensions. 1.2. Groups acting on higher categories and group cocycles. Our construction of higher group cocycles is based on obstruction theory and mathematical physics. Namely, to produce an (n + 1)-cocycle on a group, one constructs an n-category C with a group G acting on C. Provided that the n-category is sufficiently connected, this data yields an (n + 1)-cocycle on the group G with values in the automorphism group of an invertible (n − 1)-morphism. The case of a category and an associated 2-cocycle is fairly standard and will be described in detail in Section 6. For readers who are less familiar with notions of higher categories, let us explain the case of a group G acting on a (sufficiently connected) 2-category C. We denote the composition of 1-morphisms and the horizontal composition of 2-morphisms by ◦1 and we denote the vertical composition of 2-morphisms by ◦2 . The recipe is now as follows: (1) Choose an object x in C. / g(x) for each element g ∈ G. (2) Choose a 1-isomorphism αg : x / αgh for each pair of elements g, h ∈ G. (3) Choose a 2-isomorphism βg,h : g(αh ) ◦1 αg For each triple of elements g, h, k ∈ G we then have two different 2-isomorphisms relating the 1-morphisms / (ghk)(x). These two 2-isomorphisms are given by (gh)(αk ) ◦1 g(αh ) ◦1 αg and αghk : x

/ αghk . βgh,k ◦2 id(gh)(αk ) ◦1 βg,h and βg,hk ◦2 g(βh,k ) ◦1 idαg : (gh)(αk ) ◦1 g(αh ) ◦1 αg / (ghk)(x) The quotient of these 2-isomorphisms yields an automorphism of the 1-isomorphism αghk : x and this is exactly the value of our 3-cocycle on the triple (g, h, k) of group elements. The structure which arises naturally in the context of bipolarisations of group representations is slightly different from a weak 2-category. From our point of view there is no canonical way of defining the horizontal composition of 2-morphisms so we are instead replacing this operation with a family of “coproduct” functors. / µ along a specified object z. In These coproduct functors allow us to decompose any 2-morphism β : λ our context, these coproduct functors are in fact equivalences of 1-categories and a choice of inverses (up to natural isomorphisms) provides us with a “twisted” weak 2-category. The twisting arises from our systematic use of graded tensor products of categories which, in turn, is dictated by the concrete analytic examples which we are investigating. While we are unable to formulate a more precise statement at present, note that this coproduct structure is reminiscent of some of the challenges of Segal’s approach to conformal field theory [Seg04], where it is straightforward to cut a Riemann surface along a separating curve, but much more delicate to try to glue two Riemann surfaces with boundary together. In this paper we provide a detailed description of group 3-cocycles in the context of groups acting on coproduct categories. In particular, we explicitly verify the cocycle property and that the corresponding class in group cohomology is independent of various choices, see Section 8. We record that the twisted nature of the associated weak 2-category implies that our group 3-cocycle more naturally takes values in C∗ /{±1}, but we may of course square it and obtain a group 3-cocycle with values in C∗ as indicated earlier on in the introduction. 1.2.1. Polarisations, categories and extensions. The main work done in this paper concerns the construction of a twisted weak 2-category with a G-action, given the data of a bipolarisation of a representation / GL(H). As motivation, we now describe the 1-category with a G-action associated to a polariρ: G sation of ρ. It is instructive to keep in mind the analogy between the present analytic constructions and A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 5 the algebraic constructions carried out in [ADCK89, KaPe81]. In fact, one may think of all the various idempotents gP g −1 , g ∈ G, as having “commensurable” images. Strictly speaking this is of course only the case when all the differences P − gP g −1 are of finite rank
and not just Hilbert-Schmidt. For pairs of group elements g, h ∈ G, the symbol Im(hP h−1 ), Im(gP g −1 ) (which is in general not well-defined in our setting) is then replaced by the (graded) determinant of the Fredholm operator hP h−1 · gP g −1 . The analogy with determinantal theories for lattices in 1-Tate spaces as introduced in [Kap01] should now be apparent as well, see also [FrZh12] for further comments on this relationship. We start out by investigating the universal setting for the “one-dimensional story”: / H be an orthogonal projection with infinite dimensional kernel and cokernel. Consider the Let P : H universal polarised representation, i.e. the restricted general linear group  GLres (H) := g ∈ GL(H) | P − gP g −1 is Hilbert-Schmidt . / GL(H) then the representation ρ factors through a group Clearly, if P determines a polarisation of ρ : G / GLres (H) and this links the universal setting to the particular setting. homomorphism ρ : G For any g ∈ GLres (H), we define the bounded idempotent Pg := gP g −1 . For any two elements g and h of GLres (H), it then follows from Atkinson’s theorem that the composition / Ph H Ph Pg : Pg H is a Fredholm operator. Determinants of Fredholm operators play a key role in this paper, and in Section 3 and Section 4 we elaborate on the main constructions needed to describe the composition in the category here below. Notably, the condition that P − Pg be Hilbert-Schmidt for each g ∈ GLres (H) and not just compact turns out to be important, since the composition uses the Carey-Pincus perturbation isomorphism as introduced in [CaPi99b]. Definition 1.5. Let Cres be the following category with an action of GLres (H): (1) The objects in Cres are given by the set of elements in the restricted general linear group GLres (H); (2) Given two objects g, h ∈ GLres (H), the morphisms from g to h are given by the graded determinant line   Det(Ph Pg ) = Λtop Ker(Ph Pg ) ⊗ Λtop Coker(Ph Pg )∗ , Index(Ph Pg ) ; (3) The restricted general linear group GLres (H) acts on Cres by left multiplication on objects and by conjugation on morphisms. The composition of morphisms in Cres is given by a combination of torsion isomorphisms and perturbation isomorphisms in the context of determinants of Fredholm operators (see Section 6.1 for details). The group 2-cocycle on GLres (H) with values in C∗ constructed using this structure (see Section 6) appears in many places under different names. Its infinitesimal version is often called the “Japanese cocycle”, while its global version plays a prominent role in conformal field theory. One of the reasons for its importance is that it is responsible for the central extensions of loop groups. First of all, the group 2-cocycle on GLres (H) yields a central extension /E / GLres (H) /1. / C∗ 1 Now let Γ be a Lie group acting smoothly on a finite dimensional Hilbert space V (not necessarily by unitary operators). As described earlier on, we then have a polarisation of the action of the smooth single loop group G := C ∞ (T, Γ) on the Hilbert space H := L2 (T, V ). In particular, this polarisation yields a group / GLres (H). The central extension of the loop group C ∞ (T, Γ) appearing in homomorphism ρ : C ∞ (T, Γ) conformal field theory is the pull back along ρ of the above central extension of GLres (H), [PrSe86, Chapter 6]. Another avatar of the group 2-cocycle on GLres (H) turns out to be the low-dimensional version of the multiplicative character of Connes and Karoubi, [CoKa88]. To explain the link to Connes and Karoubi, remark / H yields a 2-summable Fredholm module Funi = (M1 , H, 2P − 1) that the orthogonal projection P : H where the unital ∗-algebra in question is defined by  M1 := T ∈ L (H) | [P, T ] is Hilbert-Schmidt . The restricted general linear group GLres (H) is exactly the group of invertible elements in M1 . In the one-dimensional case, the universal multiplicative character is a numerical invariant of the second algebraic / C∗ . The following holds: K-group denoted by M (Funi) : K2alg (M1 ) 6 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Theorem 1.6. Let [cres ] ∈ H 2 (GL(M1 ), C∗ ) denote the group 2-cocycle associated to the group action of GL(M1 ) on a stabilised version of Cres . Then the diagram K2alg (M1 ) / H2 (GL(M1 ), Z) ❥❥❥ ❥❥❥❥ ❥ M(Funi ) ❥ ❥ ❥❥ h[cres ],·i  ❥ ❥❥❥❥ ∗ t ❥ C h commutes, where h is the Hurewicz homomorphism and h[cres ], ·i comes from the pairing between group cohomology and group homology. Let us briefly return to the particular case where the Hilbert space H agrees with L2 (T) and the orthogonal projection P is the projection onto Hardy space. The smooth functions on the 1-torus acts on L2 (T) as multiplication operators and this action factors through M1 . Applying the functoriality of algebraic K-theory and composing with the universal multiplicative character, we obtain a numerical invariant / C∗ . This numerical invariant extends the Tate tame symbol of pairs of meromorM (FT ) : K2alg (C ∞ (T)) phic functions on a Riemann surface, see [Del91, CaPi99a, Kaa11a]. In fact, for any pair of invertible / C we may form the Steinberg symbol {u, v} ∈ K alg (C ∞ (T)) and the value smooth functions u, v : T 2 ∗ M (FT )({u, v}) ∈ C is given explicitly by the formula Z
log(u) 1 dv) · v(1)−w(u) , p∗ ( exp 2πi [0,1] v where w(u) ∈ Z denotes the winding number and p : [0, 1] / T is defined by p(t) = e2πit . 1.2.2. 2-categories associated to representations of a ring. To extend the above to a higher dimensional setting, we proceed as follows. Let R be a unital ring and let I ⊆ R be an ideal. Suppose that {πλ }λ∈Λ is a non-empty family of (not necessarily unital) representations of the ring R as bounded operators on a separable Hilbert space H satisfying the following conditions: Assumption 1.7. (1) for all x ∈ R and pairs λ, µ ∈ Λ the difference πλ (x)πµ (1) − πλ (1)πµ (x) is a trace class operator; (2) for all i ∈ I and triples λ, µ, ν ∈ Λ the difference πλ (i)πµ (1)πν (1) − πλ (i)πν (1) is a trace class operator. Let us for a moment fix two indices λ, µ ∈ Λ and explain the link between the above assumptions and relative analytic K-homology. The fact that our conditions really involve triples of indices has to do with the existence of the composition in the categories which we are going to explain in a little while. For the moment we define the Z/2Z-graded separable Hilbert space H(λ, µ) := πλ (1)H ⊕ πµ (1)H , where the first component is even and the second component is odd. We equip this Hilbert space with the even representation
/ L πλ (1)H ⊕ πµ (1)H , π(λ, µ) := πλ ⊕ πµ : R where we consider the unital ring R to be trivially graded. We then define the odd bounded operator   0 πλ (1)πµ (1) / πλ (1)H ⊕ πµ (1)H . F (λ, µ) := : πλ (1)H ⊕ πµ (1)H πµ (1)πλ (1) 0 With these definitions we record that our conditions (1) and (2) mean that the operators
[F (λ, µ), π(λ, µ)(x)] and π(λ, µ)(i) F (λ, µ)2 − idH(λ,µ) are of trace class for all x ∈ R and all i ∈ I. Comparing with [HiRo00, Definition 8.5.1] we are then justified in viewing the triple
π(λ, µ), H(λ, µ), F (λ, µ) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 7 as an even relative Fredholm module with respect to the ideal I inside the ring R. This even relative Fredholm module is then subject to the extra summability/dimensionality constraint imposed by replacing the compact operators on the Hilbert space H(λ, µ) by the trace class operators, see [Con85, Con94] for more details on these matters. Remark also that F (λ, µ) becomes self-adjoint if we impose the extra condition that πλ (1) and πµ (1) be orthogonal projections. This condition is however too restrictive for our present investigations (e.g. we are working with the whole group of invertible elements in C ∞ (T2 ) and not just the smooth functions with values in the circle). Let us now also fix two idempotents p, q ∈ R and assume that their difference p − q belongs to the ideal I. This data then yields a class in the even relative K-theory with respect to the ideal I inside the ring R. We recall that there is an index pairing between even relative K-theory and even relative K-homology with values in the group of integers. Thus, from our current data we know how to produce an integer h(p, q), (λ, µ)i ∈ Z . This integer does in fact arise as the index of an explicit Fredholm operator acting between Hilbert spaces and we denote this Fredholm operator by / πλ (q)H ⊕ πµ (p)H . F (λ, µ)(p, q) : πλ (p)H ⊕ πµ (q)H In our construction of a (twisted and weak) 2-category we are not merely interested in this integer but instead in the graded determinant line associated to the Fredholm operator F (λ, µ)(p, q) so that the degree agrees with the corresponding index. Indeed, this graded determinant line agrees with the 2-morphisms between the 1-morphisms λ, µ (the source being λ and the target µ) acting between the objects p and q (here with source p and target q). Still fixing the pair of idempotents p, q ∈ R with difference in the ideal I, we obtain a 1-category L(p, q) defined as follows: (1) The objects in L(p, q) are elements of the index set Λ; (2) Given two objects λ, µ ∈ Λ, the morphisms from λ to µ are given by the graded determinant line
Det F (λ, µ)(p, q) . As it happens for the category Cres the composition in L(p, q) is again given by a combination of torsion isomorphisms and perturbation isomorphisms (see Section 9 for details). The definition of this composition and the proof of associativity is however considerably more involved due to the complicated nature of the Fredholm operators F (λ, µ)(p, q) (see Section 15). In order to incorporate group actions we suppose that our discrete group G acts on the Hilbert space H, on the ring R (with the ideal I as an invariant subset) and on the index set Λ. Imposing the equivariance condition g(πλ (x)) = πg(λ) (g(x)) g ∈ G, λ ∈ Λ, x ∈ R we then obtain isomorphisms of categories g : L(p, q) / L(g(p), g(q)) g∈G which are compatible with the group laws. It turns out to be necessary to replace the collection of categories L with a slightly different collection of categories which we label H. The definition of H depends on the choice of a basepoint meaning that we choose an idempotent p0 ∈ R. This choice of basepoint has to be compatible with the group action in the sense that g(p0 ) − p0 ∈ I. The categories appearing in the collection H are then going to be the categories of morphisms between the objects in the associated twisted weak 2-category. The objects are defined as follows  Obj(H) := p ∈ R | p idempotent and p − p0 ∈ I and for each pair of objects p, q ∈ Obj(H) we define the category H(p, q) := L(p, p0 ) ⊗ L(q, p0 )† , where we apply a graded tensor product of categories and where the superscript “†” means dual category. The next natural step to take for constructing a twisted weak 2-category with an action of G out of our data would be to specify composition functors / H(p, q) , H(p, r) ⊗ H(r, q) 8 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON where the twisting signifies that we are applying a graded tensor product of our categories and not just the cartesian product (which would correspond to the standard definition of a weak 2-category). In our framework it turns out to be more convenient to construct coproduct functors instead of composition functors meaning that we instead construct coassociative functors / H(p, r) ⊗ H(r, q) ∆r : H(p, q) for every object r ∈ Obj(H). Contrary to the composition functors, which only satisfy twisted pentagon identities up to natural isomorphisms, our coproduct functors are in fact strictly coassociative. We refer to the collection of categories H endowed with the collection of coproduct functors ∆ as a coproduct category. The choice of a basepoint p0 implies that the coproduct category (H, ∆) does not at first sight admit an action of the group G (unless p0 happens to be a fixed point for the group action on R). We prove in the present text that the coproduct category (H, ∆) is independent of the choice of basepoint up to isomorphism of coproduct categories. More specifically we show that for any alternative idempotent p′0 ∈ R with p0 − p′0 ∈ I we have an isomorphism of coproduct categories / (H′ , ∆′ ) B(p0 , p′0 ) : (H, ∆) which we refer to as the change-of-base-point isomorphism (here (H′ , ∆′ ) denotes the coproduct category constructed using the idempotent p′0 as the base point). The main result of the present paper can now be stated as follows: Theorem 1.8. The pair (H, ∆) can be given the structure of a coproduct category with an action of the group G and this data satisfies the conditions for the construction of a group 3-cocycle on G with values in C∗ . Moreover, an explicit computation for the case where G equals the smooth double loop group C ∞ (T2 )∗ shows that this group 3-cocycle is non-trivial in general (see Section 14). Our principal example of a setting where the above theorem applies is as follows. Suppose that we are / GL(H). Let R := hqu ; u ∈ Gi be the unital ring given a bipolarisation (P, Q) of a representation ρ : G which is freely generated by idempotents {qu }u∈G and let I be the augmentation ideal defined as the kernel / R sending all qu to qe (with e ∈ G being the neutral element). The fact of the unital homomorphism R that we are considering a bipolarisation allows us to show that every bounded operator of the form Pg Qu Pg is a trace class perturbation of an idempotent on the Hilbert space Pg H. In this fashion we obtain families of representations of R parametrised by g ∈ G and these families satisfy our two main conditions (1) and (2) detailed out under Assumption 1.7. With a little bit of extra care, we may also equip all our data with an action of G and Theorem 1.8 thereby applies and produces a group 3-cocycle on the group G. Remark 1.9. In distinction to the one-dimensional case, it is not clear if there exists a universal bipolarised representation, or equivalently a unique maximal group of bounded invertible operators on which the group 3-cocycle is defined. 1.3. Relation to the Tate tame symbol and the multiplicative character. As remarked above, the first application of the results in this paper is toPthe algebra of smooth functions on the two-torus, C ∞ (T2 ). m n 2 2 The Hilbert space is L2 (T2 ) and, given ξ = (m,n)∈Z2 µ(m,n) z1 z2 ∈ L (T ), we define the orthogonal
projections P, Q ∈ L L2 (T2 ) by XX XX (1.1) Pξ = µ(m,n) z1m z2n and Qξ = µ(m,n) z1m z2n . m≥0 n∈Z ∞ m∈Z n≥0 2 ∗ The group G := C (T ) can be taken to be the group of smooth invertible functions on the 2-torus acting by multiplication on L2 (T2 ). The corresponding 3-cocycle [c] ∈ H 3 (C ∞ (T2 )∗ , C∗ ) is non-trivial. In fact, for a constant function λ ∈ C∗ we shall see that h[c], {z1 , z2 , λ}i = λ2 , where {z1 , z2 , λ} ∈ H3 (C ∞ (T2 )∗ , Z) is the class of the group 3-cycle (z1 , z2 , λ) − (z1 , λ, z2 ) + (λ, z1 , z2 ) − (λ, z2 , z1 ) + (z2 , λ, z1 ) − (z2 , z1 , λ) ∈ Z[G3 ] . This explicit computation of the pairing between group cohomology and group homology, the analogy with the one-dimensional setting as well as the situation for the formal double loop group C((t))((s)) lead us to the conjecture here below. The integral formula appearing should be compared with [BrMc96, Theorem 2.7], where this kind of integral formula is related to the product structure in Deligne cohomology, [Bl84, EsVi88]: A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 9 Conjecture 1.10. Suppose that f , g and h are smooth invertible functions on T2 . Then ! Z
log(f ) 2 p∗ · dg ∧ dh h[c], {f, g, h}i = exp (2πi)2 [0,1]2 g·h ! Z Z
w2 (f )
w1 (f ) ∗ log(g) ∗ log(g) · exp − p p · dh + · dh πi h πi h {1}×[0,1] [0,1]×{1} · h(1, 1)2w1 (f )w2 (g)−2w1 (g)w2 (f ) , / T2 is the smooth map p(t, s) := (exp(2πi · t), exp(2πi · s)) and where w1 (f ) , w2 (f ) ∈ Z are where p : [0, 1]2 the winding numbers of the smooth invertible functions on the circle z 7→ f (z, 1) and z 7→ f (1, z), respectively. While this is not the language we adopt in this paper, much of our construction of the twisted weak 2-category H can be understood as an attempt to “categorify” constructions relating to the K-theory of / L (H) a representation operator algebras. Let n ∈ N be fixed. Suppose that A is an algebra over C, π : A of A by bounded operators on a separable infinite dimensional Hilbert space H and F ∈ L (H) is a self-adjoint unitary satisfying that [F, π(a)] ∈ L n (H) ∀a ∈ A , where L n (H) denotes the nth -Schatten ideal. If n is odd, we assume that H is Z/2Z graded, F is odd and / H. We will moreover π(a) is even for all a ∈ A. In this case, we denote the grading operator by γ : H assume that the spectral subspaces for F (resp. γ) are infinite dimensional when n is even (resp. odd). The Connes-Chern character of the n-summable Fredholm module F := (A, H, F ) is the class in cyclic cohomology Ch(F ) ∈ HC n−1 (A) given by the cyclic cocycle 1 Tr(γ n F [F, π(a0 )][F, π(a1 )] . . . [F, π(an−1 )]) a0 , . . . , an−1 ∈ A , 2 / C is the operator trace, [Con85, Con94]. where γ n := idH for n even and where Tr : L 1 (H)
The pattern for secondary invariants of algebraic K-theory is as follows. Let Funi := Mn−1 , H, Funi be the universal n-summable Fredholm module. Then the topological K-theory is given by Kntop (Mn−1 ) = 0, top Kn+1 (Mn−1 ) = Z and we have a commuting diagram with exact rows: (1.2) Ch(F )(a0 , . . . , an−1 ) = / K rel(Mn−1 ) n Z / K alg (Mn−1 ) n /0 Chrel  HCn−1 (Mn−1 ) M(Funi ) Ch(Funi ) Z n (2πi)⌈ 2 ⌉  / C/(2πi)⌈ n2 ⌉ Z  /C The right-most arrow is the universal Connes-Karoubi multiplicative character and the Chern character Chrel is the relative Chern character which is defined on the relative K-theory of the Banach algebra Mn−1 , [Kar87]. The cyclic homology group appearing is constructed using the projective tensor product of Banach algebras. Any n-summable Fredholm module F := (A, H, F ) over A yields an algebra homomorphism π: A / Mn−1 with π ∗ (Funi ) = F (at least when the relevant spectral subspaces are infinite dimensional) and hence, using the functoriality of algebraic K-theory, produces the group homomorphism called the Connes-Karoubi multiplicative character, [CoKa88]: / C/(2πi)⌈ n2 ⌉ Z . M(F ) : K alg (A) n On the other hand, by the definition of the algebraic K-group K∗alg (A), we have the Hurewicz homomorphism Knalg (A) / Hn (GL(A), Z) 10 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON and the natural question is whether the multiplicative character of an n-summable Fredholm module factorises via group homology, i.e. whether the dotted arrow below exists making the diagram commute: Knalg (A) / Hn (GL(A), Z) ❥ ❥ ❥ ❥ ❥ M(F ) ❥  u❥ ❥ ⌈n ⌉ C/(2πi) 2 Z . In the case of the 2-torus we obtain a 3-summable Fredholm FT2 = C ∞ (T2 ), L2 (T2 ) ⊕ L2 (T2 ), FT2
where the self-adjoint unitary operator FT2 is obtained by taking the phase of the Dirac operator   0 i∂/∂θ1 − ∂/∂θ2 / C ∞ (T2 , C2 ) . D := : C ∞ (T2 , C2 ) i∂/∂θ1 + ∂/∂θ2 0 Comparing with Conjecture 1.10 and considering the explicit computations of the multiplicative character carried out in [Kaa11b] lead us to the following:
Conjecture 1.11. Consider the 3-summable Fredholm module FT2 and let [c] ∈ H 3 GL(C ∞ (T2 )), C∗ denote the class of the group 3-cocycle on the group of invertible matrices over C ∞ (T2 ) constructed using the bipolarisation from (1.1). Then the diagram

/ H3 GL(C ∞ (T2 )), Z K3alg C ∞ (T2 ) 1 πi ·M(FT2 )  C/(2πi)Z h[c],·i exp  / C∗ commutes. We emphasise that the spectral triple (C ∞ (T2 ), L2 (T2 ) ⊕ L2 (T2 ), D) coming from the Dirac operator on the 2-torus (and hence also the associated Fredholm module FT2 ) is deeply related to the bipolarisation / L2 (T2 ) are exactly the spectral defined in (1.1). Indeed, the two orthogonal projections P and Q : L2 (T2 ) projections onto the positive part of the spectrum of the two partial differential operators i∂/∂θ1 and i∂/∂θ2 (both acting as unbounded self-adjoint operators on the Hilbert space L2 (T2 )). In fact, one may regard the two partial differential operators i∂/∂θ1 and i∂/∂θ2 as providing a factorisation of the spectral triple (C ∞ (T2 ), L2 (T2 ) ⊕ L2 (T2 ), D) and this point of view is compatible with the unbounded Kasparov product, see [BaJu83, Mes14, KaLe13]. 1.4. The structure of the paper. Since the determinants of Fredholm operators play a major role in our constructions, we collect all the necessary definitions and results in Section 2-5. The definitions and main properties of determinant functors and their associated torsion isomorphisms are collected in Section 2 and Section 3. In Section 4, we recall the construction of the Carey-Pincus perturbation isomorphism in the context of trace class perturbations of Fredholm operators. In Section 4, we also prove a fundamental relationship between torsion isomorphisms and perturbation isomorphisms. In Section 5, we consider a particular kind of quasi-isomorphism, namely those obtained by stabilising Fredholm operators with an invertible operator. In Section 6, we review the construction of group 2-cocycles associated to group actions on categories. In particular, we apply this framework to obtain a group 2-cocycle on the restricted general linear group associated to a polarisation of a Hilbert space H. In Section 7, we then show that our group 2-cocycle on the restricted linear group recovers the low-dimensional Connes-Karoubi multiplicative character on the second algebraic K-group, and hence that our group 2-cocycle corresponds to the usual central extension of the restricted general linear group. In Section 8, we start our investigation of the two-dimensional setting, and provide an account of the category theoretic framework which we apply to exhibit group 3-cocycles. In particular, we provide the main definitions relating to coproduct categories, and explain how group actions give rise to group 3-cocycles in this particular context. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 11 In Section 9-12, we carry out the main constructions of the present paper. We thus explain how to apply representation theoretic data to obtain a coproduct category with an action of a group, and hence a group 3-cocycle on the group in question. In these sections, we only present the main constructions and state the main theorems; the actual proofs are provided later on in Section 15 and Section 16. In this respect, we would like to highlight the computation of the Fredholm determinant in Lemma 16.9, since this result can be viewed as the main reason for the functoriality of the change-of-base-point isomorphism. In Section 13 and 14, we apply our main theorem to the setting of bipolarisations of group representations and we show by an explicit computation that our group 3-cocycle is non-trivial in the case of the smooth double loop group C ∞ (T2 )∗ , thus in the context of the bipolarisation coming from the spectral projections / L2 (T2 ). P and Q : L2 (T2 ) 1.5. Acknowledgements. The first author would like to thank Ulrich Bunke for the many very nice conversations on secondary invariants, and to thank Andreas Thom and the Technische Universität Dresden for hospitality in the autumn of 2021 during the final stages of the writing of this paper. The second author would like to thank Alexander Gorokhovsky and Boris Tsygan for many helpful discussions about various aspects of this paper. The third author thanks Oliver Braunling, Ezra Getzler, Michael Groechenig, Mikhail Kapranov and Boris Tsygan for helpful conversations, and thanks the organizers of the 2016 IBS-CGP workshop on Homotopical Methods in Quantum Field Theory for the opportunity to present an early version of this work. 2. Determinants of Z/2Z-graded vector spaces We start out by reviewing some fundamental constructions relating to determinants of finite dimensional vector spaces. These constructions can be expanded and generalised in various directions leading to notions of determinant functors in different category theoretic settings, see e.g. [KnMu76, Knu02, Del87, Bre11, MTW15]. Notice that we work in a Z/2Z-graded context instead of the more common Z-graded context, where the objects arise as the cohomology groups of a bounded cochain complex, see also [Kaa12, KaNe19] for more details. Notation 2.1. Let L be a complex line, i.e. a one-dimensional complex vector space. Given a non-zero vector w ∈ L , we use w∗ to denote the vector in the dual line L ∗ = Hom(L , C) given by w∗ (w) = 1. Notation 2.2. We let Pic denote the Picard category of Z-graded complex lines. The objects are thus pairs (L , n) where L is a complex line and n is an integer. Moreover, the set of morphisms from (L , n) to (M , m) is the empty set for n 6= m and for n = m, they are exactly the vector space isomorphisms from L to M . We recall that the commutativity constraint in our Picard category is given by ǫ : (L , n) ⊗ (M , m) / (M , m) ⊗ (L , n) ǫ(s ⊗ t) := (−1)n·m · t ⊗ s . Notation 2.3. The exterior algebra of a finite dimensional complex vector space V is denoted by Λ(V ). For a homogeneous element v ∈ Λ(V ), we let ε(v) ∈ N ∪ {0} / W between finite dimensional vector spaces, we use the same denote its degree. For a linear map T : V notation / Λ(W ) T (v1 ∧ v2 ∧ . . . ∧ vk ) := T (v1 ) ∧ T (v2 ) ∧ . . . ∧ T (vk ) T : Λ(V ) for the induced algebra homomorphism between the exterior algebras. Definition 2.4. Given a finite dimensional complex vector space V, its determinant is the Z-graded complex line
Det(V ) := Λtop (V ), dim(V ) . For a finite dimensional Z/2Z-graded vector space V = V+ ⊕ V− , its determinant is the Z-graded complex line
Det(V ) := Det(V+ ) ⊗ Det(V− )∗ = Λtop (V+ ) ⊗ Λtop (V− )∗ , dim(V+ ) − dim(V− ) . We specify that Det({0}) = (C, 0) . 12 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON  Σ+ 0 / W of Z/2Z-graded vector spaces induces an isomorphism :V 0 Σ− / Det(W ) given by of the corresponding determinant lines Det(Σ) : Det(V ) An even isomorphism Σ =  ∗ v+ ⊗ v− 7→ Σ+ (v+ ) ⊗ (Σ− (v− ))∗ . We will occasionally use the notation |V | (resp. |Σ|) for Det(V ) (resp. Det(Σ)). Definition 2.5. An exact triangle of finite dimensional Z/2Z-graded vector spaces is an exact sequence ∆ of the form U `❆ ❆❆ ❆❆ ❆ ∂ ❆❆ i /V ⑥ ⑥ ⑥⑥ ⑥⑥q ~⑥ ⑥ W where i and q are even and ∂ is an odd linear map. Splitting the Z/2Z-graded vector spaces explicitly into the even and odd direct summands, such an exact triangle is given by the following six term exact sequence of finite dimensional vector spaces i+ U+ O q+ / V+ / W+ ∂− ∂+ W− o q− V− o  i− U− . For each exact triangle ∆ of finite dimensional Z/2Z-graded vector spaces, we have the torsion isomorphism of determinant lines / Det(U ) ⊗ Det(W ) . Det(∆) : Det(V ) Let us formulate its construction as a definition. Definition 2.6. Let ∆: U i /V q /W ∂ /U be an exact triangle of finite dimensional Z/2Z-graded vector spaces. Choose homogeneous elements (2.1) u+ ∈ Λdim(Im(i+ )) (U+ ) v+ ∈ Λdim(Im(q+ )) (V+ ) w+ ∈ Λdim(Im(∂+ )) (W+ ) u− ∈ Λdim(Im(i− )) (U− ) v− ∈ Λdim(Im(q− )) (V− ) w− ∈ Λdim(Im(∂− )) (W− ) such that the following wedge products yield non-trivial elements: i+ (u+ ) ∧ v+ ∈ Det(V+ ) i− (u− ) ∧ v− ∈ Det(V− ) ∂− (w− ) ∧ u+ ∈ Det(U+ ) ∂+ (w+ ) ∧ u− ∈ Det(U− ) q+ (v+ ) ∧ w+ ∈ Det(W+ ) q− (v− ) ∧ w− ∈ Det(W− ) . The torsion isomorphism Det(∆) : Det(V ) / Det(U ) ⊗ Det(W ) is then defined by i+ (u+ ) ∧ v+ ⊗ (i− (u− ) ∧ v− )∗ 7→ (−1)ε(∆) · ∂− (w− ) ∧ u+ ⊗ (∂+ (w+ ) ∧ u− )∗ ⊗ q+ (v+ ) ∧ w+ ⊗ (q− (v− ) ∧ w− )∗ , where the sign is determined by the exponent ε(∆) := dim(Im(q+ )) · dim(U+ ) + dim(Im(i− )) · dim(W+ ) + dim(Im(∂− )) · dim(V− ) ∈ N ∪ {0} . We record the following result (cf. [Kaa12, Lemma 2.1.3]): Proposition 2.7. The torsion isomorphism Det(∆) is independent of the choices of elements in the exterior algebras used in Definition 2.6. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 13 The next theorem summarises the main properties of our constructions. The first two (naturality and commutativity) follow directly from the definitions. The proof of associativity is a (much longer) exercise in finite dimensional linear algebra. We refer the interested reader to [Kaa12, Section 4], and specifically to Remark 4.3.5 and Theorem 4.3.4 of loc. cit.. Theorem 2.8. The torsion isomorphisms satisfy the following properties: (1) Naturality. Let U /V a  U′ /W b c  / V′  / W′ /U a  / U′ be a commutative diagram of finite dimensional Z/2Z-graded vector spaces, where the rows ∆ and ∆′ are exact triangles and the columns are even isomorphisms. Then the diagram Det(∆) Det(V ) / Det(U ) ⊗ Det(W ) Det(b) Det(a)⊗Det(c) Det(∆′ )  Det(V ′ )  / Det(U ′ ) ⊗ Det(W ′ ) is commutative. (2) Commutativity. Associated to two finite dimensional Z/2Z-graded vector spaces U = U+ ⊕ U− and V = V+ ⊕ V− , we can construct two exact triangles  ∆1 : U and   ∆2 : W  id 0 0 id   / U ⊕W   / U ⊕W  0 id   id 0  /W /U 0 0 /U /W . Then the diagram Det(∆1 ) / Det(U ) ⊗ Det(W ) Det(U ⊕ W ) ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ ǫ ❯❯❯❯ Det(∆2 ) *  Det(W ) ⊗ Det(U ) is commutative, where ǫ denotes the commutativity constraint from Notation 2.2. (3) Associativity. Let U U i /V q /X  /W  /Z  Y  Y /U d /U π ∂  V q  /X be a commutative diagram of finite dimensional Z/2Z-graded vector spaces, where: (a) the two first rows, ∆1 and ∆2 , and columns two and three, Γ2 and Γ3 , are exact triangles; 14 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON (b) the diagram d /U ∂  /V Z π  Y i is commutative. Then the diagram (2.2) Det(∆2 ) Det(W ) / Det(U ) ⊗ Det(Z) Det(Γ2 ) id⊗Det(Γ3 )  Det(V ) ⊗ Det(Y ) Det(∆1 )⊗id  / Det(U ) ⊗ Det(X) ⊗ Det(Y ) is commutative. 3. The torsion isomorphism of Fredholm operators The determinants and torsion isomorphisms described in the previous section in the context of finite dimensional Z/2Z-graded vector spaces can be lifted to the analytic setting of Fredholm operators acting between separable Hilbert spaces. Definition 3.1. Let H and K be separable Hilbert spaces. The determinant line of a Fredholm operator / K is the graded line T: H
|T | := Λtop Ker(T ) ⊗ Λtop Coker(T )∗ , dim(Ker(T )) − dim(Coker(T )) . Thus, upon defining the finite dimensional Z/2Z-graded vector space I(T ) := Ker(T ) ⊕ Coker(T ) we obtain that |T | = Det(I(T )). Definition 3.2. For two composable Fredholm operators T and S, the associated torsion isomorphism / |ST | T : |T | ⊗ |S| is the torsion isomorphism Det(∆(S, T ))−1 associated to the six term exact sequence ∆(S, T ): ι Ker(T ) O / Ker(ST ) T / Ker(S) S  Coker(T ) , 0 ∂ π Coker(S) o Coker(ST ) o where ι is the inclusion, π the quotient map and ∂ is given by the composition Ker(S) /H / H/Im(T ). We specify that the six term exact sequence in the above definition comes from the exact triangle  ∆(S, T ) : I(T )  ι 0 0 S    / I(ST )  T 0 0 π    / I(S)  0 ∂ 0 0   / I(T ) of finite dimensional Z/2Z-graded vector spaces. Proposition 3.3. Given three composable Fredholm operators S, T and R, the following diagram commutes: |S| ⊗ |T | ⊗ |R| T⊗id id⊗T / |T S| ⊗ |R| T  |S| ⊗ |RT | T  / |RT S| . Proof. This is an easy consequence of the associativity property of the torsion isomorphism explained in Theorem 2.8.  A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 15 / K are invertible. / H1 and Σ2 : H2 / H2 is Fredholm and that Σ1 : G Example 3.4. Suppose that T : H1 After identifying |T | ⊗ |Σ2 | and |Σ1 | ⊗ |T | with |T |, the torsion isomorphisms associated to the compositions get the following explicit description. / |Σ2 T | u+ ⊗ u∗− 7→ u+ ⊗ (Σ2 u− )∗ and L(Σ2 ) : |T | (3.1) / |T Σ1 | u+ ⊗ u∗− 7→ (Σ−1 u+ ) ⊗ u∗− , R(Σ1 ) : |T | 1 where u+ ∈ Det(Ker(T )) and u− ∈ Det(Coker(T )) are non-trivial vectors. / G ′ be Fredholm operators and let φ : H / H′ and ψ : G / G′ / G and T ′ : H′ Definition 3.5. Let T : H be bounded operators. We say that the pair (φ, ψ) is a quasi-isomorphism from T to T ′ when ψ ◦ T = / G ′ and when the induced linear maps T ′ ◦ φ: H / Ker(T ′ ) φ : Ker(T ) are isomorphisms. We let |φ, ψ| : |T | isomorphism (φ, ψ). and ψ : Coker(T ) / Coker(T ′ ) / |T ′ | denote the isomorphism of graded lines induced by a quasi- The next proposition explains the relationship between torsion isomorphisms and quasi-isomorphisms. / G, S : G / K, T ′ : H′ / G ′ and S ′ : G ′ / K′ be Fredholm operators and Proposition 3.6. Let T : H ′ suppose that (φ, ψ) and (ψ, τ ) are quasi-isomorphisms from T to T and from S to S ′ , respectively. Then (φ, τ ) is a quasi-isomorphism from ST to S ′ T ′ and the following diagram commutes: |T | ⊗ |S| T / |ST | |φ,ψ|⊗|ψ,τ |  ′ ′ |T | ⊗ |S | |φ,τ |  / |S ′ T ′ | . T Proof. The claim follows from the five lemma, the definition of the torsion isomorphism and from its naturality, i.e. from Theorem 2.8 (1).  4. The perturbation isomorphism Given a trace class operator δ : H (4.1) /H on a separable Hilbert space H, the following Fredholm determinant det(idH + δ) = 1 + ∞ X Tr(Λn δ) ∈ C n=1 is well defined and is a multiplicative generalisation of the determinant of a finite matrix, see [Sim05, Chapter / G with trace class difference, one may use the 3]. For any pair of invertible bounded operators T1 , T2 : H Fredholm determinant to define the invertible complex number det(T2 T1−1 ). The perturbation isomorphism, invented by Carey and Pincus in [CaPi99b], provides an extension of this assignment to the setting of Fredholm operators with trace class difference. For more information on these matters we also refer to the paper [KaNe19]. Theorem 4.1. [CaPi99b, Theorem 11 and Theorem 12] Let H and G be separable Hilbert spaces. There is / G with trace class difference a universal construction which, to any pair of Fredholm operators T1 , T2 : H associates an isomorphism / |T2 | P(T1 , T2 ) : |T1 | such that the following cocycle conditions hold: (1) P(T2 , T3 ) ◦ P(T1 , T2 ) = P(T1 , T3 ); (2) P(T1 , T2 ) = P(T2 , T1 )−1 . We refer to P(T1 , T2 ) as the perturbation isomorphism and we sometimes apply the notation P := P(T1 , T2 ) : |T1 | / |T2 | . The following series of examples contain an explicit description of the perturbation isomorphism in various cases which will be useful later on. 16 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON / G are two Fredholm operators such that Example 4.2. Finite rank case. Suppose that T1 and T2 : H T1 − T2 is of finite rank. Let V ⊆ H be a closed subspace of finite codimension such that T1 |V = T2 |V = / G. Let ιV denote the inclusion V ֒→ H. For i = 1, 2, the torsion isomorphism from Definition 3.2 T: V then provides the isomorphism Det(∆(Ti ,ιV )) |T | = |Ti ◦ ιV | / |ιV | ⊗ |Ti | . / |T2 | is then the unique isomorphism making the diagram The perturbation isomorphism P(T1 , T2 ) : |T1 | |T | ◗◗ ◗◗◗ ♠♠♠ ◗◗Det(∆(T 2 ,ιV )) ◗◗◗ ◗◗◗ ◗( id⊗P(T1 ,T2 ) / |ιV | ⊗ |T2 | Det(∆(T1 ,ιV ))♠♠♠♠ ♠ ♠♠♠ v♠♠♠ |ιV | ⊗ |T1 | commute. Example 4.3. Invertible case. Suppose that T1 and T2 are both invertible, but that T1 − T2 is of trace class. Then |T1 | = |T2 | = (C, 0) and, as an automorphism C / C, we have that P(T1 , T2 )(λ) = det(T2 T1−1 ) · λ . (4.2) / G are of index zero and that T1 − T2 is of Example 4.4. Index zero case. Suppose that T1 and T2 : H trace class. Then the perturbation isomorphism P(T1 , T2 ) has the following description. For i = 1, 2, choose / G such that Ti + Fi is invertible and such that Ker(Fi ) ⊆ H is a a bounded finite rank operator Fi : H / Coker(Ti ) vector space complement of Ker(Ti ) ⊆ H. Then each Fi induces an isomorphism Fi : Ker(Ti ) and the invertible operator /G Σ := (T2 + F2 )(T1 + F1 )−1 : G is of determinant class. Given basis vectors t ∈ Λtop Ker(T1 ) and s ∈ Λtop Ker(T2 ), the perturbation isomorphism is then given by P(T1 , T2 ) : t ⊗ (F1 t)∗ 7→ det(Σ) · s ⊗ (F2 s)∗ . (4.3) Example 4.5. The general case. In full generality, the perturbation isomorphism can be described as / Cm be the trivial follows. Choose n, m ∈ N0 such that m − n = Index(T2 ) = Index(T1 ) and let 0m,n : Cn n m / linear map. Then the Fredholm operators T1 ⊕ 0m,n and T2 ⊕ 0m,n : H ⊕ C G ⊕ C are of index zero and P(T1 , T2 ) ⊗ id : |T1 | ⊗ |0m,n | / |T2 | ⊗ |0m,n | agrees with the composition (4.4) P(T1 ⊕0m,n ,T2 ⊕0m,n ) |T1 | ⊗ |0m,n | ∼ = |T1 ⊕ 0m,n | / |T2 ⊕ 0m,n | ∼ = |T2 | ⊗ |0m,n | , where the first and the third isomorphism are given by the torsion isomorphisms associated to the six term exact sequences   Ker(Ti ) O 0 Cn o for i = 1, 2.  0 id 0   id / Ker(Ti ) ⊕ Cm   Coker(Ti ) ⊕ Cn o 0   id id 0    / Cm 0  Coker(Ti ) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 17 / H for T1 such that RT1 − idH and T1 R − idG are of trace class. It Remark 4.6. Choose a parametrix R : G / |T2 | follows from Theorem 4.9, which we will prove in Subsection 4.1, that the perturbation P(T1 , T2 ) : |T1 | agrees with the unique isomorphism making the following diagram commutative |T1 | ⊗ |R| T / |RT1 | P(T1 ,T2 )⊗id  |T2 | ⊗ |R| P(RT1 ,RT2 )  / |RT2 | T / H have index 0. In particular, P(T1 , T2 ) as described above commutative. Here both RT1 and RT2 : H / H. is independent of the choice of parametrix R : G 4.1. Perturbation commutes with torsion. Our goal in this subsection is to prove a fundamental relationship between torsion isomorphisms and perturbation isomorphisms. This result can be found as Theorem 4.9 and will be applied throughout this text. A similar result was proved in [KaNe19, Theorem 5.1] for finite rank perturbations of mapping cone Fredholm complexes. Since our sign conventions are slightly different from the conventions used in [KaNe19] and since we need to work with trace class perturbations, we provide a full proof for the case where the Fredholm complexes in question are the mapping cones of single Fredholm / 0. /H T /G operators, i.e. of the form 0 / K be Fredholm operators / G and S : G Lemma 4.7. Let H, G and K be separable Hilbert spaces, let T : H / G be a finite rank operator. Then the following square commutes: and let δT : H |T | ⊗ |S| (4.5) T / |ST | P P⊗id  |T + δT | ⊗ |S|  / |S(T + δT )| . T Proof. Choose a closed subspace V ⊆ H of finite codimension such that V ⊆ Ker(δT ) and let ιV : V denote the inclusion. The diagram in (4.5) then commutes if and only if the diagram |ιV | ⊗ |T | ⊗ |S| id⊗T id⊗P⊗id  |ιV | ⊗ |T + δT | ⊗ |S| /H / |ιV | ⊗ |ST | id⊗P  id⊗T / |ιV | ⊗ |S(T + δT )| commutes. To see that this latter diagram commutes, it suffices to use the description of the perturbation isomorphism given in Example 4.2 and the associativity of the torsion isomorphisms described in Proposition 3.3.  / G and S : G / K be Fredholm operators Lemma 4.8. Let H, G and K be separable Hilbert spaces, let T : H / / of index 0 and let δT : H G and δS : G K be trace class operators. Then the following square commutes: (4.6) |T | ⊗ |S| T P⊗P  |T + δT | ⊗ |S + δS | / |ST | P T  / (S + δS )(T + δT ) . / G are of index zero, we may find finite rank operators FT and Proof. Since both T and T + δT : H / G are invertible. Using Lemma 4.7 and / G such that T + FT and T + δT + FT +δT : H FT +δT : H the transitivity of the perturbation isomorphism from Theorem 4.1, we may thus assume that both T and / G are invertible. T + δT : H /K / K are of index zero, we may choose finite rank operators FS and FS+δS : G Since S and S + δS : G / K are invertible and such that Ker(FS ) and Ker(FS+δS ) are vector such that S +FS and S +δS +FS+δS : G / G are invertible, space complements of Ker(S) and Ker(S + δS ), respectively. Since T and T + δT : H / / (T + δ ) : H it then follows that ST + FS T : H K and (S + δS )(T + δT ) + FS+δS K are invertible. T 18 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Moreover, Ker(FS T ) is a vector space complement to Ker(ST ), and Ker(FS+δS (T + δT )) is a vector space complement to Ker((S + δS )(T + δT )). We now use the description of the perturbation isomorphism given in Example 4.3 and in Example 4.4, together with the description of the torsion isomorphism given in Example 3.4. The commutativity of the diagram in (4.6) then amounts to showing that the two isomorphisms LHS / |(S + δS )(T + δT )| given by and RHS : |S|

LHS : s ⊗ (FS s)∗ 7→ det (T + δT )T −1 · det (S + δS + FS+δS )(S + FS )−1
and · (T + δT )−1 (t) ⊗ (FS+δS t)∗
RHS : s ⊗ (FS s)∗ 7→ det (S + δS + FS+δS )(T + δT )((S + FS )T )−1
∗
· r ⊗ FS+δS (T + δT )r agree, where s, t and r are basis vectors for Det(Ker(S)), Det(Ker(S + δS )) and Det(Ker((S + δS )(T + δT ))), respectively. Choosing r := (T + δT )−1 (t), we only need to verify that the two Fredholm determinants

and det (T + δT )T −1 · det (S + δS + FS+δS )(S + FS )−1
det (S + δS + FS+δS )(T + δT )((S + FS )T )−1 coincide. But this follows immediately from basic properties of the Fredholm determinant: indeed, it suffices to use that the Fredholm determinant is multiplicative and that it is invariant under conjugation by invertible bounded operators (see e.g. [Sim05, Chapter 3]).  Theorem 4.9 (Perturbation commutes with torsion). Let H, G and K be separable Hilbert spaces, let / G and S : G / K be Fredholm operators and let δT : H / G and δS : G / K be trace class operators. T: H Then the following square commutes: (4.7) |T | ⊗ |S| T / |ST | P⊗P  |T + δT | ⊗ |S + δS | P  / |(S + δS )(T + δT )| . T Proof. Choose integers n, m, k ≥ 0 such that Index(T ) = m − n and Index(S) = k − m. Recall that / Ck denote the trivial maps. It can then be verified that the following / Cm and 0k,m : Cm 0m,n : Cn diagram commutes: |T | ⊗ |S| ⊗ |0m,n | ⊗ |0k,m | T⊗T  |T ⊕ 0m,n | ⊗ |S ⊕ 0k,m | / |ST | ⊗ |0k,n |  / |ST ⊕ 0k,n | . T Notice here that the horizontal maps are torsion isomorphisms of Fredholm operators, the right vertical is the torsion isomorphism coming from the six term exact sequence in Example 4.5, and the left vertical map is the composition of the commutativity constraint from Notation 2.2 and the torsion isomorphisms coming from the six term exact sequences in Example 4.5. There is of course a similar commutative diagram when T is replaced by T + δT and S is replaced by S + δS . Since T ⊕ 0m,n and S ⊕ 0k,m both have index zero, Lemma 4.8 implies that the following diagram commutes: |T ⊕ 0m,n | ⊗ |S ⊕ 0k,m | T / |ST ⊕ 0k,n | P P⊗P  |(T + δT ) ⊕ 0m,n | ⊗ |(S + δS ) ⊕ 0k,m | T  / |(S + δS )(T + δT ) ⊕ 0k,n | . A combination of these observations, together with the description of the perturbation isomorphism from Example 4.5, yields the commutativity of the diagram in (4.7).  A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 19 5. The stabilisation isomorphism In the more advanced part of this paper (when we investigate the two-dimensional setting), we shall often need an operation which allows us to change the domain and codomain of a Fredholm operator by adding an invertible operator. We call this operation stabilisation, and introduce and investigate it in this section. / H and q : G / G be idempotent Definition 5.1. Let H and G be separable Hilbert spaces, and let p : H / / qG is a Fredholm operator, and that Γ : (1−p)H (1−q)G is an invertible operators. Suppose that T : pH / G is a Fredholm operator, and the stabilisation isomorphism is the isomorphism operator. Then T + Γ : H of graded lines / |T + Γ| S : |T | induced by the quasi-isomorphism (ιp , ιq ) given by the inclusions pH ⊆ H and qG ⊆ G. The next proposition describes the relationship between the torsion isomorphism and the stabilisation isomorphism. This result is a consequence of Proposition 3.6. Proposition 5.2 (Torsion commutes with stabilisation). Let H, G and K be separable Hilbert spaces, / G and q : K / K be idempotent operators. Suppose that T : eH / pG and / H, p : G and let e : H / / / S : pG qK are Fredholm operators, and that Γ : (1 − e)H (1 − f )G and Θ : (1 − f )G (1 − q)K are invertible operators. Then the following diagram is commutative: T |T | ⊗ |S| / |ST | S⊗S S  |T + Γ| ⊗ |S + Θ| T  / |ST + ΘΓ| . We proceed by explaining how the perturbation isomorphism and the stabilisation isomorphism relate to one another. Proposition 5.3 (Perturbation commutes with stabilisation). Let H and G be two separable Hilbert / H and q : G / G be idempotent operators. Suppose that T and S : pH / qG are spaces, and let p : H / (1 − q)H is an invertible operator. Fredholm operator such that T − S is of trace class and that Γ : (1 − p)H Then the following diagram is commutative: |T | P S S  |T + Γ| / |S|  / |S + Γ| . P Proof. Upon applying Example 4.4, the claimed identity is first verified in the case where the index of T (and hence also the index of S) is equal to 0. Then, by applying Example 4.5, the general claim is verified by choosing n, m ∈ N ∪ {0} with m − n = Index(T ) = Index(S) and reducing to the index zero case by means / Cm .  of the trivial operator 0m,n : Cn We end this section with a result which elaborates on the commutativity property of the torsion isomorphism (Theorem 2.8 (2)) in the context of Fredholm operators. / H be idempotent operators such Proposition 5.4. Let H be a separable Hilbert space and let e, f, p, q : H / f H and S : pH / qH are that ep = pe = eq = qe = 0 and f p = pf = f q = qf = 0. Suppose that T : eH Fredholm operators. Then the following diagram commutes: (5.1) |T | ⊗ |S| S⊗S  |T + p| ⊗ |S + f | PPP PPPT PPP PP' ǫ / |S| ⊗ |T | S⊗S  |S + e| ⊗ |T + q| ♦♦ T ♦♦♦♦ ♦ ♦ ♦ w♦♦♦ T +S . 20 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON / (f + q)H, S + f : (p+ f )H / (q + f )H / (f + p)H, T + q : (e + q)H Proof. We remark that T + p : (e + p)H / and S + e : (p + e)H (q + e)H are Fredholm operators satisfying that (S + f )(T + p) = (T + q)(S + e) = / (f + q)H so that the isomorphisms appearing in (5.1) make sense. The result then T + S : (e + p)H follows by definition of the torsion isomorphisms and stabilisation isomorphisms appearing together with the properties (1) and (2) from Theorem 2.8. Indeed, one may identify the exact triangles ∆S+f,T +p and ∆T +q,S+e (yielding the torsion isomorphisms in question) with the exact triangles ∆1 and ∆2 relating to the direct sum decomposition I(T ) ⊕ I(S) in Theorem 2.8 (2).  6. The group 2-cocycle associated to a group action on a category We shall now see how the data of a group acting strictly on a category (subject to a few extra conditions) gives rise to a group 2-cocycle. The constructions of this section should be compared with the constructions by Brylinski given in [Bry97] (see also [Kap95] and [Str87]). The reader might at this point think that having a strict action is too restrictive a condition, but we shall see in Section 7 that this condition suffices to give a description of the Connes-Karoubi multiplicative character on the second algebraic K-group. Assumption 6.1. Assume that we are given a commutative unital ring R, a group G and a category C with at least one object satisfying the following conditions: (1) For any pair of objects x, y in C, the morphisms Mor(x, y) form a central bimodule over R, and the composition / Mor(x, z) Mor(x, y) × Mor(y, z) induces a bimodule homomorphism Mor(x, y) ⊗R Mor(y, z) / Mor(x, z) ; (2) The set of isomorphisms between any two objects is non-empty; (3) For any object x in C, the map r 7→ r · idx is an isomorphism of bimodules over R: / Mor(x, x) ; ιx : R / C, g ∈ G, induce isomorphisms (4) The group G acts strictly on C, and the associated functors g : C / of bimodules g : Mor(x, y) Mor(g(x), g(y)) whenever x, y are objects in C. We let R∗ denote the abelian group of invertible elements in R. Definition 6.2. Choose an object x in C and choose an isomorphism αg : x for every g ∈ G. Define the group 2-cochain c : Z[G2 ] / g(x) / R∗ by −1 α−1 c(g1 , g2 ) := ι−1 g1 ◦ g1 (αg2 ) ◦ αg1 g2 x
g1 , g2 ∈ G . One can visualise c(g1 , g2 ) as the difference between the two isomorphisms from x to (g1 g2 )(x) in the simplex g1 (x) ❏❏ ⑤= ❏❏g1 (αg2 ) ⑤ ❏❏ ⑤⑤ ⑤ ❏❏ ⑤ ❏% ⑤ ⑤⑤ / (g1 g2 )(x) x αg g αg1 1 2 / R∗ is a group 2-cocycle, and its class in group cohomology Lemma 6.3. The group 2-cochain c : Z[G2 ] 2 ∗ [c] ∈ H (G, R ) is independent of the choices made in Definition 6.2. / R∗ is a group 2-cocycle can be verified by a straightforward computation. Proof. The fact that c : Z[G2 ] In order to show that the corresponding class in group cohomology is independent of the choices made, we / g(y) be an alternative choice of isomorphism for each g ∈ G. let y be an alternative object in C and βg : y / Choosing an isomorphism φ : x y, we then obtain that the quotient of the two 2-cochains involved can
−1 α−1 ) ◦ βg ◦ φ .  be expressed as the coboundary of the 1-cochain given by g 7→ ι−1 g ◦ g(φ x A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 21 / H be an orthogonal 6.1. The group 2-cocycle on the restricted general linear group. Let P : H 2 projection acting on a separable Hilbert space H. Let L (H) ⊆ L (H) denote the ideal of Hilbert-Schmidt operators inside the bounded operators L (H). The restricted general linear group, GLres (H), is defined as the group of bounded invertible linear transformations u of H satisfying P − uP u−1 ∈ L 2 (H) . We are now going to construct a category Cres satisfying the conditions of Assumption 6.1, for the unital commutative ring R := C and the restricted general linear group G := GLres (H). In particular, we obtain a class
[cres ] ∈ H 2 GLres (H), C∗ in the second group cohomology of the restricted general linear group. For an element u ∈ GLres (H), we apply the notation Pu := uP u−1 for the idempotent operator on H obtained by conjugating the orthogonal projection P by u. We begin by recording a few standard lemmas: Lemma 6.4. For every triple of elements u, v, w ∈ GLres (H), the difference / Im(Pw ) Pw Pv Pu − Pw Pu : Im(Pu ) is of trace class. Proof. This follows from the identities Pw Pv Pu − Pw Pu = Pw (Pv − Pu )Pu = Pw v[P, v −1 u]u−1 Pu = w[P, w−1 v][P, v −1 u]P u−1 and the fact that the product of two Hilbert-Schmidt operators is a trace class operator.  Corollary 6.5. For every pair of elements u, v ∈ GLres (H), the bounded operator / Im(Pv ) Pv Pu : Im(Pu ) is a Fredholm operator. Proof. By Lemma 6.4 we have that Pu Pv : Im(Pv ) / Im(Pu ) is a parametrix.  Definition 6.6. We define a category Cres by the following criteria: (1) The objects of Cres are the elements in the restricted general linear group; (2) For any pair of objects u, v ∈ GLres (H), the set of morphisms from u to v is given by the graded line Mor(u, v) := |Pv Pu | , / where Pv Pu : Im(Pu ) Im(Pv ) is the Fredholm operator from Lemma 6.5. (3) For any triple of objects u, v, w ∈ GLres (H), the composition of morphisms is given by the isomorphism |Pv Pu | ⊗ |Pw Pv | T / |Pw Pv Pu | P / |Pw Pu | , obtained from composing the torsion isomorphism of Fredholm operators and the perturbation isomorphism from Section 3 and Section 4. We continue by defining the action of GLres (H): / Cres by the following: Definition 6.7. For each g ∈ GLres (H), we define a functor g : Cres (1) On the objects in C, the functor g is given by left multiplication with the group element g ∈ GLres (H); (2) For any pair of objects u, v ∈ GLres (H), the isomorphism / Mor(g · u, g · v) g : Mor(u, v) is given by the torsion isomorphism L(g)R(g −1 ) : |Pv Pu | from Example 3.4. / |gPv Pu g −1 | = |Pgv Pgu | 22 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON / |Pgv Pgu | can equally well be described as Remark that the above isomorphism L(g)R(g −1 ) : |Pv Pu | / / Pgu H and the quasi-isomorphism |g, g| : |Pv Pu | |Pgv Pgu | induced by the bounded operators g : Pu H / g : Pv H Pgv H. The rest of this section is devoted to proving that the above definition indeed provides us with a category equipped with a strict action of the restricted general linear group satisfying the conditions of Assumption 6.1. Proposition 6.8. Cres is a category satisfying Assumption 6.1 (1), (2) and (3). Proof. The only non-trivial claim is that the composition is associative. For every four objects u, v, w, x ∈ GLres (H), it suffices to show that every square in the following diagram commutes: |Pv Pu | ⊗ |Pw Pv | ⊗ |Px Pw | id⊗T / |Pv Pu | ⊗ |Px Pw Pv | id⊗P / |Pv Pu | ⊗ |Px Pv | T⊗id T T  |Pw Pv Pu | ⊗ |Px Pw | T  / |Px Pw Pv Pu | P P P P⊗id  |Pw Pv | ⊗ |Px Pw |  / |Px Pw Pv | T  / |Px Pv Pu | P  / |Px Pu | . The upper left square commutes by the associativity of the torsion isomorphism from Proposition 3.3. The lower right square commutes by the cocycle property of the perturbation isomorphism from Theorem 4.1. The lower left and upper right squares commute since perturbation commutes with torsion, see Theorem 4.9.  Proposition 6.9. The group GLres (H) acts strictly on the category Cres . Proof. The only non-trivial assertion is that the group action on morphisms respects the composition of morphisms. Thus, let g ∈ GLres (H). By the definition of the composition and the group action, it suffices to check that we have a commuting diagram |Pv Pu | ⊗ |Pw Pv | L(g)R(g−1 )⊗L(g)R(g−1 ) T / |Pw Pv Pu | P L(g)R(g−1 )  |Pgv Pgu | ⊗ |Pgw Pgv | L(g)R(g−1 )  T / |Pw Pu | / |Pgw Pgv Pgu | P  / |Pgw Pgu | . But the commutativity of the first square follows from Proposition 3.6, and the commutativity of the second square follows since perturbation commutes with torsion, see Theorem 4.9.  Corollary 6.10. The formula from Definition 6.2 defines a class in the second group cohomology of the restricted general linear group, [cres ] ∈ H 2 (GLres (H), C∗ ). 7. Comparison with the multiplicative character / H on the separable Hilbert space H with infinite dimensional Let us fix an orthogonal projection P : H kernel and image. Define the unital subalgebra M1 ⊆ L (H) by  M1 := x ∈ L (H) | [P, x] is Hilbert-Schmidt . Notice that M1 becomes a unital Banach algebra when equipped with the norm kxk := kxk∞ + k[P, x]k2 , / [0, ∞) is the Hilbert-Schmidt / [0, ∞) is the operator norm and k · k2 : L 2 (H) where k · k∞ : L (H) 1 ∗ norm. We remark that the group of invertible elements (M ) is exactly the restricted general linear group (M1 )∗ = GLres (H) . The universal 2-summable Fredholm module is given by the triple Funi := (M1 , H, 2P − 1). In order to describe the Connes-Karoubi multiplicative character / C∗ M (Funi) : K alg (M1 ) 2 A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 23 using our category theoretic approach, we need to construct a polarisation for the general linear group GL(M1 ) consisting of invertible matrices with entries in M1 . To this end, we represent GL(M1 ) on the infinite Hilbert space direct sum H∞ := ⊕∞ j=1 H by defining the representation GLn (M1 ) ∞ n ∞ X X X ξi · ei . gij (ξj ) · ei + ξj · ej ) := g( / GL(H∞ ) i=n+1 i,j=1 j=1 for each n ∈ N and verifying that these representations are compatible with the direct limit structure of / GL(H∞ ) is then polarised by GL(M1 ) = lim/ GLn (M1 ). The corresponding representation GL(M1 ) n ∞ the infinite direct sum of orthogonal projections / H∞ P ∞ : H∞ ∞ ∞ X X P (ξj ) · ej . ξj · ej ) := P ∞( j=1 j=1 2 1 ∗ By a slight abuse of notation, we let [cres ] ∈ H (GL(M ), C ) denote the class in group cohomology coming / H∞ . from the action of GL(M1 ) on the category Cres associated with the orthogonal projection P ∞ : H∞ By a result of Connes and Karoubi, [CoKa88, Theorem 5.6], we may compute the multiplicative character using a central extension of the connected component of the identity, GL0 (M1 ) ⊆ GL(M1 ): (7.1) 1 / C∗ / GL0 (M1 ) /Γ / 1. This central extension is described in detail by Pressley and Segal in [PrSe86, Chapter 6] and arises from a combination of the Fredholm determinant and the exact sequence of Banach algebras 0 / L 1 (P H) /T1 / M1 / 0. Recall here that the unital Banach algebra T 1 ⊆ M1 × L (P H) is defined by  T 1 := (x, y) | x ∈ M1 , y ∈ L (P H) , P xP − y ∈ L 1 (P H) and equipped with the norm k(x, y)k := kxk + kP xP − yk1 , where k · k1 : L 1 (P H) norm defined on all trace class operators. / [0, ∞) is the trace Theorem 7.1. The following diagram is commutative: K2alg (M1 ) h / H2 (GL(M1 ), Z) ❥❥ ❥ ❥❥ ❥ ❥ ❥ M(Funi ) ❥ ❥❥❥ h[cres ],·i  ❥ ❥❥❥❥ ∗ t ❥ C where h is the Hurewicz homomorphism and h[cres ], ·i comes from the pairing between group cohomology and group homology. Proof. By the results of Connes and Karoubi mentioned just before the statement of this theorem and by the fact that K2alg (M1 ) is isomorphic to the second group homology of the elementary matrices E(M1 ) (see for example [Ros94, Theorem 5.2.7]), it suffices to show that the restriction of [cres ] ∈ H 2 (GL(M1 ), C∗ ) to H 2 (GL0 (M1 ), C∗ ) agrees with the group cohomology class provided by the central extension in (7.1). Moreover, since the passage from M1 to the (n × n)-matrices Mn (M1 ) can be described by passing from the polarised Hilbert space P H ⊆ H to the polarised Hilbert space P ⊕n H⊕n ⊆ H⊕n we may restrict our attention to the group of invertible elements in M1 . That is, to the restricted general linear group GLres (H). An element g ∈ GLres (H) lies in the connected component of the identity, GL0res (H), precisely when the / P H has index zero. For each such g we may thus choose a bounded finite Fredholm operator P gP : P H / H is invertible. The group 2-cocycle coming from / P H such that P gP + Fg : H rank operator Fg : P H the central extension in (7.1) is then given by
muni (g, h) := det (P ghP + Fgh )(P hP + Fh )−1 (P gP + Fg )−1 g, h ∈ GL0res (H) . Let us now turn to the description of the restriction of [cres ] to H 2 (GL0res (H), C∗ ). We start by choosing the unit 1 ∈ GL0res (H) as our object. For each g ∈ GL0res (H) we have the perturbation isomorphism P(Pg · P, g(P gP + Fg )−1 ) : |Pg · P | / |g(P gP + Fg )−1 | = (C, 0) 24 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON and we may thus choose the isomorphism
αg := P g(P gP + Fg )−1 , Pg · P (1C ) ∈ Mor(1, g) = |Pg · P | . Using that perturbation commutes with torsion, Theorem 4.9, it can then be verified that the inverse is given by
−1 α−1 , P · Pg (1C ) ∈ |P · Pg | = Mor(g, 1) . g = P (P gP + Fg )g For a pair of elements g, h ∈ GL0res (H), the evaluation cres (g, h) ∈ C∗ is therefore given by the image of −1 −1 αgh ⊗ g(α−1 | ⊗ |P · Pg | h ) ⊗ αg ∈ |Pgh · P | ⊗ |g · P Ph · g under the composition |Pgh · P | ⊗ |g · P Ph · g −1 | ⊗ |P · Pg | T P / |P Pg Pgh P | / |P | = (C, 0) . We record that g(αh−1 ) ∈ |g · P Ph · g −1 | = |Pg Pgh | agrees with the value
P g(P hP + Fh )h−1 g −1 , g · P Ph · g −1 (1C ) as can be verified by using the argument in the proof of Proposition 6.9. However, using that perturbation commutes with torsion, Theorem 4.9, together with the transitivity property of the perturbation isomorphism, Theorem 4.1, we obtain that the diagram gh(P ghP + Fgh )−1 ⊗ g(P hP + Fh )h−1 g −1 ⊗ (P gP + Fg )g −1 P / |Pgh · P | ⊗ |Pg · Pgh | ⊗ |P · Pg | T T  (P gP + Fg )(P hP + Fh )(P ghP + Fgh )−1  P / P · Pg · Pgh · P ❞❞ ❞ ❞ ❞ ❞ ❞ ❞❞❞❞ ❞❞❞❞❞❞ P ❞❞❞❞❞P ❞ ❞ ❞ ❞ ❞  ❞❞❞❞❞ ❞❞❞❞❞❞ |P | r❞ is commutative. This entails that cres (g, h) ∈ C∗ is given by the image of the unit 1C ∈ C under the perturbation isomorphism P : (P gP + Fg )(P hP + Fh )(P ghP + Fgh )−1 / |P | = (C, 0) . But by Example 4.3, this is exactly the Fredholm determinant
det (P ghP + Fgh )(P hP + Fh )−1 (P gP + Fg )−1 . This shows that cres (g, h) = muni (g, h) for all g, h ∈ GL0res (H) and the theorem is proved.  We record the following corollary: Corollary 7.2. Let F = (A, H, F ) be a unital 2-summable Fredholm module such that the orthogonal / M1 denote an algebra projection P := (F + 1)/2 has infinite dimensional image and kernel. Let π : A homomorphism such that F agrees with the pull back of the universal 2-summable Fredholm module along π. Then the diagram K2alg (A) h / H2 (GL(A), Z) ❦❦ ❦ ❦ ❦❦❦ ❦ M(F ) ❦ ❦❦ ∗  u❦❦❦❦❦❦ hπ [cres ],·i ∗ C is commutative, where π ∗ [cres ] denotes the pull back of the group cohomology class [cres ] along the induced / GL(M1 ). group homomorphism π : GL(A) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 25 8. The group 3-cocycle associated to a group action on a coproduct category In this section we start our discussion of the two-dimensional situation and we shall explain how group actions on categories endowed with some extra structure give rise to group 3-cocycles. The idea of constructing group cocycles from groups acting on higher categories appears in many places, see for example [Str87, JoSt93, BrSp76, BaLa04, FrZh12]. We are here taking a slightly different route and develop the notion of a group acting on a “coproduct category”, since this is the structure which appears naturally in our analytic applications. Coproduct categories are related to weak 2-categories, but our systematic use of graded tensor products makes this relationship less straightforward. The extra sign appearing in the composition of graded tensor products of morphisms also has consequences for our construction of group 3-cocycles: we only obtain well-defined group 3-cohomology classes with values in the quotient group C∗ /{±1}. 8.1. Coproduct categories. We are going to consider categories where the morphisms between two objects are vectors in a Z-graded complex line. Definition 8.1. A category of Z-graded complex lines is a category C with at least one object such that (1) The set of morphisms Mor(a, b) is a Z-graded complex line whenever a, b are objects in C; / Mor(a, c) induces an isomorphism of (2) The composition of morphisms ◦ : Mor(b, c) × Mor(a, b) Z-graded complex lines: / Mor(a, c) ; M : Mor(a, b) ⊗ Mor(b, c) (3) For each object a in H(x, y), the Z-graded complex line (C, 0) is isomorphic to Mor(a, a) via the map λ 7→ λ · ida . In particular, Mor(a, a) has degree 0 ∈ Z. For a morphism α ∈ Mor(a, b) = (L , m) we write ε(α) := m ∈ Z for the degree. Lemma 8.2. If C is a category of Z-graded complex lines, then any two objects a, b ∈ C are isomorphic. / Mor(a, a) is an isomorphism of Z-graded complex lines Proof. Let a, b ∈ C. Since ◦ : Mor(a, b) ⊗ Mor(b, a) we may choose α ∈ Mor(a, b) and β ∈ Mor(b, a) such that β ◦ α = ida . Since the map λ 7→ λ · idb is an isomorphism of Z-graded complex lines (and the composition is associative and linear) it follows easily that α ◦ β = idb as well.  Let C and D be categories of Z-graded complex lines. The graded tensor product C⊗D is also a category of Z-graded complex lines. The objects in this category are given by pairs of objects (a, b), where a ∈ C and b ∈ D. It is convenient already at this point to introduce the notation ′ The morphisms Mor(a ⊗ b, a ⊗ b ′
a ⊗ b := (a, b) ∈ Obj(C ⊗ D) . are given by the Z-graded complex line:
Mor a ⊗ b, a′ ⊗ b′ := Mor(a, a′ ) ⊗ Mor(b, b′ ) A morphism from a ⊗ b to a′ ⊗ b′ is thus given by a tensor product / a′ ⊗ b ′ , α ⊗ β: a⊗ b / a′ has degree ε(α) ∈ Z and β : b / b′ has degree ε(β) ∈ Z. The unit where the morphism α : a / ida⊗b : a ⊗ b a ⊗ b is the tensor product of the units ida ∈ Mor(a, a) and idb ∈ Mor(b, b). The composition of morphisms is given by the formula
′ M (α ⊗ β) ⊗ (α′ ⊗ β ′ ) := (−1)ε(β)·ε(α ) · M (α ⊗ α′ ) ⊗ M (β ⊗ β ′ ). For an extra category E of Z-graded complex lines we may identify the graded tensor products (C ⊗ D) ⊗ E and C ⊗ (D ⊗ E) via an obvious isomorphism of categories, which we shall from now on tacitly suppress. Let C and D be categories of Z-graded complex lines. A linear functor F from C to D is a functor / D such that the induced map F: C / Mor(F (a), F (b)) F : Mor(a, b) a, b ∈ C is a linear isomorphism of Z-graded complex lines. 26 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Definition 8.3. Let X be a non-empty  set, and let H(x, y) be a category of Z-graded complex lines for every x, y ∈ X. We say that the collection H(x, y) x,y∈X is a coproduct category when it is equipped with linear functors / H(x, z) ⊗ H(z, y) x, y, z ∈ X ∆z : H(x, y) referred to as coproducts such that the coassociativity relation (id ⊗ ∆z )∆w = (∆w ⊗ id)∆z : H(x, y) / H(x, w) ⊗ H(w, z) ⊗ H(z, y) . holds for all elements x, y, z, w ∈ X. A morphism of coproduct categories Φ : H for each x, y ∈ X, a linear functor ϕ : H(x, y) / X ′ of the underlying sets and / H′ consists of a map ϕ : X / H′ (ϕ(x), ϕ(y)) such that the diagram ∆z H(x, y) / H(x, z) ⊗ H(z, y) ϕ ϕ⊗ϕ   ∆′ϕ(z) / H′ (ϕ(x), ϕ(z)) ⊗ H′ (ϕ(z), ϕ(y)) H′ (ϕ(x), ϕ(y)) commutes for all x, y, z ∈ X. The composition of morphisms of coproduct categories is carried out in the obvious way (composing the underlying maps between sets and the linear functors). An action of a group G on a coproduct category H is a group homomorphism G / Aut(H) . We now discuss the link between coproduct categories and “twisted” weak 2-categories. The twisting comes from our use of graded tensor products. We refer to [B6́7] for generalities regarding weak 2-categories (bicategories). Suppose that (H, ∆, X) is a coproduct category. We start out by “inverting” the coproducts ∆z : H(x, y) / H(x, z) ⊗ H(z, y) . Lemma 8.4. For each triple of elements x, y, z ∈ X, it holds that the coproduct ∆z : H(x, y) / H(x, z) ⊗ H(z, y) is an equivalence of categories. / H(x, z) ⊗ H(z, y) is fully faithful Proof. Let x, y, z ∈ X be given. By definition, the functor ∆z : H(x, y) so we only need to argue that ∆z is essentially surjective. This does in fact hold since any two objects a ⊗ b and a′ ⊗ b′ are isomorphic in H(x, z) ⊗ H(z, y) as can be seen by applying Lemma 8.2.  / H(x, y) Using the above lemma, for each x, y, z ∈ X, we may choose a functor ∗ : H(x, z) ⊗ H(z, y) / / together with natural isomorphisms ξ : ∗ ∆z idH(x,y) and η : ∆z ∗ idH(x,z)⊗H(z,y). By construction we have that ∗ is linear in the sense that the induced map

/ Mor ∗ (a ⊗ b), ∗(a′ ⊗ b′ ) ∗ : Mor a ⊗ b, a′ ⊗ b′ is linear at the level of the underlying complex lines. It need however not be the case that ∗ preserves the degree. We now define our “twisted” weak 2-category H(2): (1) The objects in H(2) are the elements in the set X; (2) The 1-morphisms from an object x to an object y are the objects in the category H(x, y) and the composition of 1-morphisms a ∈ H(x, z) and b ∈ H(z, y) is defined by b ◦1 a := ∗(a ⊗ b); (3) The 2-morphisms from a 1-morphism a ∈ H(x, y) to a 1-morphism a′ ∈ H(x, y) are the vectors / a′ and in the Z-graded complex line Mor(a, a′ ). The vertical composition of 2-morphisms α : a ′′ ′ ′′ ′ / a with a, a , a ∈ H(x, y) is given by the composition in the category H(x, y). The horizontal β: a / a′ and β : b / b′ with a, a′ ∈ H(x, z) and b, b′ ∈ H(z, y) is defined composition of 2-morphisms α : a ′ ′ / by β ◦1 α := ∗(α ⊗ β) : b ◦1 a b ◦1 a . A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 27 The extra adjective “twisted” primarily signifies that the assignment ◦1 : H(z, y) × H(x, z) / H(x, y) defined on objects by (b, a) 7→ b ◦1 a and on morphisms by (β, α) 7→ β ◦1 α is not a functor because of a / a′ and / b′ and β ′ : b′ / b′′ in H(z, y) and 2-morphisms α : a sign defect. Indeed, for 2-morphisms β : b ′′ ′ ′ / a in H(x, y) we instead have that α:a ′ (β ′ ◦2 β) ◦1 (α′ ◦2 α) = ∗((α′ ◦2 α) ⊗ (β ′ ◦2 β)) = ∗((α′ ⊗ β ′ ) ◦2 (α ⊗ β)) · (−1)ε(β)·ε(α ) ′ = (β ′ ◦1 α′ ) ◦2 (β ◦1 α) · (−1)ε(β)·ε(α ) . One may define unitors and associators for our 2-category theoretic data but these operations are only satisfying twisted versions of the usual naturality, pentagon and triangle identities, meaning that these identities are only satisfied up to specified signs. For objects a ∈ H(x, z), b ∈ H(z, w) and c ∈ H(w, y) one may for example define the associator Aa,b,c : c ◦1 (b ◦1 a) / (c ◦1 b) ◦1 a as the unique isomorphism making the diagram here below commute: (∆z ⊗ id)∆w (c ◦1 (b ◦1 a)) (∆z ⊗id)(η(b◦1 a)⊗c ) (∆z ⊗id)∆w (Aa,b,c )  (id ⊗ ∆w )∆z ((c ◦1 b) ◦1 a) (id⊗∆w )(ηa⊗(c◦1 b) ) / ∆z (b ◦1 a) ⊗ c PPP PPηPa⊗b ⊗idc PPP PP' a⊗b⊗c ♥♥7 ♥ ♥ ♥♥ ♥♥♥ ♥♥♥ ida ⊗ηb⊗c / a ⊗ ∆w (c ◦1 b) Notice in this respect how the coassociativity of our coproduct functors is applied in order to make sense of the above diagram. We are not currently aware of a standardised treatment of weak 2-categories which are twisted in the above sense and believe that this is a subject for further investigations. In the present text we work in the context of coproduct categories since these appear naturally in our example driven approach. Moreover, our coproduct categories arise without making any auxiliary choices and as a consequence all our identities are strict instead of the usual “up to natural isomorphisms” familiar from a 2-category theoretic framework. / Aut(H) is a group action. 8.2. 3-cocycles. Suppose that (H, ∆, X) is a coproduct category and that G We shall now see how to construct a group 3-cocycle from this data. Because of our systematic use of graded tensor products, this group 3-cocycle takes values in the quotient group C∗ /{±1}. Definition 8.5. Choose an element x ∈ X. For each g ∈ G, choose an object ag ∈ H(x, gx) and for each pair of elements g, h ∈ G, choose an isomorphism / ag ⊗ g(ah ) . βg,h : ∆gx (agh ) For each triple of elements g, h, k ∈ G, define the automorphism γ(g, h, k) : ag ⊗ g(ah ) ⊗ (gh)(ak ) / ag ⊗ g(ah ) ⊗ (gh)(ak ) , in the category H(x, gx) ⊗ H(gx, ghx) ⊗ H(ghx, ghkx), such that the diagram here below commutes: ∆gx (agh ) ⊗ (gh)(ak ) ❥❥4 (∆gx ⊗id)(βgh,k ) ❥❥❥❥ ❥ ❥ ❥❥❥❥ ❥❥❥❥ (∆gx ⊗ id)∆ghx (aghk ) ❚❚❚❚ ❚❚❚❚ ❚❚❚❚ ❚❚❚❚ (id⊗∆ghx )(βg,hk ) * ag ⊗ ∆ghx (g(ahk )) βg,h ⊗id / ag ⊗ g(ah ) ⊗ (gh)(ak ) γ(g,h,k) id⊗(g⊗g)(βh,k )  / ag ⊗ g(ah ) ⊗ (gh)(ak ) 28 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We define the group 3-cochain c : Z[G3 ] / C∗ by c(g, h, k) · idag ⊗g(ah )⊗(gh)(ak ) := γ(g, h, k) Proposition 8.6. The group 3-cochain c : Z[G3 ] (8.1) g, h, k ∈ G . / C∗ satisfies the following twisted group 3-cocycle relation: (−1)ε(βg,h )ε(βk,l ) · c(gh, k, l) · c(g, h, kl) = c(h, k, l) · c(g, hk, l) · c(g, h, k) for all g, h, k, l ∈ G. Proof. Let g, h, k, l ∈ G be given. We start by moving each of the automorphisms γ(gh, k, l), γ(g, h, kl), γ(h, k, l), γ(g, hk, l) and γ(g, h, k) so that they all become automorphisms of the object ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) in the category H(x, gx) ⊗ H(gx, ghx) ⊗ H(ghx, ghkx) ⊗ H(ghkx, ghklx) . We notice that this operation does not change the associated elements c(gh, k, l), c(g, h, kl), c(h, k, l), c(g, hk, l) and c(g, h, k) in C∗ . Indeed, for the left hand side of (8.1), we consider the following two automorphisms: −1 (id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) ◦ (id ⊗ id ⊗ ∆ghkx )(γ(g, h, kl)) ◦ (id ⊗ id ⊗ (gh ⊗ gh)(βk,l ))
= (id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) ◦ (id ⊗ id ⊗ ∆ghkx ) (id ⊗ (g ⊗ g)(βh,kl )) ◦ (id ⊗ ∆ghx )(βg,hkl ) −1 −1 −1 ◦ (∆gx ⊗ ∆ghkx )(βgh,kl ) ◦ (βg,h ⊗ id ⊗ id) ◦ (id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) / ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) : ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) and −1 (βg,h ⊗ id ⊗ id) ◦ (∆gx ⊗ id ⊗ id)(γ(gh, k, l)) ◦ (βg,h ⊗ id ⊗ id) = (βg,h ⊗ id ⊗ id) ◦ (id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) ◦ (∆gx ⊗ ∆ghkx )(βgh,kl )
−1 −1 −1 ◦ (∆gx ⊗ id ⊗ id) (∆ghx ⊗ id)(βghk,l ) ◦ (βgh,k ⊗ id) ◦ (βg,h ⊗ id ⊗ id) / : ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) . We record that their composition is given by the automorphism (8.2)
(id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) ◦ (id ⊗ id ⊗ ∆ghkx ) (id ⊗ (g ⊗ g)(βh,kl )) ◦ (id ⊗ ∆ghx )(βg,hkl )
−1 −1 −1 ◦ (∆gx ⊗ id ⊗ id) (∆ghx ⊗ id)(βghk,l ) ◦ (βgh,k ⊗ id) ◦ (βg,h ⊗ id ⊗ id) · (−1)ε(βg,h )·ε(βk,l ) , where the extra sign (−1)ε(βg,h )·ε(βk,l ) comes from the definition of the composition in the graded tensor product of categories H(x, gx) ⊗ H(gx, ghx) ⊗ H(ghx, ghkx) ⊗ H(ghkx, ghklx) . For the right hand side of (8.1), we consider the following three automorphisms: (id ⊗ g ⊗ g ⊗ g)(id ⊗ γ(h, k, l)) = (id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) ◦ (id ⊗ id ⊗ ∆ghkx )(id ⊗ (g ⊗ g)(βh,kl )) −1 −1 ◦ (id ⊗ ∆ghx ⊗ id)(id ⊗ (g ⊗ g)(βhk,l )) ◦ (id ⊗ (g ⊗ g)(βh,k ) ⊗ id) / ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) : ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) and −1 (id ⊗ (g ⊗ g)(βh,k ) ⊗ id) ◦ (id ⊗ ∆ghx ⊗ id)(γ(g, hk, l)) ◦ (id ⊗ (g ⊗ g)(βh,k ) ⊗ id) = (id ⊗ (g ⊗ g)(βh,k ) ⊗ id) ◦ (id ⊗ ∆ghx ⊗ id)(id ⊗ (g ⊗ g)(βhk,l )) ◦ (id ⊗ id ⊗ ∆ghkx )(id ⊗ ∆ghx )(βg,hkl ) −1 ◦ (∆gx ⊗ id ⊗ id)(∆ghx ⊗ id)(βghk,l ) −1 −1 ◦ (id ⊗ ∆ghx ⊗ id)(βg,hk ⊗ id) ◦ (id ⊗ (g ⊗ g)(βh,k ) ⊗ id) / ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) : ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 29 and γ(g, h, k) ⊗ id = (id ⊗ (g ⊗ g)(βh,k ) ⊗ id) ◦ (id ⊗ ∆ghx ⊗ id)(βg,hk ⊗ id) −1 −1 ◦ (∆gx ⊗ id ⊗ id)(βgh,k ⊗ id) ◦ (βg,h ⊗ id ⊗ id) / ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) . : ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) We record that their composition is given by (id ⊗ id ⊗ (gh ⊗ gh)(βk,l )) ◦ (id ⊗ id ⊗ ∆ghkx )(id ⊗ (g ⊗ g)(βh,kl )) (8.3) −1 ◦ (id ⊗ id ⊗ ∆ghkx )(id ⊗ ∆ghx )(βg,hkl ) ◦ (∆gx ⊗ id ⊗ id)(∆ghx ⊗ id)(βghk,l ) −1 −1 ◦ (∆gx ⊗ id ⊗ id)(βgh,k ⊗ id) ◦ (βg,h ⊗ id ⊗ id) . Since the two automorphisms of ag ⊗ g(ah ) ⊗ (gh)(ak ) ⊗ (ghk)(al ) given in (8.2) and (8.3) agree up to the sign (−1)ε(βg,h )·ε(βk,l ) , we have proved the proposition.  As a consequence of the above proposition, we obtain a group 3-cocycle after passing to the quotient of C∗ by the subgroup {±1} ⊆ C∗ . We denote the associated class in group cohomology by
[c] ∈ H 3 G, C∗ /{±1} . For any finite number of elements x1 , x2 , . . . , xn ∈ X, we introduce the notation ∆x1 ,x2 ,...,xn := (∆x1 ⊗ id⊗(n−1) ) ◦ (∆x2 ⊗ id⊗(n−2) ) ◦ . . . ◦ ∆xn / H(x, x1 ) ⊗ H(x1 , x2 ) ⊗ . . . ⊗ H(xn , y) : H(x, y) x, y ∈ X .
Lemma 8.7. The class [c] ∈ H 3 G, C∗ /{±1} is independent of the choices made in Definition 8.5. Proof. Let x′ ∈ X be an alternative element, let a′g ∈ H(x′ , gx′ ), g ∈ G, be alternative objects and let ′ / C∗ denote the / a′g ⊗ g(a′ ), g, h ∈ G be alternative isomorphisms. We let c′ : Z[G3 ] βg,h : ∆gx′ (a′gh ) h / C∗ up to signs and a coboundary. alternative 3-cochain and we will show that c′ agrees with c : Z[G3 ] We start by choosing a set of intertwiners in the following way: Choose an object ζ ∈ H(x, x′ ). For each g ∈ G, ζ ⊗ ∆gx (a′g ) and ∆x′ (ag ) ⊗ g(ζ) are then both objects in the category H(x, x′ ) ⊗ H(x′ , gx) ⊗ H(gx, gx′ ) and we may thus choose an isomorphism σg : ζ ⊗ ∆gx (a′g ) / ∆x′ (ag ) ⊗ g(ζ) . For any finite number of group elements g1 , g2 , . . . , gn ∈ G, we then have the intertwining isomorphism σg1 ,g2 ,...,gn : ζ ⊗ ∆g1 x (a′g1 ) ⊗ g1⊗2 ∆g2 x (a′g2 ) ⊗ . . . ⊗ (g1 · . . . · gn−1 )⊗2 ∆gn x (a′gn ) / ∆x′ (ag1 ) ⊗ g ⊗2 ∆x′ (ag2 ) ⊗ . . . ⊗ (g1 · . . . · gn−1 )⊗2 ∆x′ (agn ) ⊗ (g1 g2 · . . . · gn )(ζ) , 1 defined as the composition
σg1 ,g2 ,...,gn := id⊗2(n−1) ⊗ (g1 g2 · . . . · gn−1 )⊗3 (σgn )

◦ . . . ◦ id⊗2 ⊗ g1⊗3 (σg2 ) ⊗ id⊗2(n−2) ◦ σg1 ⊗ id⊗2(n−1) . For each g, h ∈ G, we define the automorphism δ(g, h) : ∆x′ ,gx,gx′ (agh ) ⊗ (gh)(ζ) / ∆x′ ,gx,gx′ (agh ) ⊗ (gh)(ζ) as the composition of isomorphisms: ∆x′ ,gx,gx′ (agh ) ⊗ (gh)(ζ) (∆x′ ⊗∆gx′ )(βg,h )⊗id −1 σg,h / ∆x′ (ag ) ⊗ ∆gx′ g(ah ) ⊗ (gh)(ζ) / ζ ⊗ ∆gx (a′ ) ⊗ g ⊗2 ∆hx (a′ ) g h ′ id⊗(∆gx ⊗∆ghx )(βg,h )−1 (id⊗∆gx,gx′ ⊗id)(σgh ) / ζ ⊗ ∆gx,gx′ ,ghx (a′gh ) / ∆x′ ,gx,gx′ (agh ) ⊗ (gh)(ζ) . 30 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We identify δ(g, h) with the number d(g, h) ∈ C∗ and thereby obtain a 2-cochain / C∗ . d : Z[G2 ] We claim that the coboundary of d agrees with the quotient of c and c′ , thus that d(h, k) · d(gh, k)−1 · d(g, hk) · d(g, h)−1 = c(g, h, k) · c′ (g, h, k)−1 , for all g, h, k ∈ G. Thus, let g, h, k ∈ G be given. We are going to compare automorphisms of the object ∆x′ (ag ) ⊗ ∆gx′ g(ah ) ⊗ ∆ghx′ (gh)(ak ) ⊗ (ghk)(ζ) . We thus start by representing the numbers c(g, h, k) and c′ (g, h, k)−1 as automorphisms of this object. The number c(g, h, k) ∈ C∗ can be represented by the automorphism
(∆x′ ⊗ ∆gx′ ⊗ ∆ghx′ ) γ(g, h, k) ⊗ id

= id⊗2 ⊗ g ⊗4 (∆x′ ⊗ ∆hx′ )(βh,k ) ⊗ id ◦ (∆x′ ⊗ ∆gx′ ,ghx,ghx′ )(βg,hk ) ⊗ id

−1 −1 ) ⊗ id⊗3 , ) ⊗ id ◦ (∆x′ ⊗ ∆gx′ )(βg,h ◦ (∆x′ ,gx,gx′ ⊗ ∆ghx′ )(βgh,k whereas the number c′ (g, h, k)−1 ∈ C∗ can be represented by the automorphism
−1 σg,h,k ◦ id ⊗ (∆gx ⊗ ∆ghx ⊗ ∆ghkx )(γ ′ (g, h, k)−1 ) ◦ σg,h,k

′ ′ ) ⊗ id⊗2 ◦ id ⊗ (∆gx,gx′ ,ghx ⊗ ∆ghkx )(βgh,k ) = σg,h,k ◦ id ⊗ (∆gx ⊗ ∆ghx )(βg,h

−1 ′ ′ . )−1 ◦ σg,h,k ◦ id ⊗ (∆gx ⊗ ∆ghx,ghx′ ,ghkx )(βg,hk )−1 ◦ id⊗3 ⊗ g ⊗4 (∆hx ⊗ ∆hkx )(βh,k d(h,k)·d(g,hk) In order to compare the quotient cc(g,h,k) ′ (g,h,k) with the quotient d(gh,k)·d(g,h) , we now represent the numbers d(g, h)−1 , d(gh, k)−1 and d(g, hk), d(h, k) as automorphisms of convenient objects. We represent d(g, h)−1 ∈ C∗ as the automorphism of the object
(∆x′ ,gx,gx′ ⊗ ∆ghx′ ) agh ⊗ (gh)(ak ) ⊗ (ghk)(ζ) , given by

id⊗4 ⊗ (gh)⊗3 (σk ) ◦ (δ(g, h)−1 ⊗ id⊗2 ) ◦ id⊗4 ⊗ (gh)⊗3 (σk−1 )

−1 ) ⊗ id⊗3 ◦ id⊗4 ⊗ (gh)⊗3 (σk ) ◦ (σg,h ⊗ id⊗2 ) = (−1)ε(σk )·ε(βg,h ) · (∆x′ ⊗ ∆gx′ )(βg,h

−1 ′ ) ⊗ id⊗2 ◦ id ⊗ (∆gx ⊗ ∆ghx )(βg,h ) ⊗ id⊗2 ◦ (id ⊗ ∆gx,gx′ ⊗ id)(σgh
◦ id⊗4 ⊗ (gh)⊗3 (σk−1 )
−1 ) ⊗ id⊗3 ◦ σg,h,k = (−1)ε(σk )·ε(βg,h ) · (∆x′ ⊗ ∆gx′ )(βg,h
−1 ′ ◦ id ⊗ (∆gx ⊗ ∆ghx )(βg,h ) ⊗ id⊗2 ◦ (id ⊗ ∆gx,gx′ ⊗ id⊗3 )(σgh,k ). We represent d(gh, k)−1 ∈ C∗ as the automorphism of the object ∆x′ ,gx,gx′ ,ghx,ghx′ (aghk ) ⊗ (ghk)(ζ) , given by (id ⊗ ∆gx,gx′ ⊗ id⊗3 )(δ(gh, k)−1 )
−1 ) ⊗ id ◦ (id ⊗ ∆gx,gx′ ⊗ id⊗3 )(σgh,k ) = (∆x′ ,gx,gx′ ⊗ ∆ghx′ )(βgh,k
−1 ′ ◦ id ⊗ (∆gx,gx′ ,ghx ⊗ ∆ghkx )(βgh,k ) ◦ (id ⊗ ∆gx,gx′ ,ghx,ghx′ ⊗ id)(σghk ). Similarly, we represent the number d(g, hk) ∈ C∗ by the automorphism of the object ∆x′ ,gx,gx′ ,ghx,ghx′ (aghk ) ⊗ (ghk)(ζ) , given by (id⊗3 ⊗ ∆ghx,ghx′ ⊗ id)(δ(g, hk)) ′ )−1 = (id ⊗ ∆gx,gx′ ,ghx,ghx′ ⊗ id)(σghk ) ◦ id ⊗ (∆gx ⊗ ∆ghx,ghx′ ,ghkx )(βg,hk
−1 ) ◦ (∆x′ ⊗ ∆gx′ ,ghx,ghx′ )(βg,hk ) ⊗ id . ◦ (id⊗3 ⊗ ∆ghx,ghx′ ⊗ id)(σg,hk
A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 31 We finally represent the number d(h, k) ∈ C∗ by the automorphism of the object ∆x′ (ag ) ⊗ ∆gx′ ,ghx,ghx′ g(ahk ) ⊗ (ghk)(ζ) , determined by id⊗2 ⊗ g ⊗5 δ(h, k)

′ )−1 = id⊗2 ⊗ (id ⊗ ∆ghx,ghx′ ⊗ id)g ⊗3 σhk ◦ id⊗3 ⊗ (∆ghx ⊗ ∆ghkx )g ⊗2 (βh,k

−1 ◦ id⊗2 ⊗ g ⊗4 (∆x′ ⊗ ∆hx′ )(βh,k ) ⊗ id . ◦ id⊗2 ⊗ g ⊗5 σh,k With these identifications, we may represent the product d(gh, k)−1 · d(g, h)−1 ∈ C∗ by the automorphism

−1 −1 ) ⊗ id⊗3 ◦ σg,h,k ) ⊗ id ◦ (∆x′ ⊗ ∆gx′ )(βg,h (−1)ε(σk )·ε(βg,h ) · (∆x′ ,gx,gx′ ⊗ ∆ghx′ )(βgh,k

′ ′ ◦ id ⊗ (∆gx ⊗ ∆ghx )(βg,h ) ⊗ id⊗2 ◦ id ⊗ (∆gx,gx′ ,ghx ⊗ ∆ghkx )(βgh,k ) −1 ) ◦ (id ⊗ ∆gx,gx′ ,ghx,ghx′ ⊗ id)(σghk of the object ∆x′ ,gx,gx′ ,ghx,ghx′ (aghk ) ⊗ (ghk)(ζ) . Furthermore, we may represent the product d(g, hk) · d(h, k) ∈ C∗ by the automorphism
(∆x′ ⊗ ∆gx′ ,ghx,ghx′ )(βg,hk ) ⊗ id ◦ (id ⊗ ∆gx,gx′ ,ghx,ghx′ ⊗ id)(σghk )

′ ′ ◦ id ⊗ (∆gx ⊗ ∆ghx,ghx′ ,ghkx )(βg,hk )−1 ◦ (σg−1 ⊗ id⊗4 ) ◦ id⊗3 ⊗ (∆ghx ⊗ ∆ghkx )g ⊗2 (βh,k )−1

−1 ◦ id⊗2 ⊗ g ⊗4 (∆x′ ⊗ ∆hx′ )(βh,k ) ⊗ id ◦ id⊗2 ⊗ g ⊗5 σh,k
′ = (−1)ε(σg )·ε(βh,k ) · (∆x′ ⊗ ∆gx′ ,ghx,ghx′ )(βg,hk ) ⊗ id ◦ (id ⊗ ∆gx,gx′ ,ghx,ghx′ ⊗ id)(σghk )

′ ′ )−1 ◦ id ⊗ (∆gx ⊗ ∆ghx,ghx′ ,ghkx )(βg,hk )−1 ◦ id⊗3 ⊗ (∆ghx ⊗ ∆ghkx )g ⊗2 (βh,k
−1 ◦ σg,h,k ◦ id⊗2 ⊗ g ⊗4 (∆x′ ⊗ ∆hx′ )(βh,k ) ⊗ id of the object ∆x′ (ag ) ⊗ ∆gx′ ,ghx,ghx′ g(ahk ) ⊗ (ghk)(ζ) . ∗ Combining these observations, we may represent the quotient cc(g,h,k) by the automorphism ′ (g,h,k) ∈ C

id⊗2 ⊗ g ⊗4 (∆x′ ⊗ ∆hx′ )(βh,k ) ⊗ id ◦ (∆x′ ⊗ ∆gx′ ,ghx,ghx′ )(βg,hk ) ⊗ id ◦ (−1)ε(σk )·ε(βg,h ) · d(gh, k)−1 d(g, h)−1 · (id ⊗ ∆gx,gx′ ,ghx,ghx′ ⊗ id)(σghk )

−1 ′ ′ , )−1 ◦ σg,h,k ◦ id ⊗ (∆gx ⊗ ∆ghx,ghx′ ,ghkx )(βg,hk )−1 ◦ id⊗3 ⊗ g ⊗4 (∆hx ⊗ ∆hkx )(βh,k acting on the object ∆x′ (ag ) ⊗ ∆gx′ g(ah ) ⊗ ∆ghx′ (gh)(ak ) ⊗ (ghk)(ζ) . But this latter automorphism can now be seen to represent the quotient ′ (−1)ε(σk )·ε(βg,h )+ε(σg )·ε(βh,k ) · d(h, k) · d(g, hk) ∈ C∗ . d(g, h) · d(gh, k) This proves the lemma. Lemma 8.8. Let Φ : H1  / H2 be a morphism of coproduct categories. Let G be a group and let / Aut(Hi ) ρi : G be a group homomorphism for i = 1, 2. Assume that for all g ∈ G, ρ2 (g)Φ = Φρ1 (g). Then
[c1 ] = [c2 ] ∈ H 3 G, C∗ /{±1} . Proof. This is a consequence of Lemma 8.7. Let Xi be the set underlying Hi for i = 1, 2. Choose x ∈ X1 . For each g ∈ G, choose an object ag ∈ H1 (x, gx) and for each g, h ∈ G choose an isomorphism / ag ⊗ g(ah ) . βg,h : ∆gx (agh ) These choices then yield the group 3-cochain c1 : Z[G3 ] / C∗ . 32 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Applying Φ, we obtain x′ := ϕ(x) ∈ X2 . Because Φ is G-equivariant, for each g ∈ G, a′g := ϕ(ag ) ∈ H2 (x′ , gx′ ). Moreover, because Φ is a morphism of coproduct categories, for each g, h ∈ G, we obtain an isomorphism ′ / a′g ⊗ g(a′h ) . βg,h := ϕ(βg,h ) : ∆gx′ (a′gh ) / C∗ . These choices then yield the group 3-cochain c2 : Z[G3 ] For all g, h, k ∈ G, our choices determine automorphisms / ag ⊗ g(ah ) ⊗ gh(ak ) γ(g, h, k): ag ⊗ g(ah ) ⊗ gh(ak ) γ ′ (g, h, k) : a′g ⊗ g(a′h ) ⊗ gh(a′k ) / a′g ⊗ g(a′h ) ⊗ gh(a′k ) as in Definition 8.5. In particular, we have that c1 (g, h, k) · idag ⊗g(ah )⊗gh(ak ) = γ(g, h, k) and c2 (g, h, k) · ida′g ⊗g(a′h )⊗gh(a′k ) = γ ′ (g, h, k) . Further, because Φ is a G-equivariant morphism of coproduct categories, / a′g ⊗ g(a′h ) ⊗ gh(a′k ) . ϕ(γ(g, h, k)) = γ ′ (g, h, k) : a′g ⊗ g(a′h ) ⊗ gh(a′k ) Finally, because Φ is a morphism of categories, it is unital and we obtain that c1 (g, h, k) = c2 (g, h, k). This proves the lemma.  9. Categories associated to representations of a ring Throughout this section we fix a separable Hilbert space H, a unital ring R and a two-sided ideal I in R. We recall that L 1 (H) ⊆ L (H) denotes the ideal of trace class operators inside the bounded operators on the Hilbert space H. The unit in R is denoted by 1. We work under the following assumption: Assumption 9.1. Suppose that we are given a family Rep = {πλ }λ∈Λ , indexed by a non-empty set Λ, of (not necessarily unital) representations of the unital ring R as bounded operators on H satisfying the following conditions: (1) for all x ∈ R and pairs λ, µ ∈ Λ, πλ (x)πµ (1) − πλ (1)πµ (x) ∈ L 1 (H) ; (2) for all i ∈ I and triples λ, µ, ν ∈ Λ, πλ (i)πµ (1)πν (1) − πλ (i)πν (1) ∈ L 1 (H) . Under Assumption 9.1 and given a fixed idempotent p0 ∈ R serving as a “base point”, we shall in this section see how to build two categories Lp and L†p , whenever p ∈ R is an idempotent with p − p0 ∈ I. The category L†p will play the role of a dual to the category Lp . We start by introducing some notation: Notation 9.2. For λ, µ ∈ Λ we define the following bounded operators: (1) For an idempotent p ∈ R, define   πλ (p) − πλ (p)πµ (p)πλ (p) πλ (p)πµ (p) Ω(λ, µ)(p) := 2πµ (p)πλ (p) − (πµ (p)πλ (p))2 πµ (p)πλ (p)πµ (p) − πµ (p) / πλ (p)H ⊕ πµ (p)H . : πλ (p)H ⊕ πµ (p)H We sometimes consider Ω(λ, µ)(p) as a bounded operator on H ⊕ H by putting it equal to zero on the range of the idempotent id − πλ (p) ⊕ πµ (p). (2) For idempotents p, q ∈ R with p − q ∈ I, define

F (λ, µ)(p, q) := πλ (q) ⊕ πµ (p) Ω(λ, µ)(1) πλ (p) ⊕ πµ (q) / πλ (q)H ⊕ πµ (p)H . : πλ (p)H ⊕ πµ (q)H Notation 9.3. For n ≥ 3, a tuple of indices λ = (λ1 , λ2 , . . . , λn ) in Λ and 1 ≤ i < j ≤ n we apply the following notation: A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 33 (1) For every tuple of idempotents p = (p1 , p2 , . . . , pn ) in R with mutual differences in I and with pi = pj := q, we define Ωij (λ)(p) := Ω(λi , λj )(q) + X πλk (pk ) : k6=i,j n M πλk (pk )H k=1 / n M πλk (pk )H . k=1 (2) For every tuple of idempotents p = (p1 , p2 , . . . , pn ) in R with mutual differences in I, we define F ij (λ)(p) := F (λi , λj )(pi , pj ) + X k6=i,j where τ : {1, 2, . . . , n} πλk (pk ) : n M πλk (pk )H / k=1 n M πλk (pτ (k) )H , k=1 / {1, 2, . . . , n} is the transposition which interchanges i and j. In the following lemmas we present a few consequences of our assumptions, in particular we shall see that / πλ (q)H ⊕ πµ (p)H are in fact Fredholm operators. the bounded operators F (λ, µ)(p, q) : πλ (p)H ⊕ πµ (q)H Lemma 9.4. Suppose that λ, µ ∈ Λ, and that x, y ∈ R are two elements with x − y ∈ I. Then the difference

πλ (y) ⊕ πµ (x) Ω(λ, µ)(1) − Ω(λ, µ)(1) πλ (x) ⊕ πµ (y) is a trace class operator on H ⊕ H. Proof. We compute that

πλ (y) ⊕ πµ (x) Ω(λ, µ)(1) − Ω(λ, µ)(1) πλ (x) ⊕ πµ (y)   πλ (y) − πλ (y)πµ (1)πλ (1) πλ (y)πµ (1) = 2πµ (x)πλ (1) − πµ (x)πλ (1)πµ (1)πλ (1) πµ (x)πλ (1)πµ (1) − πµ (x)   πλ (x) − πλ (1)πµ (1)πλ (x) πλ (1)πµ (y) − , 2πµ (1)πλ (x) − πµ (1)πλ (1)πµ (1)πλ (x) πµ (1)πλ (1)πµ (y) − πµ (y) and notice that the off diagonal terms lie in L 1 (H) by Assumption 9.1 (1) and the diagonal terms lie in L 1 (H) by Assumption 9.1 (1) and (2).  Lemma 9.5. Suppose that λ, µ ∈ Λ, and that p ∈ R is an idempotent. Then F (λ, µ)(p, p) − Ω(λ, µ)(p) ∈ L 1 (πλ (p)H ⊕ πµ (p)H) and
2 Ω(λ, µ)(p) = πλ (p) ⊕ πµ (p) , hence Ω(λ, µ)(p) is invertible on the range of the idempotent πλ (p) ⊕ πµ (p). Proof. Using Assumption 9.1 and the fact that p ∈ R is an idempotent, we see that the difference πλ (p)πµ (1)− πλ (p)πµ (p) is of trace class (and similarly for λ and µ reversed). The first statement of the lemma now follows by noticing that   πλ (p) − πλ (p)πµ (1)πλ (p) πλ (p)πµ (p) F (λ, µ)(p, p) = . 2πµ (p)πλ (p) − πµ (p)πλ (1)πµ (1)πλ (p) πµ (p)πλ (1)πµ (p) − πµ (p) The second statement can be proved by setting T := πλ (p)πµ (p) and T † := πµ (p)πλ (p) and verifying that   πλ (p) − T T † T Ω(λ, µ)(p) = 2T † − T † T T † T † T − πµ (p)       πλ (p) 0 πλ (p) T πλ (p) 0 = · · T† −πµ (p) 0 πµ (p) −T † πµ (p)       πλ (p) 0 πλ (p) −T πλ (p) 0 = · · .  T† πµ (p) 0 πµ (p) T† −πµ (p) One may also consider the case where λ = µ ∈ Λ, and p, q ∈ R are two idempotents with p − q ∈ I. Then   0 πλ (1) Ω(λ, λ)(1) = πλ (1) 0 34 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON and the bounded operator F (λ, λ)(p, q) = is therefore invertible.  0 πλ (q) πλ (p) 0  : πλ (p)H ⊕ πλ (q)H / πλ (q)H ⊕ πλ (p)H Lemma 9.6. Suppose that λ, µ ∈ Λ, and that p, q ∈ R are idempotents with p − q ∈ I. Then / πλ (q)H ⊕ πµ (p)H F (λ, µ)(p, q) : πλ (p)H ⊕ πµ (q)H is a Fredholm operator with parametrix F (λ, µ)(q, p) : πλ (q)H ⊕ πµ (p)H have that (9.1) / πλ (p)H ⊕ πµ (q)H. In fact we F (λ, µ)(q, p)F (λ, µ)(p, q) − πλ (p) ⊕ πµ (q) ∈ L 1 (πλ (p)H ⊕ πµ (q)H) . Proof. Since we may interchange the roles of the idempotents p and q, it suffices by Atkinson’s theorem to prove the inclusion in (9.1). But this inclusion follows in a straightforward way from Lemma 9.4 and Lemma 9.5.  9.1. The category Lp and its dual L†p . We continue working under Assumption 9.1, and on top of that we now fix an idempotent p0 ∈ R which will play the role of a base point. We shall moreover fix an idempotent p ∈ R with p − p0 ∈ I. We construct a category of Z-graded complex lines Lp as follows (see Definition 8.1): (1) the non-empty set of objects of Lp is the index set Λ for the family of representations Rep = {πλ }λ∈Λ , (2) the morphisms from λ to µ are the elements in the graded determinant line of the Fredholm operator F (λ, µ)(p, p0 ), thus Lp (λ, µ) := F (λ, µ)(p, p0 ) , / λ is the element (3) for each λ ∈ Λ, the unit idλ : λ   0 πλ (p0 ) 1 ∈ (C, 0) = = Lp (λ, λ) . πλ (p) 0 The composition of morphisms will be given below in Definition 9.10. We also construct a category of Z-graded complex lines L†p having the same objects as Lp and with morphisms from λ to µ given by the graded determinant line L†p (λ, µ) := F (λ, µ)(p0 , p) . The unit idλ : λ / λ is the multiplicative unit in C,  0 1 ∈ (C, 0) = πλ (p0 ) πλ (p) 0  = L†p (λ, λ) . The composition of morphisms in the category L†p is given below in Definition 9.11. For each pair of elements (λ, µ) in the index set Λ, there are isomorphisms of Z-graded lines (C, 0) ϕ / L†p (λ, µ) ⊗ Lp (λ, µ) Lp (λ, µ) ⊗ L†p (λ, µ) and ψ / (C, 0) , defined as follows: Using Lemma 9.6 we define ϕ as the composition (9.2) ϕ : (C, 0) = |πλ (p0 ) ⊕ πµ (p)| P / |F (λ, µ)(p, p0 ) · F (λ, µ)(p0 , p)| T−1 / |F (λ, µ)(p0 , p)| ⊗ |F (λ, µ)(p, p0 )| , and similarly, we define ψ as the composition (9.3) ψ : |F (λ, µ)(p, p0 )| ⊗ |F (λ, µ)(p0 , p)| T P / |F (λ, µ)(p0 , p) · F (λ, µ)(p, p0 )| / |πλ (p) ⊕ πµ (p0 )| = (C, 0) . / L†p (λ, µ) ⊗ Lp (λ, µ) and ψ(λ, µ) : Lp (λ, µ) ⊗ L†p (λ, µ) We shall sometimes write ϕ(λ, µ) : (C, 0) instead of ϕ and ψ when we want to be more specific about the objects involved. / (C, 0) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 35 Lemma 9.7. Let (λ, µ) be a pair of elements in Λ. It holds that L†p (λ, µ) is right dual to Lp (λ, µ) in the sense that the following two diagrams Lp (λ, µ) id⊗ϕ / Lp (λ, µ) ⊗ L†p (λ, µ) ⊗ Lp (λ, µ) ❙❙❙❙ ❙❙❙❙ ❙❙❙❙ ψ⊗id ❙❙❙❙ id  ) Lp (λ, µ) and L†p (λ, µ) ϕ⊗id / L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ L†p (λ, µ) ❙❙❙ ❙❙❙❙ ❙❙❙ id⊗ψ ❙❙❙❙ id ❙❙)  L†p (λ, µ) commute. Proof. We only verify that (ψ ⊗ id) ◦ (id ⊗ ϕ) = id; the proof in the remaining case is similar. We suppress the pair (λ, µ) from the notation and consider the following diagram id⊗T−1 id⊗P / |F (p, p0 )| ⊗ |F (p0 , p)| ⊗ |F (p, p0 )| / |F (p, p0 )| ⊗ |F (p, p0 ) · F (p0 , p)| |F (p, p0 )| ❱ ❱❱❱❱ ❱❱❱❱ ❱❱❱❱ T⊗id T ❱❱❱❱ P ❱❱❱+   −1 T / |F (p0 , p) · F (p, p0 )| ⊗ |F (p, p0 )| |F (p, p0 ) · F (p0 , p) · F (p, p0 )| ❨❨❨❨❨❨ ❨❨❨❨❨❨ ❨❨❨❨❨❨ P⊗id ❨❨❨❨❨❨ P ❨❨❨❨,  |F (p, p0 )| The upper and lower triangles commute because torsion commutes with perturbation, see Theorem 4.9. The square commutes by the associativity of torsion, see Proposition 3.3. The composition of the diagonal maps is equal to the identity on Lp (λ, µ) by the associativity of perturbation, see Theorem 4.1 (1). The composition of the upper horizontal maps with the right vertical maps is (ψ ⊗ id) ◦ (id ⊗ ϕ) by definition.  9.2. The composition of morphisms. We continue working under Assumption 9.1 and we will moreover fix two idempotents p and p0 ∈ R with p − p0 ∈ I. The idempotent p0 will serve as a base point, whereas the idempotent p labels the tentative categories of Z-graded complex lines Lp and L†p . For the remainder of this section, we define the composition of morphisms in Lp and L†p , see Definition 9.10 and Definition 9.11, and state the required properties. The main proofs are given in Section 15. Throughout this subsection (λ, µ, ν) is a fixed triple of elements in the index set Λ and this triple will therefore often be suppressed from the notation. We define the Fredholm operators F 12 := F (λ, µ)(p, p0 ) + πν (p0 ) : πλ (p)H ⊕ πµ (p0 )H ⊕ πν (p0 )H / πλ (p0 )H ⊕ πµ (p)H ⊕ πν (p0 )H F†12 := F (λ, µ)(p0 , p) + πν (p0 ) : πλ (p0 )H ⊕ πµ (p)H ⊕ πν (p0 )H / πλ (p)H ⊕ πµ (p0 )H ⊕ πν (p0 )H and similarly for the remaining cases, F 23 := F (µ, ν)(p, p0 ) + πλ (p0 ), F†23 := F (µ, ν)(p0 , p) + πλ (p0 ) and F 13 := F (λ, ν)(p, p0 ) + πµ (p0 ), F†13 := F (λ, ν)(p0 , p) + πµ (p0 ). Comparing with Notation 9.3, we thus have that F 12 = F 12 (λ, µ, ν)(p, p0 , p0 ), F†12 = F 12 (λ, µ, ν)(p0 , p, p0 ) and so on. Lemma 9.8. For any i ∈ I, it holds that

(πλ (i) ⊕ 0 ⊕ 0) Ω13 (1)Ω23 (1)Ω12 (1) − πλ (1) ⊕ πµ (1) ⊕ πν (1) ∈ L 1 πλ (1)H ⊕ πµ (1)H ⊕ πν (1)H . Proof. We compute modulo the trace ideal inside the bounded operators, using the notation ∼1 for the corresponding equivalence relation. Recalling the definitions from Notation 9.2 and Notation 9.3 and applying 36 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Assumption 9.1, we obtain that The result of  0  0 0  0 0 πλ (1)πν (1) 0 (πλ (i) ⊕ 0 ⊕ 0) · Ω13 (1) ∼1  0 0 0 0 0  0 0 0 (0 ⊕ 0 ⊕ πν (i)) · Ω23 (1) ∼1  0 0 0 0 πν (1)πµ (1) 0  0 0 0 (0 ⊕ πµ (i) ⊕ 0) · Ω12 (1) ∼1  πµ (1)πλ (1) 0 0 0 0 0   · (0 ⊕ 0 ⊕ πν (i))   · (0 ⊕ πµ (i) ⊕ 0) and   · (πλ (i) ⊕ 0 ⊕ 0) . the lemma then follows from the computation     0 πλ (1)πν (1) 0 0 0 0 0 · 0 0 0 0 0  ·  πµ (1)πλ (1) 0 0 0 0 πν (1)πµ (1) 0 0 0 = πλ (1)πν (1)πµ (1)πλ (i) ⊕ 0 ⊕ 0 ∼1 πλ (i) ⊕ 0 ⊕ 0 .  0 0  · (πλ (i) ⊕ 0 ⊕ 0) 0  Lemma 9.9. The following two bounded operators on πλ (1)H ⊕ πµ (1)H ⊕ πν (1)H, F†13 F 23 F 12 + πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) 13 23 and 12 Ω (p0 )Ω (p0 )Ω (p0 ) + πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) , agree modulo the trace ideal L 1 (πλ (1)H ⊕ πµ (1)H ⊕ πν (1)H). Proof. According to Notation 9.3 (and the conventions explained in the beginning of this subsection), we clarify that the composition of Fredholm operators F†13 F 23 F 12 makes sense and yields a bounded endomorphism of the Hilbert space πλ (p)H ⊕ πµ (p0 )H ⊕ πν (p0 )H. Similarly, we record that Ω13 (p0 )Ω23 (p0 )Ω12 (p0 ) yields an invertible bounded endomorphism of the Hilbert space πλ (p0 )H ⊕ πµ (p0 )H ⊕ πν (p0 )H. Using Lemma 9.4, Lemma 9.5 and Lemma 9.8, we obtain the result from the following computation
modulo L 1 πλ (1)H ⊕ πµ (1)H ⊕ πν (1)H : F†13 F 23 F 12 − πλ (p) ⊕ πµ (p0 ) ⊕ πν (p0 ) 
 ∼1 πλ (p) ⊕ πµ (p0 ) ⊕ πν (p0 ) · Ω13 (1)Ω23 (1)Ω12 (1) − πλ (1) ⊕ πµ (1) ⊕ πν (1) 
 ∼1 πλ (p0 ) ⊕ πµ (p0 ) ⊕ πν (p0 ) · Ω13 (1)Ω23 (1)Ω12 (1) − πλ (1) ⊕ πµ (1) ⊕ πν (1) ∼1 Ω13 (p0 )Ω23 (p0 )Ω12 (p0 ) − πλ (p0 ) ⊕ πµ (p0 ) ⊕ πν (p0 ) . Using Lemma 9.9, we may define an isomorphism of Z-graded lines: µp : F (λ, µ)(p, p0 ) ⊗ F (µ, ν)(p, p0 ) ⊗ F (λ, ν)(p0 , p) (9.4) S S / F 12 ⊗ F 23 ⊗ F 13 † T / F 13 F 23 F 12 † P / Ω13 (p0 ) · Ω23 (p0 ) · Ω12 (p0 ) + πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) = (C, 0) . / F 13 F 23 F 12 + πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) † We use the above trivialisation to define the composition of morphisms in Lp : Definition 9.10. For λ, µ, ν ∈ Λ the isomorphism of Z-graded lines / Lp (λ, ν) Mp : Lp (λ, µ) ⊗ Lp (µ, ν) is defined as the composition F (λ, µ)(p, p0 ) ⊗ F (µ, ν)(p, p0 ) id⊗id⊗ϕ µp ⊗id / F (λ, µ)(p, p0 ) ⊗ F (µ, ν)(p, p0 ) ⊗ F (λ, ν)(p0 , p) ⊗ F (λ, ν)(p, p0 ) / F (λ, ν)(p, p0 ) .  A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 37 Using Lemma 9.9 together with Lemma 9.5 and Lemma 9.6, we may also define the following isomorphism of Z-graded lines: µ†p : F (λ, ν)(p, p0 ) ⊗ F (µ, ν)(p0 , p) ⊗ F (λ, µ)(p0 , p) (9.5) S / F 13 ⊗ F†23 ⊗ F†12 T / F†12 F†23 F 13 P / Ω12 (p0 ) · Ω23 (p0 ) · Ω13 (p0 ) + πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) = (C, 0) . S / F†12 F†23 F 13 + πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) We now define the composition of morphisms in L†p : Definition 9.11. For λ, µ, ν ∈ Λ the isomorphism of Z-graded complex lines / L†p (λ, ν) M†p : L†p (µ, ν) ⊗ L†p (λ, µ) is defined as the composition F (µ, ν)(p0 , p) ⊗ F (λ, µ)(p0 , p) ϕ⊗id⊗id id⊗µ†p / F (λ, ν)(p0 , p) ⊗ F (λ, ν)(p, p0 ) ⊗ F (µ, ν)(p0 , p) ⊗ F (λ, µ)(p0 , p) / F (λ, ν)(p0 , p) . The next result explains the relationship between the compositions in the Z-graded categories Lp and L†p and we present a full proof later on in Section 15. Proposition 9.12. For each λ, µ, ν ∈ Λ, the following diagram of isomorphisms of Z-graded lines is commutative: id⊗ψ⊗id / Lp (λ, µ) ⊗ L†p (λ, µ) Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (µ, ν) ⊗ L†p (λ, µ) Mp ⊗M†p  Lp (λ, ν) ⊗ L†p (λ, ν) ψ ψ  / (C, 0) The next two theorems, which we also prove in Section 15, show that Lp is indeed a category of Z-graded complex lines (see Definition 8.1). Similar theorems can be stated and proved for L†p , in fact these unitality and associativity theorems can be verified using the duality relation in Proposition 9.12 together with the corresponding theorems for Lp . Theorem 9.13. Suppose that the conditions in Assumption 9.1 are satisfied and that p0 ∈ R is a fixed idempotent. For any idempotent p ∈ R with p − p0 ∈ I, the pair (Lp , Mp ) satisfies the unitality condition, i.e. the following diagrams are commutative: id⊗idµ / Lp (λ, µ) ⊗ Lp (µ, µ) Lp (λ, µ) ❚ ❚❚❚❚ ❚❚❚❚ ❚❚❚❚ Mp ❚❚❚❚ id  * Lp (λ, µ) idλ ⊗id / Lp (λ, λ) ⊗ Lp (λ, µ) Lp (λ, µ) ❚ ❚❚❚❚ ❚❚❚❚ ❚❚❚❚ Mp ❚❚❚❚ id  * Lp (λ, µ) Theorem 9.14. Suppose that the conditions in Assumption 9.1 are satisfied and that p0 ∈ R is a fixed idempotent. For any idempotent p ∈ R with p − p0 ∈ I, the pair (Lp , Mp ) satisfies the associativity condition, i.e. the following diagram is commutative: Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ Lp (ν, τ ) ❱❱❱❱ ✐✐ ❱❱❱id⊗M ❱❱❱❱ p ✐ ❱❱❱❱ ✐✐✐ ✐ ✐ ✐ ❱❱* t✐✐✐ Lp (λ, ν) ⊗ Lp (ν, τ ) Lp (λ, µ) ⊗ Lp (µ, τ ) ❯❯❯❯ ❯❯❯❯ ✐✐✐✐ ❯❯❯❯ ✐✐✐✐ ✐ ✐ ✐ ❯❯❯❯ Mp ✐✐✐✐ Mp ❯❯❯* t✐✐✐✐ Lp (λ, τ ) Mp ⊗id✐✐✐✐✐ 38 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON 10. The coproduct category associated to representations of a ring Throughout this section we fix a unital ring R and an ideal I in R. We assume that Rep = {πλ }λ∈Λ is a family of (not necessarily unital) algebra homomorphisms from R to the bounded operators on a fixed separable Hilbert space H. This family of representations is assumed to satisfy Assumption 9.1 and we fix an idempotent p0 in the unital ring R. To ease the notation, we let Idem(R) ⊆ R denote the subset of idempotent elements in R. We refer back to Section 9 for the definition of the categories of Z-graded complex lines Lp and L†q appearing here below. The generalities needed regarding graded tensor products and coproduct categories can be found in Subsection 8.1. Definition 10.1. We define a coproduct category (H, ∆, X) as follows:  (1) The underlying set X is given by X := p ∈ Idem(R) | p − p0 ∈ I . (2) For each p, q ∈ X we define the category of Z-graded complex lines H(p, q) to be the full subcategory of the graded tensor product of Lp and L†q on the subset of objects  DΛ := λ ⊗ λ | λ ∈ Λ ⊆ Obj(Lp ⊗ L†q ) . Thus, we define H(p, q) := Lp ⊗ L†q
DΛ , where (−)|DΛ denotes full subcategory on DΛ . We identify the set of objects in H(p, q) with Λ via / DΛ given by λ 7→ λ ⊗ λ. the isomorphism Λ (3) Given p, e, q ∈ X, the coproduct ∆e : H(p, q) / H(p, e) ⊗ H(e, q) is given by λ 7→ λ ⊗ λ on objects and on morphisms it is determined by the isomorphism Lp (λ, µ) ⊗ L†q (λ, µ) id⊗ϕ⊗id / Lp (λ, µ) ⊗ L†e (λ, µ) ⊗ Le (λ, µ) ⊗ L†q (λ, µ) , of Z-graded complex lines, where we recall that the isomorphism ϕ : (C, 0) introduced in Equation (9.2). / L†e (λ, µ) ⊗ Le (λ, µ) was We refer to the fixed idempotent p0 ∈ R as the base point of the coproduct category H. We remark that our tentative coproduct category H depends on the choice of the idempotent p0 ∈ R. The dependency on this choice of base point will be examined in detail in Section 11. We now prove that Definition 10.1 yields a coproduct category in the sense of Subsection 8.1. For later reference, we state this result as a theorem and note that the proof is a consequence of Lemma 10.3, Lemma 10.4 and Proposition 10.5. Theorem 10.2. The assignments in Definition 10.1 form a coproduct category H. We start by showing that H(p, q) is indeed a category for every p, q ∈ X: Lemma 10.3. Let p, q ∈ X. Then H(p, q) is a category of Z-graded complex lines. Proof. The fact that H(p, q) is a category of Z-graded complex lines follows from Theorem 9.13 and Theorem 9.14 together with the corresponding theorems for the dual category L†q . See also the generalities regarding graded tensor products from Subsection 8.1.  For p, q ∈ X and λ, µ, ν ∈ Λ we let Mp,q : H(p, q)(λ, µ) ⊗ H(p, q)(µ, ν) / H(p, q)(λ, ν) denote the isomorphism of Z-graded complex lines coming from the composition in H(p, q). The next lemma has a straightforward proof, which we therefore omit. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 39 Lemma 10.4. The coproduct operation ∆ of Definition 10.1 is coassociative in the sense that / H(p, e) ⊗ H(e, f ) ⊗ H(f, q) , (id ⊗ ∆f )∆e = (∆f ⊗ id)∆e : H(p, q) whenever p, e, f, q ∈ X. The proof that the various coproducts are functorial requires more care. We state the result here, but the proof will occupy the remainder of this section. Proposition 10.5. The coproduct ∆ of Definition 10.1 is functorial in the sense that / H(p, e) ⊗ H(e, q) ∆e : H(p, q) is a linear functor, whenever p, e, q ∈ X. To prove Proposition 10.5, we need the following lemmas. Lemma 10.6. Let λ, µ, ν ∈ Λ and p ∈ X be given. Then the trivialisation (10.1) µ†p ⊗ µp : Lp (λ, ν) ⊗ L†p (µ, ν) ⊗ L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) / (C, 0) agrees with the trivialisation (10.2) ψ ◦ (id ⊗ ϕ−1 ⊗ id) ◦ (id⊗2 ⊗ ϕ−1 ⊗ id⊗2 ) : Lp (λ, ν) ⊗ L†p (µ, ν) ⊗ L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) / (C, 0) , where the trivialisations µp and µ†p are defined in Equation (9.4) and Equation (9.5), respectively. Proof. To ease the notation in this proof we suppress the idempotents p and p0 and put F (λ, µ) := F (λ, µ)(p, p0 ) and F† (λ, µ) := F (λ, µ)(p0 , p) . We are sometimes also suppressing the tuple of indices (λ, µ, ν) and hence applying the notation from the beginning of Subsection 9.2. By the definition of ϕ and ψ from Equation (9.2) and (9.3), the isomorphism in Equation (10.2) is given by the composition |F (λ, ν)| ⊗ |F† (µ, ν)| ⊗ |F† (λ, µ)| ⊗ |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| id⊗2 ⊗(P◦T)⊗id⊗2 id⊗(P◦T)⊗id / |F (λ, ν)| ⊗ |F† (µ, ν)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| / |F (λ, ν)| ⊗ |F† (λ, ν)| P◦T / |πλ (p) ⊕ πν (p0 )| = (C, 0) . Without altering this isomorphism, we could have stabilised beforehand (see Proposition 5.2 and Proposition 5.3), thus achieving the following alternative description of the above isomorphism: |F (λ, ν)| ⊗ |F† (µ, ν)| ⊗ |F† (λ, µ)| ⊗ |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| S / |F 13 | ⊗ |F 23 | ⊗ |F 12 | ⊗ |F 12 | ⊗ |F 23 | ⊗ |F 13 | † id ⊗2 ⊗(P◦T)⊗id ⊗2 † † / |F 13 | ⊗ |F 23 | ⊗ |F 23 | ⊗ |F 13 | † id⊗(P◦T)⊗id / |F 13 | ⊗ |F 13 | † † P◦T / |πλ (p) ⊕ πµ (p0 ) ⊕ πν (p0 )| = (C, 0) . Gathering all the torsion isomorphisms and all the perturbation isomorphisms (using Theorem 4.9) and using the associativity of these operations (Theorem 2.8 and Theorem 4.1), we then obtain the formula |F (λ, ν)| ⊗ |F† (µ, ν)| ⊗ |F† (λ, µ)| ⊗ |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| (T◦S)⊗(T◦S) T P / F 12 · F 23 · F 13 ⊗ F 13 · F 23 · F 12 † † † / F 13 · F 23 · F 12 · F 12 · F 23 · F 13 † † † / |πλ (p) ⊕ πµ (p0 ) ⊕ πν (p0 )| = (C, 0) , 40 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON for the isomorphism in Equation (10.2). However, using that stabilisation commutes with torsion and perturbation (Proposition 5.2 and Proposition 5.3), and that torsion commutes with perturbation (Theorem 4.9) we now see that the isomorphism in Equation (10.2) agrees with the isomorphism |F (λ, ν)| ⊗ |F† (µ, ν)| ⊗ |F† (λ, µ)| ⊗ |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| (T◦S)⊗(T◦S) / |F 12 · F 23 · F 13 | ⊗ |F 13 · F 23 · F 12 | † S⊗S † † / |F 12 · F 23 · F 13 + πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πµ (1 − p0 )| † † ⊗ |F†13 · F 23 · F 12 + πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πµ (1 − p0 )| P⊗P / Ω12 (p0 ) · Ω23 (p0 ) · Ω13 (p0 ) + (πλ ⊕ πµ ⊕ πν )(1 − p0 ) ⊗ Ω13 (p0 ) · Ω23 (p0 ) · Ω12 (p0 ) + (πλ ⊕ πµ ⊕ πν )(1 − p0 ) T / (πλ ⊕ πµ ⊕ πν )(1) = (C, 0) . But this is exactly the isomorphism in Equation (10.1) and the lemma is therefore proved. Before stating the next lemma, we recall from Notation 2.2 that ǫ : (L , n) ⊗ (M , m) denotes the commutativity constraint in the context of Z-graded complex lines.  / (M , m) ⊗ (L , n) Lemma 10.7. Let λ, µ, ν ∈ Λ and let p ∈ X. Then the composition L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ L†p (µ, ν) ⊗ Lp (µ, ν) (10.3) M†p ⊗Mp ǫ / L† (µ, ν) ⊗ L† (λ, µ) ⊗ Lp (λ, µ) ⊗ Lp (µ, ν) p p / L†p (λ, ν) ⊗ Lp (λ, ν) agrees with the composition L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ L†p (µ, ν) ⊗ Lp (µ, ν) ϕ−1 ⊗ϕ−1 / (C, 0) ϕ / L†p (λ, ν) ⊗ Lp (λ, ν) . Proof. By Lemma 10.6 and the definition of the isomorphisms M†p and Mp , the isomorphism in (10.3) agrees with the isomorphism L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ L†p (µ, ν) ⊗ Lp (µ, ν) ǫ / L†p (µ, ν) ⊗ L†p (λ, µ) ⊗ Lp (λ, µ) ⊗ Lp (µ, ν) ϕ⊗id⊗4 ⊗ϕ id⊗ / L† (λ, ν) ⊗ Lp (λ, ν) ⊗ L† (µ, ν) ⊗ L† (λ, µ) ⊗ Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L† (λ, ν) ⊗ Lp (λ, ν) p p p p
−1 ⊗2 −1 ⊗2 ψ◦(id⊗ϕ ⊗id)◦(id ⊗ϕ ⊗id ) ⊗id / L†p (λ, ν) ⊗ Lp (λ, ν) . Since the sign related to the above commutativity constraint is equal to one, we obtain the result of the lemma by an application of the duality relation from Lemma 9.7.  / H(p, e) ⊗ Proof of Proposition 10.5. Let p, q and e be elements in X. We must show that ∆e : H(p, q) H(e, q) is unital and respects the composition of morphisms. For λ ∈ Λ, one may verify that the unitality condition follows from the identity     0 πλ (e) 0 πλ (p0 ) † ϕ(λ, λ)(1) = idλ ⊗ idλ ∈ Le (λ, λ) ⊗ Le (λ, λ) = ⊗ , πλ (p0 ) 0 πλ (e) 0 / L† (λ, λ)⊗ Le (λ, λ) (see (9.2)). which in turn is a straightforward consequence of the definition of ϕ : (C, 0) e To show that ∆e respects the composition of morphisms, we let λ, µ and ν be indices in Λ. We need to prove that the following diagram commutes (10.4)

∆e ⊗∆e / H(p, e)(λ, µ) ⊗ H(e, q)(λ, µ) ⊗ H(p, e)(µ, ν) ⊗ H(e, q)(µ, ν) H(p, q)(λ, µ) ⊗ H(p, q)(µ, ν) Mp,q  H(p, q)(λ, ν) Mp,e,q ∆e  / H(p, e)(λ, ν) ⊗ H(e, q)(λ, ν) , A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 41 where the vertical arrow to the right is the isomorphism of Z-graded complex lines induced by the composition in the graded tensor product H(p, e) ⊗ H(e, q). Let sp ∈ Lp (λ, µ), sq ∈ L†q (λ, µ) and tp ∈ Lp (µ, ν), tq ∈ L†q (µ, ν) be given and let np , nq , mp and mq ∈ Z denote their respective degrees. By definition, the evaluation of the left-hand side of the diagram in (10.4) on the morphism sp ⊗ sq ⊗ tp ⊗ tq ∈ H(p, q)(λ, µ) ⊗ H(p, q)(µ, ν) is given by
∆e ◦ Mp,q ) (sp ⊗ sq ) ⊗ (tp ⊗ tq ) = (−1)nq ·(mp +mq ) · Mp (sp ⊗ tp ) ⊗ ϕ(λ, ν)(1) ⊗ M†q (tq ⊗ sq ) . On the other hand, let us write ϕ(λ, µ)(1) = ϕ− (λ, µ) ⊗ ϕ+ (λ, µ) ∈ L†e (λ, µ) ⊗ Le (λ, µ) ϕ(µ, ν)(1) = ϕ− (µ, ν) ⊗ ϕ+ (µ, ν) ∈ L†e (λ, µ) and ⊗ Le (λ, µ) . The evaluation of the right-hand side of (10.4) on the same morphism is then given by

Mp,e,q ◦ (∆e ⊗ ∆e ) (sp ⊗ sq ) ⊗ (tp ⊗ tq ) = (−1)nq ·(mp +mq ) · Mp (sp ⊗ tp ) ⊗ M†e (ϕ− (µ, ν) ⊗ ϕ− (λ, µ)) ⊗ Me (ϕ+ (λ, µ) ⊗ ϕ+ (µ, ν)) ⊗ M†q (tq ⊗ sq ) . But this proves the proposition, since Lemma 10.7 shows that M†e (ϕ− (µ, ν) ⊗ ϕ− (λ, µ)) ⊗ Me (ϕ+ (λ, µ) ⊗ ϕ+ (µ, ν)) = ϕ(λ, ν)(1) .  11. Change of base point Throughout this section we fix a separable Hilbert space H, a unital ring R and a two-sided ideal I in R. We assume that Rep = {πλ }λ∈Λ is a family of (not necessarily unital) representations of R as bounded operators on H, satisfying Assumption 9.1 with respect to the ideal I ⊆ R. Consider two idempotents p0 , p′0 ∈ R with p0 − p′0 ∈ I. In this section, we define a linear isomorphism of coproduct categories / H′ , B(p0 , p′0 ) : H where H and H′ are the coproduct categories constructed using the base points p0 and p′0 , respectively (see Definition 10.1). We refer to this isomorphism as the change-of-base-point isomorphism. The isomorphism B(p0 , p′0 ) is the identity on the set X = {p ∈ Idem(R) | p − p0 ∈ I} = {p ∈ Idem(R) | p − p′0 ∈ I} = X ′ . Given p, q ∈ X = X ′ , we define the linear functor B(p0 , p′0 ) : H(p, q) / H′ (p, q) to be the identity map on objects (thus elements in the index set Λ), and to be given on morphisms by an isomorphism / H′ (p, q)(λ, µ) B(p0 , p′0 ) : H(p, q)(λ, µ) λ, µ ∈ Λ of Z-graded complex lines, see Definition 11.2. From now on, we fix the two indices λ, µ ∈ Λ and the two idempotents p, q ∈ X = X ′ . We often suppress the tuples of indices (λ, µ) and (λ, µ, µ) and refer the reader to Notation 9.3 for an explanation of the notation applied. Lemma 11.1. The difference of the two Fredholm operators  0 0 0 F 12 (p0 , q, p)F 13 (p, q, p0 ) +  0 0 πµ (1 − p0 ) 0 0 0  0 0 0 F 12 (p′0 , q, p)F 13 (p, q, p′0 ) +  0 0 πµ (1 − p′0 ) 0 0 0   and  , both acting from the Hilbert space πλ (p)H⊕πµ (q)H⊕πµ (1)H to the Hilbert space πλ (q)H⊕πµ (1)H⊕πµ (p)H, is of trace class. 42 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Proof. Using Lemma 9.4, we compute modulo trace class operators: F 12 (p0 , q, p)F 13 (p, q, p0 ) ∼1 (πλ (q) ⊕ πµ (p0 ) ⊕ πµ (p)) · Ω12 (1)Ω13 (1) . Since a similar computation holds with p0 replaced by p′0 , it suffices to show that the difference   0 0 0 (0 ⊕ πµ (p0 − p′0 ) ⊕ 0) · Ω12 (1)Ω13 (1) −  0 0 πµ (p0 − p′0 )  0 0 0 is of trace class. But this follows from Assumption 9.1 since p0 − p′0 ∈ I. Indeed, as in Lemma 9.8 we see that (0 ⊕ πµ (p0 − p′0 ) ⊕ 0) · Ω12 (1)Ω13 (1)    0 0 0 0 0 πλ (1)πµ (1)  (0 ⊕ 0 ⊕ πµ (p0 − p′0 )) 0 ∼1  πµ (1)πλ (1) 0 0   0 0 0 0 0 0 0 0   0 0 0  ∼1  0 0 πµ (p0 − p′0 )  . 0 0 0 It follows from the above Lemma 11.1 that the perturbation isomorphism appearing in our definition of the change-of-base-point isomorphism makes sense: Definition 11.2. For λ, µ ∈ Λ and idempotents p, q ∈ X = X ′ , the isomorphism of Z-graded complex lines / H′ (p, q)(λ, µ) B(p0 , p′ ) : H(p, q)(λ, µ) 0 is defined as the composition |F (p, p0 )| ⊗ |F (p0 , q)| S−1 ◦P◦S T◦S / F 12 (p0 , q, p) · F 13 (p, q, p0 ) / F 12 (p′0 , q, p) · F 13 (p, q, p′0 ) (T◦S)−1 / |F (p, p′0 )| ⊗ |F (p′0 , q)| . Applying the cocycle property of the perturbation isomorphism from Theorem 4.1, we obtain that the change-of-base-point isomorphism satisfies the following properties: Lemma 11.3. The change-of-base-point isomorphism is reflexive, skew-symmetric and transitive in the sense that / H(p, q); (1) B(p0 , p0 ) = id : H(p, q) ′ ′ −1 / H′ (p, q); (2) B(p0 , p0 ) = B(p0 , p0 ) : H(p, q) ′ ′′ ′ ′′ / H′′ (p, q), (3) B(p0 , p0 ) ◦ B(p0 , p0 ) = B(p0 , p0 ) : H(p, q) ′ ′′ whenever p0 , p0 , p0 are idempotents in R which agree modulo the ideal I ⊆ R. It is moreover not hard to see that the change-of-base-point satisfies the unitality condition, thus that it sends units in the category H(p, q) to the corresponding units in the category H′ (p, q) for all p, q ∈ X. Recall in this respect that the unit morphism in H(p, q)(λ, λ) is defined by the unit 1 ∈ C under the isomorphism     0 πλ (p0 ) 0 πλ (q) ∼ (C, 0) = ⊗ . πλ (p) 0 πλ (p0 ) 0 The proofs of the following two results are more involved, and will be given in Subsection 16.1 and Subsection 16.5. Proposition 11.4. The change-of-base-point isomorphism commutes with the coproducts. Thus, for each λ, µ ∈ Λ and each p, e, q ∈ X = X ′ , the following diagram is commutative: H(p, q)(λ, µ) ∆e B(p0 ,p′0 )  H′ (p, q)(λ, µ) / H(p, e)(λ, µ) ⊗ H(e, q)(λ, µ) B(p0 ,p′0 )⊗B(p0 ,p′0 ) ∆′e  / H′ (p, e)(λ, µ) ⊗ H′ (e, q)(λ, µ) . A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 43 Proposition 11.5. The change-of-base-point isomorphism commutes with the products. Thus, for λ, µ, ν ∈ Λ and p, q ∈ X = X ′ , the following diagram is commutative: H(p, q)(λ, µ) ⊗ H(p, q)(µ, ν) Mp,q / H(p, q)(λ, ν) B(p0 ,p′0 ) B(p0 ,p′0 )⊗B(p0 ,p′0 )  H′ (p, q)(λ, µ) ⊗ H′ (p, q)(µ, ν) M′p,q  / H′ (p, q)(λ, ν) . We summarise the results of this section into the following statement: Theorem 11.6. The change-of-base-point isomorphism B(p0 , p′0 ) : H categories. / H′ is an isomorphism of coproduct 12. Group actions and main result We continue working under Assumption 9.1 and we fix an idempotent p0 ∈ R. Associated to the separable Hilbert space H, we have the group GL(H) of invertible bounded operators on H, and associated to the unital ring R, we have the group of automorphisms Aut(R). Finally we have the group of permutations Perm(Λ) of the index set Λ (indexing the representations of R on the separable Hilbert space H). On top of Assumption 9.1 we impose the following: Assumption 12.1. Suppose that we are given a group G together with three group homomorphisms / GL(H) / Aut(R) / Perm(Λ) , α: G β: G γ: G which satisfy the following conditions: (1) β(g)(i) ∈ I for all i ∈ I and g ∈ G; (2) β(g)(p0 ) − p0 ∈ I for all g ∈ G;
(3) α(g)πλ (r)α(g)−1 = πγ(g)(λ) β(g)(r) for all r ∈ R, λ ∈ Λ and g ∈ G. We emphasise that we do not impose any continuity conditions on the group homomorphisms α, β and γ. We now define a strict action of G on the coproduct category Hp0 given in Definition 10.1, where the subscript p0 indicates the base point. For each g ∈ G, we first define an isomorphism of coproduct categories: / Hβ(g)(p ) . tg : Hp0 0 On the underlying set Xp0 = {p ∈ Idem(R) | p − p0 ∈ I} we have the bijection / Xβ(g)(p ) = Xp0 (12.1) tg : Xp0 p 7→ β(g)(p) . 0 / Note that our assumptions on β(g) : R R imply that Xp0 = Xβ(g)(p0 ) and that this set is invariant under the automorphism β(g). For each p, q ∈ X, we have the assignment
/ Hβ(g)(p ) β(g)(p), β(g)(q) tg : Hp0 (p, q) 0 defined as follows: on objects, i.e. on the index set Λ, we put /Λ (12.2) tg := γ(g) : Λ and on morphisms we have the isomorphism of Z-graded complex lines tg := Ad(α(g) ⊕ α(g)) ⊗ Ad(α(g) ⊕ α(g))
(12.3) / Hβ(g)(p ) β(g)(p), β(g)(q) (γ(g)(λ), γ(g)(µ)) , : Hp0 (p, q)(λ, µ) 0 where Ad(α(g)⊕α(g)) is notation for the isomorphism of Z-graded complex lines given by Ad(α(g)⊕α(g)) := L(α(g) ⊕ α(g))R(α(g −1 ) ⊕ α(g −1 )), see Example 3.4. Note that the equivariance condition from Assumption 12.1 implies that    

α(g) 0 α(g)−1 0 F (λ, µ)(p, p0 ) = F γ(g)(λ), γ(g)(µ) β(g)(p), β(g)(p0 ) , 0 α(g) 0 α(g)−1 and similarly with (p, p0 ) replaced by (p0 , q). 44 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Lemma 12.2. For each g ∈ G, the above assignments tg defined in Equation (12.1), (12.2) and (12.3) yield / Hp0 / Hβ(g)(p ) . Moreover, it holds that te = id : Hp0 an isomorphism of coproduct categories tg : Hp0 0 and that the following diagram is commutative for all g, h ∈ G: tg (12.4) / Hβ(g)(p ) Hp0 ❏ 0 ❏❏ ❏❏ ❏❏ th thg ❏❏❏ $  Hβ(hg)(p0 ) . Proof. We suppress the group homomorphisms α, β and γ from the notation. The fact that te = id and that the diagram in Equation (12.4) commutes is a straightforward consequence of the definition of tg for / Hg(p ) (g(p), g(q)) g ∈ G. So for each p, q, e ∈ X and each g ∈ G, we focus on proving that tg : Hp0 (p, q) 0 is a functor and that the following diagram is commutative: ∆e Hp0 (p, q) (12.5) / Hp0 (p, e) ⊗ Hp0 (e, q) tg ⊗tg tg Hg(p0 ) 
g(p), g(q) ∆g(e) / Hg(p ) 0

g(p), g(e) ⊗ Hg(p0 ) g(e), g(q) . Let us fix three objects λ, µ, ν ∈ Λ. Using the definition of the coproduct from Definition 10.1, we see that verifying the commutativity of the diagram in Equation (12.5) amounts to showing that the diagram here below is commutative ϕ / F (λ, µ)(p0 , e) ⊗ F (λ, µ)(e, p0 ) (C, 0) ❲❲❲ ❲❲❲❲❲ ❲❲❲❲❲ ϕ ❲❲❲❲❲ Ad(g⊕g)⊗Ad(g⊕g) ❲❲❲❲❲ ❲+  F (g(λ), g(µ))(g(p0 ), g(e)) ⊗ F (g(λ), g(µ))(g(e), g(p0 )) . (12.6) This commutativity result in turn follows from the definition of the duality operation ϕ (see Equation (9.2)) together with the associativity of torsion and the fact that torsion commutes with perturbation (Proposition 3.3 and Theorem 4.9). We continue by investigating the functoriality of tg . The unitality condition is straightforward to verify, so we focus on showing that tg is compatible with the compositions. We now argue that the following composition of isomorphisms of Z-graded complex lines (12.7) F (λ, µ)(p, p0 ) ⊗ F (µ, ν)(p, p0 ) ⊗ F (λ, ν)(p0 , p) Ad(g⊕g)⊗3 µg(p) / F (g(λ), g(µ))(g(p), g(p0 )) ⊗ F (g(µ), g(ν))(g(p), g(p0 )) ⊗ F (g(λ), g(ν))(g(p0 ), g(p)) / (C, 0) agrees with the trivialisation µp : F (λ, µ)(p, p0 ) ⊗ F (µ, ν)(p, p0 ) ⊗ F (λ, ν)(p0 , p) / (C, 0) . Recalling the definition of the trivialisation µp (and µg(p) ) from Equation (9.4) we see that the identity between the above two isomorphisms follows since torsion is associative and commutes with both perturbation and stabilisation (Proposition 3.3, Theorem 4.9 and Proposition 5.2). A similar argument shows that the two trivialisations in the dual case / (C, 0) µ† ◦ Ad(g ⊕ g)⊗3 and µ†p : F (λ, ν)(p, p0 ) ⊗ F (µ, ν)(p0 , p) ⊗ F (λ, µ)(p0 , p) g(p) also agree, see Equation (9.5). The fact that tg is compatible with compositions is now a consequence of these observations together with the commutativity of the diagram in Equation (12.6).  The action of the group G on the coproduct category Hp0 is then defined by the composition: (12.8) ρ(g) : Hp0 tg / Hβ(g)(p 0) B(β(g)(p0 ),p0 ) / Hp0 for all g ∈ G , A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 45 where we recall that B(β(g)(p0 ), p0 ) denotes the change-of-base-point isomorphism from Section 11. It follows from Lemma 12.2 and Theorem 11.6 that ρ(g) is an automorphism of the coproduct category Hp0 for every g ∈ G. / Aut(Hp0 ) given in Equation (12.8) is a group homomorphism. Lemma 12.3. The assignment ρ : G Proof. Given the properties of the change-of-base-point isomorphism from Lemma 11.3 and the properties / Hβ(g)(p ) , g ∈ G, established in Lemma 12.2, we only need to show that the of the isomorphisms tg : Hp0 0 following diagram of isomorphisms is commutative: (12.9) Hβ(g)(p0 ) th B(β(g)(p0 ),p0 )  Hp0 / Hβ(hg)(p ) 0 B β(hg)(p0 ),β(h)(p0 ) th  / Hβ(h)(p ) , 0
for all g, h ∈ G. However, recalling the definition of the change-of-base-point isomorphism from Definition 11.2, we see that the commutativity of the diagram in Equation (12.9) again follows by an application of the associativity of torsion and the commutativity property of torsion with respect to stabilisation and perturbation (Proposition 3.3, Proposition 5.2 and Theorem 4.9).  We summarise what we have achieved so far, thus stating the main result of this paper, combining Proposition 8.6, Theorem 10.2 and Lemma 12.3: Theorem 12.4. Let R be a unital ring equipped with a fixed idempotent element p0 ∈ R, let I ⊆ R be an ideal, let G be a group and let H be a separable Hilbert space. Suppose that we have a non-empty family Rep = {πλ }λ∈Λ / L (H), λ ∈ Λ, satisfying the conditions consisting of (not necessarily unital) ring homomorphisms πλ : R of Assumption 9.1. Suppose moreover that we have three group homomorphisms α: G / GL(H) β: G / Aut(R) γ: G / Perm(Λ) satisfying the conditions of Assumption 12.1. Then the constructions presented in Definition 10.1 yield a coproduct category Hp0 and the assignment in Equation (12.8) yields a strict action of the group G on Hp0 . In particular, we have a group cohomology class [cHp0 ] ∈ H 3 (G, C∗ /{±1}) given by the constructions in Definition 8.5. 13. Coproduct categories and group cocycles from bipolarised representations In this section, we provide a general operator theoretic framework, which gives rise to coproduct categories with group actions. In particular, this framework gives rise to group 3-cocycles. We fix a separable Hilbert space H and recall that we have the unital C ∗ -algebra L (H) of bounded operators on H, the trace ideal L 1 (H) ⊆ L (H) and the group of invertible bounded operators GL(H) ⊆ L (H). / GL(H). Definition 13.1. Let G be a group equipped with a group homomorphism α : G We say that a pair (P, Q) of idempotents in L (H) is a bipolarisation of the group homomorphism / GL(H), when the following holds for all u, v, g, h ∈ G: α: G   1 (1) α(g)P α(g −1 ), α(u)Qα(u−1 ) ∈
L (H); −1
−1 −1 (2) α(g)P α(g ) − α(h)P α(h ) α(u)Qα(u ) − α(v)Qα(v −1 ) ∈ L 1 (H). We apply the notation Pg = α(g)P α(g −1 ) and Qu = α(u)Qα(u−1 ) for all g, u ∈ G . We shall in this section see how to construct a coproduct category H(P, Q, α) for any group homomorphism /GL(H) equipped with a bipolarisation (P, Q). Moreover, this coproduct category will carry an action α: G of the group G, and it therefore yields a group 3-cocycle on G with values in C∗ /{±1}. By construction, the cohomology class of this group 3-cocycle is canonically associated to our data. We start by giving a different description of a bipolarisation: 46 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON / GL(H) be a group homomorphism and let P, Q : H Lemma 13.2. Let α : G / GL(H) if and only if The pair (P, Q) is a bipolarisation of α : G [P, α(g)][Q, α(u)] ∈ L 1 (H) and / H be bounded idempotents. [α(g)P α(g −1 ), Q] ∈ L 1 (H) for all g, u ∈ G. / GL(H) is a group homomorphism and that L 1 (H) ⊆ L (H) is an ideal our claim Proof. Using that α : G becomes a consequence of the identities   α(u−1 ) α(g)P α(g −1 ), α(u)Qα(u−1 ) α(u) = [α(u−1 g)P α(g −1 u), Q] and (α(g)P α(g −1 ) − P )(α(u)Qα(u−1 ) − Q) = −α(g)[P, α(g −1 )][Q, α(u)]α(u−1 ) , which are valid for all g, u ∈ G.  In order to proceed, we need the following operator theoretic result, where we recall that L p (H) ⊆ L (H) denotes the pth Schatten ideal (for p ∈ [1, ∞)): / H is a bounded operator with T 2 − T ∈ L p (H). Lemma 13.3. Let p ∈ [1, ∞) and suppose that T : H / H with T − E ∈ L p (H). Then there exists a bounded idempotent E : H Proof. We apply the notation Br (z) := {w ∈ C | |w − z| < r} for the open ball around z ∈ C with radius r > 0. Furthermore, we let Sp(T ) ⊆ C denote the spectrum of T . The operator T 2 − T ∈ L p (H) is in particular compact and hence, by the Riesz-Schauder theorem, it has discrete spectrum with 0 ∈ C as the only possible limit point. Moreover, by holomorphic functional calculus, it holds that Sp(T 2 − T ) = {z 2 − z | z ∈ Sp(T )}, and we conclude that the set
Sp(T ) \ Sp(T ) ∩ (Bε (0) ∪ Bε (1)) is finite for all ε ∈ (0, 1). Let us choose a δ ∈ (0, 1/2) such that Sp(T ) ∩ {δ · exp(2πit) | t ∈ [0, 1]} = ∅ = Sp(T ) ∩ {1 + δ · exp(2πit) | t ∈ [0, 1]} . Using the holomorphic functional calculus we then obtain two idempotents E0 := χBδ (0) (T ) , E1 := χBδ (1) (T ) ∈ L (H) , where χU : C / [0, 1] refers to the indicator function associated with a subset U ⊆ C. We claim that T − E1 = T E0 − (1 − T )E1 + T (1 − E0 − E1 ) ∈ L p (H) . / H is of finite rank. Indeed, this follows since To see this, we first remark that T (1 − E0 − E1 ) : H 2 the compact operator T − T restricts to an invertible operator on the image of the bounded idempotent / H and this image must therefore be finite dimensional. We may thus focus our attention 1 − E0 − E1 : H on showing that X := T E0 ∈ L p (H) and Y := (1 − T )E1 ∈ L p (H) . / H. We have that To this end we consider the invertible operators exp(2πiX) and exp(2πiY ) : H exp(2πiX) − 1 = ∞ ∞ X X (2πi)n (2πi)n E0 T n = E0 (T n − T ) . n! n! n=1 n=1 For every n ≥ 2 we notice that T n − T = (T 2 − T )(1 + . . . + T n−2 ). It follows that T n − T ∈ L p (H) and that we have the estimate
kT n − T kp ≤ kT 2 − T kp · 1 + kT k + . . . + kT kn−2 / [0, ∞) is the operator norm). The sum / [0, ∞) (here k · k : L (H) for the p-norm k · kp : L p (H) P∞ (2πi)n n n=1 n! E0 (T − T ) therefore converges absolutely in the p-norm and we may thus conclude that exp(2πiX) − 1 ∈ L p (H) (since L p (H) is a Banach space when equipped with the p-norm). A similar computation shows that exp(2πiY ) − 1 ∈ L p (H) as well. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 47 Notice now that the holomorphic function f : z 7→ (exp(2πiz) − 1)/z is invertible on the open ball B1/2 (0). Since both Sp(X) ⊆ B1/2 (0) and Sp(Y ) ⊆ B1/2 (0) we must have that f (X) and f (Y ) are invertible. But this implies that X, Y ∈ L p (H) since exp(2πiX) − 1 = X · f (X) and exp(2πiY ) − 1 = Y · f (Y ) .  As a corollary we have the following: Corollary 13.4. Suppose that E, F : H there exists a bounded idempotent E ′ : EH / H are two bounded idempotents with [E, F ] ∈ L 2 (H). Then / EH such that EF E − E ′ ∈ L 1 (EH). Proof. This follows from Lemma 13.3 since (EF E)2 − EF E = EF EF E − EF E = EF [E, F ]E = E[E, F ][E, F ]E ∈ L 1 (EH)  Notation 13.5. For any group G, we denote by RG the free unital ring with one generator qu for each u ∈ G subject to the relation qu2 = qu . We let IG denote the two-sided ideal in RG generated by the subset {qu − qv | u, v ∈ G} ⊆ RG . We emphasise that the unit 1 in RG is different from the idempotent qe ∈ RG , where e ∈ G is the neutral element in G. / Aut(RG ) defined by Clearly, the group G acts on RG via the group homomorphism β : G β(g) : qu 7→ qgu (13.1) for all g, u ∈ G , and this action is compatible with the ideal IG ⊆ RG in the sense that β(g)(i) ∈ IG and β(g)(qu ) − qu ∈ IG for all i ∈ IG and all g, u ∈ G. / GL(H). We will say that Definition 13.6. Let (P, Q) be a bipolarisation of a group homomorphism α : G a family σ = {σg }g∈G of (not necessarily unital) representations of the unital ring RG as bounded operators on H is admissible when the relations (1) σg (1) = Pg ; (2) σg (qu ) − Pg Qu Pg ∈ L 1 (H); (3) σhg (qhu ) = α(h)σg (qu )α(h−1 ) hold for all g, h, u ∈ G. / GL(H) we define the set For any bipolarisation (P, Q) of a group representation α : G  Adm(P, Q, α) := σ | σ = {σg }g∈G is an admissible family of representations of RG on H and we will consider the index set defined as the cartesian product: (13.2) Λ(P, Q, α) := G × Adm(P, Q, α) = {(h, σ) | h ∈ G , σ ∈ Adm(P, Q, α)} . We define an action of G on the index set Λ = Λ(P, Q, α) by permutations: (13.3) γ: G / Perm(Λ) γ(k)(h, σ) := (kh, σ) . For each λ = (h, σ) ∈ Λ(P, Q, α) we then have the representation πλ := σh : RG family of representations of RG as bounded operators on H: / L (H). This yields the Rep(P, Q, α) := {πλ }λ∈Λ . We shall now see that our index set Λ := Λ(P, Q, α) is non-empty: Lemma 13.7. Suppose that (P, Q) is a bipolarisation of a group representation α : G each g ∈ G, we may choose a ring homomorphism σg : RG / GL(H). Then for / L (H) such that the associated family σ = {σg }g∈G is admissible. In particular, we have an element (h, σ) ∈ Λ(P, Q, α) for all h ∈ G. 48 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Proof. Using Corollary 13.4, we may choose idempotents Eu ∈ L (P H), u ∈ G, satisfying Eu − P Qu P ∈ L 1 (P H). / L (H) by setting For each g ∈ G, we define the representation σg : RG σg (qu ) := α(g) · Eg−1 u · α(g −1 ) : H /H for all u ∈ G and σg (1) := Pg . Remark in this respect that σg (1) · σg (qu ) = σg (qu ) = σg (qu ) · σg (1) since P Eg−1 u = Eg−1 u = Eg−1 u P for all u ∈ G. We leave it to the reader to verify that the relations in Definition 13.6 are satisfied.  The next theorem is now mainly a consequence of Theorem 12.4: Theorem 13.8. Suppose that (P, Q) is a bipolarisation of a group homomorphism α : G we have an associated group 3-cohomology class / GL(H). Then [c(P, Q, α)] ∈ H 3 (G, C∗ /{±1}) . More precisely, our family of representations Rep(P, Q, α) = {πλ }λ∈Λ(P,Q,α) of RG satisfies Assumption 9.1 with respect to the ideal IG ⊆ RG . Moreover, the three group homomorphisms / GL(H) / Aut(RG ) / Perm(Λ(P, Q, α)) α: G β: G γ: G satisfy the conditions in Assumption 12.1 with respect to the fixed idempotent qe ∈ RG (and the ideal IG ⊆ RG ). In particular, we obtain a coproduct category H(P, Q, α) equipped with a strict action of the group G and this data yields our group 3-cohomology class [c(P, Q, α)] ∈ H 3 (G, C∗ /{±1}) for the group G with values in C∗ /{±1}. Proof. By Theorem 12.4 and Lemma 13.7 we only need to verify condition (1) and (2) in Assumption 9.1 / Aut(RG ) satisfies and condition (1), (2) and (3) in Assumption 12.1. We have already argued that β : G condition (1) and (2) in Assumption 12.1. To check the equivariance condition (3), we let (h, σ) ∈ Λ(P, Q, α) and k, u ∈ G be given and notice that α(k)π(h,σ) (qu )α(k)−1 = α(k)σh (qu )α(k)−1 = σkh (qku ) = π(kh,σ) (qku ) = πγ(k)(h,σ) (β(k)(qu )) , where the second equality sign follows by the admissibility of the family of representations σ = {σg }g∈G . We are thus left with condition (1) and (2) in Assumption 9.1. Let (h, σ), (k, ρ) and (l, τ ) ∈ Λ be given. Since L 1 (H) ⊆ L (H) is an ideal (and we are working with ring homomorphisms), it suffices to verify the commutator condition (1) on the generators qu ∈ RG , u ∈ G. But for each u ∈ G we have that σh (qu )ρk (1) = σh (qu )Pk ∼1 Ph Qu Ph Pk ∼1 Ph Qu Pk ∼1 Ph Pk Qu Pk ∼1 Ph ρk (qu ) = σh (1)ρk (qu ) , where the first and fourth equivalence follow by admissibility and the second and third equivalence follow / GL(H) (see Definition 13.6 and Definition 13.1). since (P, Q) is a bipolarisation of α : G Since we have already verified the commutator condition (1), it suffices to check condition (2) of Assumption 9.1 for the generators qu − qv ∈ IG , u, v ∈ G. But for each u, v ∈ G we have that σh (qu − qv )ρk (1)τl (1) ∼1 Ph (Qu − Qv )Ph Pk Pl ∼1 Ph (Qu − Qv )Pk Pl ∼1 Ph (Qu − Qv )Ph Pl ∼1 σh (qu − qv )τl (1), where we are again using admissibility and the fact that (P, Q) is a bipolarisation of α : G / GL(H).  / GL(V ) 13.1. Group 3-cocycles on double loop groups. Let Γ be a Lie group and suppose that ρ : Γ is a representation of Γ by invertible operators on a finite dimensional complex Hilbert space V . We suppose / C, ρv,w : g 7→ hv, ρ(g)wi is smooth for that the representation is smooth in the sense that the map Γ all v, w ∈ V . We are using the convention that inner products on complex Hilbert spaces are linear in the second variable and conjugate linear in the first variable. We are interested in the double loop group G := C ∞ (T2 , Γ) consisting of smooth maps from the 2-torus with values in Γ (where the group structure is given by point-wise application of the group law in Γ) and we / GL(V ). are going to construct a group 3-cocycle on G associated to the smooth representation ρ : Γ A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 49 We let L2 (T2 ) denote the separable Hilbert space of equivalence classes of square integrable functions on the 2-torus and we put H := L2 (T2 ) ⊗ V . The unital ∗-algebra of smooth functions on the 2-torus C ∞ (T2 ) acts on the Hilbert space L2 (T2 ) as multiplication operators and we denote the corresponding unital ∗/ L (L2 (T2 )). Let {ei }n denote an orthonormal basis for V and define homomorphism by m : C ∞ (T2 ) i=1 the group homomorphism (13.4) αρ : G / GL(H) αρ (f )(ξ ⊗ ej ) := n X m(ρei ,ej ◦ f )(ξ) ⊗ ei , i=1 / GL(H) is independent for all ξ ∈ L2 (T2 ), f ∈ G and j ∈ {1, 2, . . . , n}. The group homomorphism α : G of the choice of orthonormal basis {ei }ni=1 for V . The Hilbert space L2 (T2 ) has an orthonormal basis {z1n1 z2n2 }(n1 ,n2 )∈Z2 , where the functions z1 and / C denote the projections onto the two factors in the Cartesian product T2 = T × T followed z2 : T2 / L2 (T2 ) by by the inclusion T ⊆ C. We may then define two orthogonal projections P and Q : L2 (T2 )  n1 n2 z1 z2 for n1 ≥ 0 P (z1n1 z2n2 ) := and 0 for n1 < 0  n1 n2 (13.5) z1 z2 for n2 ≥ 0 Q(z1n1 z2n2 ) := 0 for n2 < 0 . / L2 (T2 ) is then exactly It holds that P Q = QP and the image of the orthogonal projection P Q : L2 (T2 ) 2 2 2 2 the Hardy-space for the bi-disc H (D ) ⊆ L (T ). From the orthogonal projections P and Q we obtain orthogonal projections P ⊗ idV and Q ⊗ idV on the Hilbert space H = L2 (T2 ) ⊗ V . The main application of our work can now be summarised in the following result: Theorem 13.9. The pair of orthogonal projections (P ⊗ idV , Q ⊗ idV ) defined in Equation (13.5) is a bipo/ GL(H) defined in Equation (13.4). In particular, larisation of the group homomorphism αρ : C ∞ (T2, Γ)  we obtain an associated group 3-cohomology class c(P ⊗ idV , Q ⊗ idV , αρ ) ∈ H 3 (C ∞ (T2 , Γ), C∗ /{±1}). Proof. Using Lemma 13.2 and the relation P Q = QP  we see that the first part of the theorem follows if we can verify that [P, m(f )][Q, m(g)] ∈ L 1 (L2 (T2 )) and [P, m(f )], Q ∈ L 1 (L2 (T2 )) for all f, g ∈ C ∞ (T2 ). Since the functions f and g are both smooth, we know that their Fourier coefficients, respectively {an1 ,n2 }n1 ,n2 ∈Z and {bm1 ,m2 }m1 ,m2 ∈Z , are rapidly decreasing in the sense that the series X X and |an1 ,n2 ||n1 |k |n2 |l |bm1 ,m2 ||m1 |k |m2 |l n1 ,n2 ∈Z m1 ,m2 ∈Z are convergent for all k, l ∈ N ∪ {0}. We compute as follows: X [P, m(f )][Q, m(g)] = an1 ,n2 bm1 ,m2 · z2n2 [P, z1n1 ][Q, z2m2 ]z1m1  and n1 ,n2 ,m1 ,m2 ∈Z X  [P, m(f )], Q = − an1 ,n2 [P, z1n1 ][Q, z2n2 ] n1 ,n2 ∈Z so that it now suffices to show that [P, z1n ][Q, z2m ] 1 = |n| · |m| for all n, m ∈ Z (where we recall that / [0, ∞) denotes the trace norm). This identity however follows from the fact that k · k1 : L 1 (L2 (T2 )) |n| −|n| |m| −|m| (P − z1 P z1 )(Q − z2 Qz2 ) is an orthogonal projection of rank |n| · |m| (and hence has trace norm equal to |n| · |m|). The second part of the theorem is an immediate consequence of the first part and of Theorem 13.8.  We end this subsection by giving one more example of a bipolarised group homomorphism, but this time in the context of the smooth noncommutative 2-torus, so that we obtain a group 3-cocycle on the invertible elements in this unital ∗-algebra. Thus, we fix an irrational number θ ∈ R \ Q and consider the unitary / L2 (T2 ) defined by operators U and V : L2 (T2 ) U (z1n1 z2n2 ) := z1n1 +1 z2n2 and V (z1n1 z2n2 ) := exp(2πin1 θ)z1n1 z2n2 +1 . 50 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We notice the relation U V = exp(−2πiθ)V U . The smooth noncommutative 2-torus C ∞ (T2θ ) is then defined as the unital ∗-subalgebra of L (L2 (T2 )) consisting of all bounded operators of the form X an1 ,n2 U n1 V n2 ∈ L (L2 (T2 )) , n1 ,n2 ∈Z where the sequence of complex numbers {an1 ,n2 }n1 ,n2 ∈Z is rapidly decreasing, i.e. X |an1 ,n2 ||n1 |k |n2 |l < ∞ , n1 ,n2 ∈Z for all k, l ∈ N ∪ {0}. The smooth noncommutative 2-torus is in fact a unital Fréchet ∗-algebra where the corresponding countable family of semi-norms {k · kk,l }k,l∈N∪{0} is given by X X an1 ,n2 U n1 V n2 k,l := k an1 ,n2 nk1 nl2 U n1 V n2 k∞ . n1 ,n2 ∈Z n1 ,n2 ∈Z ∞ let C (T2θ )∗ C ∞ (T2θ )∗ We denote the group of invertible elements in C ∞ (T2θ ) and define the group homomorphism / αθ : GL(L2 (T2 )) induced by the unital inclusion C ∞ (T2θ ) ⊆ L (L2 (T2 )). The proof of the next proposition is then very similar to the proof of Theorem 13.9 and will therefore not be repeated here. Proposition 13.10. The pair of orthogonal projections (P, Q) defined in Equation (13.5) is a bipolarisation / GL(L2 (T2 )). In particular, we obtain an associated group of the group homomorphism αθ : C ∞ (T2θ )∗  3 ∞ 2 ∗ 3-cohomology class c(P, Q, αθ ) ∈ H (C (Tθ ) , C∗ /{±1}). 14. Non-triviality of group 3-cocycles on double loop groups We now analyse in more detail the case of the double loop group G = C ∞ (T2 )∗ of invertible smooth / GL(L2 (T2 )) denote the group homomorphism determined by the unital functions on the 2-torus. Let α : G 2 ∞ 2 / L (L (T2 )) of C ∞ (T2 ) as multiplication operators on H := L2 (T2 ). We recall representation m : C (T ) / L2 (T2 ) defined in Equation from Theorem 13.9 that the two orthogonal projections P and Q : L2 (T2 ) (13.5) yield a bipolarisation of α. We thus have an associated group 3-cohomology class [c(P, Q, α)] ∈ H 3 (G, C∗ /{±1}). For a fixed λ ∈ C∗ we consider the class in group 3-homology {z1 , z2 , λ} ∈ H3 (G, Z) coming from the group 3-cycle (14.1) (z1 , z2 , λ) − (z1 , λ, z2 ) + (λ, z1 , z2 ) − (λ, z2 , z1 ) + (z2 , λ, z1 ) − (z2 , z1 , λ) ∈ Z[G3 ] We are going to compute the pairing between our group 3-cohomology class [c(P, Q, α)] and the group 3/ C∗ /{±1} homology class {z1 , z2 , λ}. This yields the result h[c(P, Q, α)], {z1 , z2 , λ}i = [λ], where [·] : C∗ denotes the quotient map. In particular, we obtain that our group 3-cohomology class [c] = [c(P, Q, α)] is non-trivial. The computation of the pairing h[c(P, Q, α)], {z1 , z2 , λ}i is carried out in several steps and the reader can in these steps see how the various category theoretic constructions appearing in this paper are working in practice. We recall from Notation 13.5 that RG denotes the free unital ring with one generator qu for every u ∈ G and that these generators are subject to the relation qu2 = qu so that they become elements in Idem(RG ). The ideal IG ⊆ RG is the smallest two-sided ideal containing the difference qu − qv for every u, v ∈ G. According to Theorem 12.4, Theorem 13.8 and Definition 10.1, the coproduct category H := H(P, Q, α) is based on the underlying set  X := p ∈ Idem(RG ) | p − qe ∈ IG together with the choice of basepoint qe ∈ X. For every pair of elements p, q ∈ X we have the category H(p, q). The objects in H(p, q) are elements in the index set Λ := Λ(P, Q, α) := G × Adm(P, Q, α), where Adm(P, Q, α) is the set of admissible families of representations of RG , see Definition 13.6. For every pair of elements (g, σ), (h, τ ) ∈ Λ the morphisms from (g, σ) to (h, τ ) are given by the Z-graded complex line


H(p, q) (g, σ), (h, τ ) := Lp (g, σ), (h, τ ) ⊗ L†q (g, σ), (h, τ ) . We remind the reader of the notation H := L2 (T2 ) Pg := α(g)P α(g −1 ) and Qu := α(u)Qα(u−1 ) . A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 51

We then recall that the Z-graded complex lines Lp (g, σ), (h, τ ) and L†q (g, σ), (h, τ ) are respectively the determinants of the two Fredholm operators


F (g, σ), (h, τ ) (p, qe ) := σg (qe ) ⊕ τh (p) Ω((g, σ), (h, τ ))(1) σg (p) ⊕ τh (qe ) and


F (g, σ), (h, τ ) (qe , q) := σg (q) ⊕ τh (qe ) Ω((g, σ), (h, τ ))(1) σg (qe ) ⊕ τh (q) , where the matrix in the middle is given by Ω((g, σ), (h, τ ))(1) :=  Pg − Pg Ph Pg 2Ph Pg − (Ph Pg )2 Pg Ph Ph Pg Ph − Ph  , see Notation 9.2. The composition in the category H(p, q) is described in detail in Subsection 9.2. For an extra element f ∈ X we have the coproduct functor / H(p, f ) ⊗ H(f, q), ∆f : H(p, q) introduced in Definition 10.1. The group action of the group G on the coproduct category H(P, Q, α) is determined by the three group homomorphisms / GL(L2 (T2 )) , β : G α: G / Aut(RG ) , γ : G / Perm(Λ(P, Q, α)), where we recall that β(k)(qu ) = qk·u and γ(k)(g, σ) = (k · g, σ). At the level of the underlying set X, the / H(P, Q, α) is induced by β(k) : RG / RG and for every pair (p, q) ∈ X ×X automorphism ρ(k) : H(P, Q, α) / H(β(k)(p), β(k)(q)) is given in Equation (12.3) and Equation (12.8). the automorphism ρ(k) : H(p, q) Notice that this automorphism involves the change-of-base-point isomorphism in an essential way. In order to find a representative for the class [c(P, Q, α)] ∈ H 3 (G, C∗ /{±1}) we follow the recipe from Definition 8.5. We start by choosing the element qe ∈ X. For each g ∈ G we should then choose an object ag ∈ H(qe , qg ), so we need to choose ag ∈ Λ(P, Q, α) = G × Adm(P, Q, α). We put ag := (e, σ), where σ is a specific admissible family of representations, which we now construct following the procedure described in Lemma 13.7. For every u ∈ G we choose an idempotent Eu ∈ L (P H) with Eu − P Qu P ∈ L 1 (P H) and we arrange that Eu := P Qξ·zt1 ·zt2 P = P Qzt2 ∈ L (P H), 1 2 2 whenever u = ξ · z1t1 · z2t2 is an invertible monomial. This provides us with the non-unital representation / L (H) given by σe : RG σe : qu 7→ Eu and σe (1) := P . / L (H) given by For each k ∈ G, we then have the non-unital representation σk : RG σk (qu ) := α(k)(Ek−1 u )α(k −1 ) and σk (1) := Pk . In particular, it holds that (14.2) σk (qu ) = Pzl1 Qzt2 , 1 z1l1 z2l2 2 z1t1 z2t2 whenever k = κ · and u = ξ · are invertible monomials. According to Lemma 13.7 we then have our admissible family of representations σ := {σk }k∈G and hence the objects ag := (e, σ) ∈ H(qe , qg ). Remark that our choices entail that the group element g ∈ G only influences which category the object ag = (e, σ) belongs to. According to Definition 8.5, the next step is to choose an isomorphism βg,h : ∆qg (agh ) / ag ⊗ g(ah ) for every pair of elements (g, h) ∈ G× G. The isomorphism βg,h belongs to the category H(qe , qg )⊗ H(qg , qgh ) and travels from the object ∆qg (ag ) = (e, σ) ⊗ (e, σ) to the object ag ⊗ g(ah ) = (e, σ) ⊗ (g, σ). We may thus choose βg,h of the form βg,h := id(e,σ) ⊗ bg,h , / (g, σ) is an isomorphism in the category H(qg , qgh ) so that bg,h is a non-zero vector in where bg,h : (e, σ) the graded determinant line



Lqg (e, σ), (g, σ) ⊗ L†qgh (e, σ), (g, σ) = F (e, σ), (g, σ) (qg , qe ) ⊗ F (e, σ), (g, σ) (qe , qgh ) . 52 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We now compute the kernel and cokernel of the involved Fredholm operators for specific choices of g, h ∈ G. It will in this respect be convenient to introduce the finite rank orthogonal projection Γn,m,t,s := (Pz1m − Pz1n )(Qz2s − Qz2t ) : H /H defined for n, m, t, s ∈ N ∪ {0} with n ≥ m and t ≥ s. Lemma 14.1. Let g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and u = ξ · z1t1 z2t2 , v = η · z1s1 z2s2 be invertible monomials with n1 , m1 , t2 , s2 ∈ N ∪ {0} and n1 ≥ m1 , t2 ≥ s2 . It holds that   Ker F ((h, σ), (g, σ))(qv , qu ) = Im(Γn1 ,m1 ,t2 ,s2 ) ⊕ {0}   Coker F ((h, σ), (g, σ))(qv , qu ) = {0} and   Ker F ((g, σ), (h, σ))(qu , qv ) = {0} ⊕ Im(Γn1 ,m1 ,t2 ,s2 )   Coker F ((g, σ), (h, σ))(qu , qv ) = {0} . Moreover, it holds that and hence that
∗ F ((h, σ), (g, σ))(qu , qv ) = F ((h, σ), (g, σ))(qv , qu )
∗ F ((g, σ), (h, σ))(qv , qu ) = F ((g, σ), (h, σ))(qu , qv ) and   Ker F ((h, σ), (g, σ))(qu , qv ) = {0}   Coker F ((h, σ), (g, σ))(qu , qv ) ∼ = Im(Γn1 ,m1 ,t2 ,s2 ) ⊕ {0}   Ker F ((g, σ), (h, σ))(qv , qu ) = {0}   Coker F ((g, σ), (h, σ))(qv , qu ) ∼ = {0} ⊕ Im(Γn1 ,m1 ,t2 ,s2 ) . and Proof. We will only treat the case where the object (h, σ) appears to the left of the object (g, σ) since the remaining identities follow by similar considerations. Since n1 ≥ m1 ≥ 0 and thus Pz1n1 Pz1m1 = Pz1n1 = Pz1m1 Pz1n1 we have that   Pz1m1 − Pz1n1 Pz1n1 . Ω((h, σ), (g, σ))(1) = Pz1n1 0 In particular Ω((h, σ), (g, σ))(1) is self-adjoint. Recall next that σg (qu ) := Pz1n1 Qzt2 and hence 2 F ((h, σ), (g, σ))(qv , qu ) = (Pz1m1 Qzt2 ⊕ Pz1n1 Qz2s2 )Ω((h, σ), (g, σ))(1)(Pz1m1 Qz2s2 ⊕ Pz1n1 Qzt2 ) 2 2 and F ((h, σ), (g, σ))(qu , qv ) = (Pz1m1 Qz2s2 ⊕ Pz1n1 Qzt2 )Ω((h, σ), (g, σ))(1)(Pz1m1 Qzt2 ⊕ Pz1n1 Qz2s2 ) . 2 2 Since the idempotents in the above formulae are all self-adjoint we see that
∗ F ((h, σ), (g, σ))(qu , qv ) = F ((h, σ), (g, σ))(qv , qu ) . It thus suffices to compute the kernel and cokernel of F ((h, σ), (g, σ))(qv , qu ). Since t2 ≥ s2 ≥ 0 as well, we obtain that   Pz1m1 − Pz1n1 Pz1n1 (Pz1m1 Qz2s2 ⊕ Pz1n1 Qzt2 ) F ((h, σ), (g, σ))(qv , qu ) = (Pz1m1 Qzt2 ⊕ Pz1n1 Qz2s2 ) 2 2 0 Pz1n1 ! (Pz1m1 − Pz1n1 )Qzt2 Pz1n1 Qzt2 2 2 = Pz1n1 Qz2s2 0 : (Pz1m1 Qz2s2 )H ⊕ (Pz1n1 Qzt2 )H 2 / (P m1 z1 Qzt2 )H ⊕ (Pz1n1 Qz2s2 )H . 2 But this Fredholm operator is surjective and has kernel given by the image of the finite rank projection / H ⊕ H. The lemma is proved.  (Pz1m1 − Pz1n1 )(Qz2s2 − Qzt2 ) ⊕ 0 : H ⊕ H 2 A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 53 For every n, m, t, s ∈ N ∪ {0} with n > m and t > s, we choose the non-zero vector

ωn,m,t,s := [z1m z2s ] ∧ . . . ∧ [z1n−1 z2s ] ∧ [z1m z2s+1 ] ∧ . . . ∧ [z1n−1 z2s+1 ] (14.3)

∧ . . . ∧ [z1m z2t−1 ] ∧ . . . ∧ [z1n−1 z2t−1 ] ∈ Λtop Im(Γn,m,t,s ) and for n, m, t, s ∈ N ∪ {0} with n = m or t = s we choose ωn,m,t,s = 1 ∈ C = Λtop ({0}). In particular, for invertible monomials g = µ · z1n1 z2n2 and h = ν · z1m1 z2m2 with n1 , n2 , m1 , m2 ∈ N ∪ {0} we choose the isomorphism

(14.4) bg,h := ωn∗ 1 ,0,n2 ,0 ⊗ ωn1 ,0,n2 +m2 ,0 ∈ F (e, σ), (g, σ) (qg , qe ) ⊗ F (e, σ), (g, σ) (qe , qgh ) , where the gradings of the two Z-graded complex lines are given by the indices −n1 · n2 and n1 · (n2 + m2 ), respectively. Whenever g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and k = κ · z1l1 z2l2 are invertible monomials with exponents n1 , n2 , m1 , m2 , l1 , l2 in N ∪ {0}, we are interested in computing the number c(g, h, k) ∈ C∗ coming from the automorphism (14.5) (g, σ) ⊗ (gh, σ) −1 b−1 g,h ⊗bgh,k / (e, σ) ⊗ (e, σ) ∆qgh (bg,hk ) / (g, σ) ⊗ (g, σ) id⊗g(bh,k ) / (g, σ) ⊗ (gh, σ) in the category H(qg , qgh ) ⊗ H(qgh , qghk ), see Definition 8.5. In order to carry out this computation we need to have a better understanding of the coproduct, the group action and the composition relating to the coproduct category H(P, Q, α). We are now going to investigate these operations. 14.1. The coproduct. We consider invertible monomials u = ξ · z1t1 z2t2 , v = η · z1s1 z2s2 and w = ζ · z1r1 z2r2 as well as g = µ · z1n1 z2n2 and h = ν · z1m1 z2m2 . Recall from Definition 10.1 that the coproduct functor / H(qu , qw ) ⊗ H(qw , qv ) is given by ∆qw : H(qu , qv ) ∆qw : (g, σ) 7→ (g, σ) ⊗ (g, σ) and ∆qw : ω+ ⊗ ω− 7→ ω+ ⊗ ϕ(1) ⊗ ω− on objects and morphisms, respectively. We specify that ω+ ⊗ ω− ∈ F ((g, σ), (h, σ))(qu , qe ) ⊗ F ((g, σ), (h, σ))(qe , qv ) and recall that / F ((g, σ), (h, σ))(qe , qw ) ⊗ F ((g, σ), (h, σ))(qw , qe ) ϕ : (C, 0) is the duality isomorphism from Equation (9.2). Computing the coproduct functor thus really amounts to computing the duality isomorphism: Lemma 14.2. Let g = µ·z1n1 z2n2 , h = ν ·z1m1 z2m2 and w = ζ ·z1r1 z2r2 be invertible monomials with n1 ≥ m1 ≥ 0 / F ((h, σ), (g, σ))(qe , qw ) ⊗ F ((h, σ), (g, σ))(qw , qe ) and r2 ≥ 0. Then the duality isomorphism ϕ : (C, 0) is given explicitly by ϕ(1) = ωn1 ,m1 ,r2 ,0 ⊗ ωn∗ 1 ,m1 ,r2 ,0 . / F ((g, σ), (h, σ))(qe , qw ) ⊗ F ((g, σ), (h, σ))(qw , qe ) takes the Similarly, the duality isomorphism ϕ : (C, 0) form ϕ(1) = ωn∗ 1 ,m1 ,r2 ,0 ⊗ ωn1 ,m1 ,r2 ,0 . Proof. We restrict our attention to the first of the two duality isomorphisms since the computation of the second one is similar but easier. We recall from Equation (9.2) that the duality isomorphism in question is defined as the composition (14.6) (C, 0) = σh (qe ) ⊕ σg (qw ) P / F ((h, σ), (g, σ))(qw , qe ) · F ((h, σ), (g, σ))(qe , qw ) T−1 / F ((h, σ), (g, σ))(qe , qw ) ⊗ F ((h, σ), (g, σ))(qw , qe ) . The product of the Fredholm operators appearing is given by F ((h, σ), (g, σ))(qw , qe ) · F ((h, σ), (g, σ))(qe , qw )    (Pz1m1 − Pz1n1 )Qz2r2 Pz1n1 Q (Pz1m1 − Pz1n1 )Qz2r2 = · r n 0 Pz1 1 Qz22 Pz1n1 Q   0 Pz1m1 Q − (Pz1m1 − Pz1n1 )(Q − Qz2r2 ) = 0 Pz1n1 Qz2r2 Pz1n1 Qz2r2 0  54 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON acting on the Hilbert space (Pz1m1 Q)H ⊕ (Pz1n1 Qz2r2 )H. To ease the notation we put Γ := Γn1 ,m1 ,r2 ,0 = (Pz1m1 − Pz1n1 )(Q − Qz2r2 ) and the perturbation isomorphism appearing in Equation (14.6) can then be computed using Example 4.4:

P : 1 7→ ωn1 ,m1 ,r2 ,0 ⊗ ωn∗ 1 ,m1 ,r2 ,0 ∈ Λtop (Im(Γ)), (n1 − m1 )r2 ⊗ Λtop (Im(Γ))∗ , (m1 − n1 )r2 . Moreover, it can be verified using Lemma 14.1 that the torsion isomorphism appearing in Equation (14.6) comes from the six term exact sequence Im(Γ) ⊕ {0} O Im(Γ) ⊕ {0} / {0} Im(Γ) ⊕ {0} o {0} 0 Im(Γ) ⊕ {0} so that in fact T−1 (ωn1 ,m1 ,r2 ,0 ⊗ ωn∗ 1 ,m1 ,r2 ,0 ) = ωn1 ,m1 ,r2 ,0 ⊗ ωn∗ 1 ,m1 ,r2 ,0 , see Definition 3.2 and Definition 2.6. This proves the lemma.  We apply Lemma 14.2 to advance our computation of the number c(g, h, k) ∈ C∗ , where g = µ·z1n1 z2n2 , h = ν · z1m1 z2m2 and k = κ · z1l1 z2l2 are invertible monomials with n1 , n2 , m1 , m2 , l1 , l2 ∈ N0 . Indeed, we obtain that ∆qgh (bg,hk ) = ωn∗ 1 ,0,n2 ,0 ⊗ ϕ(1) ⊗ ωn1 ,0,n2 +m2 +l2 ,0 = ωn∗ 1 ,0,n2 ,0 ⊗ ωn1 ,0,n2 +m2 ,0 ⊗ ωn∗ 1 ,0,n2 +m2 ,0 ⊗ ωn1 ,0,n2 +m2 +l2 ,0 / (g, σ) ⊗ (g, σ) = bg,h ⊗ ωn∗ ,0,n +m ,0 ⊗ ωn1 ,0,n2 +m2 +l2 ,0 : (e, σ) ⊗ (e, σ) 1 2 2 as an isomorphism in the category H(qg , qgh ) ⊗ H(qgh , qghk ). The automorphism in Equation (14.5) yielding the number c(g, h, k) ∈ C∗ then reduces to the composition of isomorphisms (14.7) (gh, σ) b−1 gh,k / (e, σ) ∗ ωn ⊗ωn1 ,0,n2 +m2 +l2 ,0 1 ,0,n2 +m2 ,0 / (g, σ) g(bh,k ) / (gh, σ) inside the category H(qgh , qghk ) at least up to the sign (−1)n1 ·m2 ·l2 ·(m1 +1) . This sign comes from commuting bg,h past b−1 gh,k since these isomorphisms lie in Z-graded complex lines with gradings n1 ·m2 and −(n1 +m1 )·l2 , respectively. / H(P, Q, α) in 14.2. The group action. Our aim is now to compute the group action ρ(k) : H(P, Q, α) l1 l2 the case where k = κ · z1 z2 is an invertible monomial with l1 , l2 ≥ 0. We are interested in the situation where the elements in the underlying set X are idempotents of the form qu , qv ∈ X for invertible monomials u = ξ · z1t1 z2t2 and v = η · z1s1 z2s2 . This means that we are looking at the isomorphism of categories / H(qku , qkv ) . ρ(k) : H(qu , qv ) For two invertible monomials g = µ · z1n1 z2n2 and h = ν · z1m1 z2m2 we have the objects (h, σ) and (g, σ) in the category H(qu , qv ) with corresponding Z-graded complex line of morphisms given by H(qu , qv )((h, σ), (g, σ)) = F ((h, σ), (g, σ))(qu , qe ) ⊗ F ((h, σ), (g, σ))(qe , qv ) . Supposing moreover that n1 ≥ m1 ≥ 0 and that s2 , t2 ≥ 0 we have computed the above graded determinant line explicitly in Lemma 14.1 and we may choose the non-trivial vector ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 ∈ H(qu , qv )((h, σ), (g, σ)), see Equation (14.3) for the notation. Recalling the definition of ρ(k) from Equation (12.3) and Equation (12.8) and the computation in Example 3.4 we obtain that (14.8)

ρ(k) ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 = B(qk , qe ) tk (ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 )
= B(qk , qe ) κ(n1 −m1 )(s2 −t2 ) · ωn∗ 1 +l1 ,m1 +l1 ,t2 +l2 ,l2 ⊗ ωn1 +l1 ,m1 +l1 ,s2 +l2 ,l2 , where the non-trivial vector
∗ α(k)(ωn1 ,m1 ,t2 ,0 ) ⊗ α(k)(ωn1 ,m1 ,s2 ,0 ) = (κ(n1 −m1 )t2 · ωn1 +l1 ,m1 +l1 ,t2 +l2 ,l2 )∗ ⊗ κ(n1 −m1 )s2 · ωn1 +l1 ,m1 +l1 ,s2 +l2 ,l2 = κ(n1 −m1 )(s2 −t2 ) · ωn∗ 1 +l1 ,m1 +l1 ,t2 +l2 ,l2 ⊗ ωn1 +l1 ,m1 +l1 ,s2 +l2 ,l2 A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 55 belongs to the graded determinant line F ((kh, σ), (kg, σ))(qku , qk ) ⊗ F ((kh, σ), (kg, σ))(qk , qkv ) . To finish our computation of the group action we therefore need to investigate the change-of-basepoint isomorphism more carefully in the present context. Lemma 14.3. Let g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and u = ξ · z1t1 z2t2 , v = η · z1s1 z2s2 , w = ζ · z1r1 z2r2 be invertible monomials with n1 ≥ m1 ≥ 0 and t2 , s2 ≥ r2 ≥ 0. The change-of-basepoint isomorphism B(qw , qe ) : F ((h, σ), (g, σ))(qu , qw ) ⊗ F ((h, σ), (g, σ))(qw , qv ) / F ((h, σ), (g, σ))(qu , qe ) ⊗ F ((h, σ), (g, σ))(qe , qv ) is given explicitly by B(qw , qe ) : ωn∗ 1 ,m1 ,t2 ,r2 ⊗ ωn1 ,m1 ,s2 ,r2 7→ (−1)(n1 −m1 )(t2 −s2 )r2 · ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 . Proof. We focus on the case where t2 ≥ s2 since the case where s2 ≥ t2 follows from a similar computation. Applying a slight abuse of notation we write σg instead of (g, σ). We are also going to suppress the triple of representations (σh , σg , σg ) from our formulae. Recall from Section 11 that the change-of-basepoint isomorphism fits in the commutative diagram (14.9) B(qw ,qe ) F (σh , σg )(qu , qw ) ⊗ F (σh , σg )(qw , qv ) / F (σh , σg )(qu , qe ) ⊗ F (σh , σg )(qe , qv ) STS STS  P F 12 (qw , qv , qu )F 13 (qu , qv , qw ) + e23 (σg (1 − qw ))  / F 12 (qe , qv , qu )F 13 (qu , qv , qe ) + e23 (σg (1 − qe )) where e23 (σg (1 − qw )) refers to the matrix of operators which zeroes everywhere except for the operator σg (1 − qw ) in position (2, 3). The first Fredholm operator in the lower line is given by F 12 (qw , qv , qu )F 13 (qu , qv , qw ) + e23 (σg (1 − qw ))   (Pz1m1 − Pz1n1 )Qzt2 0 (Pz1m1 − Pz1n1 )Qz2s2 Pz1n1 Qz2s2 2    Pz1n1 Qz2r2 0 0 0 =  0 0 Pz1n1 Qzt2 Pzn1 Qzt2 1 2  + e23 (σg (1 − qw )) (Pz1m1 − Pz1n1 )Qzt2 2  0 = Pz1n1 Qzt2 2 Pz1n1 Qz2s2 0 0 0 2 0 Pz1n1 Qz2s2 0  Pz1n1 Qz2r2  0  0  Pz1n1  . 0 A similar computation shows that the same formula holds for the second Fredholm operator in the lower line and hence that F 12 (qw , qv , qu )F 13 (qu , qv , qw ) + e23 (σg (1 − qw )) = F 12 (qe , qv , qu )F 13 (qu , qv , qe ) + e23 (σg (1 − qe )) . The perturbation isomorphism appearing in the expression for the change-of-basepoint isomorphism is therefore equal to the identity map and we may focus on computing the vertical isomorphisms in Equation (14.9). The stabilisation isomorphisms appearing are invisible in the final expressions since they merely assure that the involved Fredholm operators act on the correct Hilbert spaces without interfering with kernels and cokernels in an essential way. We therefore restrict attention to the torsion isomorphisms. We record that the kernel and cokernel of the Fredholm operator   0 (Pz1m1 − Pz1n1 )Qzt2 Pz1n1 Qz2s2 2  0 0 Pz1n1  G :=   0 0 Pz1n1 Qzt2 2 : (Pz1m1 Qzt2 )H ⊕ (Pz1n1 Qz2s2 )H ⊕ Pz1n1 H 2 are given by Ker(G) = {0} and / (P m1 z1 Qz2s2 )H ⊕ Pz1n1 H ⊕ (Pz1n1 Qzt2 )H
Coker(G) ∼ = Im Γn1 ,m1 ,t2 ,s2 ⊕ {0} ⊕ {0} . 2 56 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Combining this computation of the kernel and the cokernel with the expressions given in Lemma 14.1 we see from Definition 3.2 that the relevant six term exact sequence for the torsion isomorphism in the left hand side of Equation (14.9) is given by
/ Im Γn1 ,m1 ,s2 ,r2 {0} {0} {0} o
Im Γn1 ,m1 ,t2 ,s2 o Qz s2 2 
Im Γn1 ,m1 ,t2 ,r2 Using that ωn1 ,m1 ,t2 ,r2 = ωn1 ,m1 ,s2 ,r2 ∧ ωn1 ,m1 ,t2 ,s2 (see Equation (14.3)) and comparing with Definition 2.6 we obtain that the left hand side of Equation (14.9) operates as follows: STS : ωn∗ 1 ,m1 ,t2 ,r2 ⊗ ωn1 ,m1 ,s2 ,r2 7→ (−1)(n1 −m1 )(t2 −s2 )(s2 −r2 ) · ωn∗ 1 ,m1 ,t2 ,s2 . Similarly, we find that the relevant six term exact sequence with regards to the right hand side is given by
/ Im Γn1 ,m1 ,s2 ,0 {0} {0} {0} o
Im Γn1 ,m1 ,t2 ,s2 o Qz s2 2 
Im Γn1 ,m1 ,t2 ,0 and hence that the right hand side of Equation (14.9) takes the form STS : ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 7→ (−1)(n1 −m1 )(t2 −s2 )s2 · ωn∗ 1 ,m1 ,t2 ,s2 . We conclude that the change-of-basepoint isomorphism is given explicitly by B(qw , qe ) : ωn∗ 1 ,m1 ,t2 ,r2 ⊗ ωn1 ,m1 ,s2 ,r2 7→ (−1)(n1 −m1 )(t2 −s2 )r2 · ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 .  We now combine the above Lemma 14.3 with the computation in Equation (14.8) and obtain the expression: ρ(k)(ωn∗ 1 ,m1 ,t2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 ) = B(qk , qe )(κ(n1 −m1 )(s2 −t2 ) · ωn∗ 1 +l1 ,m1 +l1 ,t2 +l2 ,l2 ⊗ ωn1 +l1 ,m1 +l1 ,s2 +l2 ,l2 ) = (−1)(n1 −m1 )(t2 −s2 )l2 · κ(n1 −m1 )(s2 −t2 ) · ωn∗ 1 +l1 ,m1 +l1 ,t2 +l2 ,0 ⊗ ωn1 +l1 ,m1 +l1 ,s2 +l2 ,0 . To finish our discussion of the group action in this concrete setting we continue our computation of the number c(g, h, k) ∈ C∗ in the case where g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and k = κ · z1l1 z2l2 are invertible monomials with n1 , n2 , m1 , m2 , l1 , l2 ≥ 0. Recalling the definition of bh,k from Equation (14.4) we record that ∗ ∗ ⊗ ωm1 +n1 ,n1 ,m2 +l2 +n2 ,0 . g(bh,k ) = g(ωm ⊗ ωm1 ,0,m2 +l2 ,0 ) = (−1)m1 l2 n2 · µm1 l2 · ωm 1 +n1 ,n1 ,m2 +n2 ,0 1 ,0,m2 ,0 Hence, comparing with Equation (14.7), we obtain that the automorphism yielding the number c(g, h, k) ∈ C∗ is given by (14.10) (gh, σ) b−1 gh,k / (e, σ) ∗ ωn ⊗ωn1 ,0,n2 +m2 +l2 ,0 1 ,0,n2 +m2 ,0 / (g, σ) ∗ µm1 l2 ·ωn ⊗ωn1 +m1 ,n1 ,n2 +m2 +l2 ,0 1 +m1 ,n1 ,n2 +m2 ,0 / (gh, σ) up to the sign (−1)l2 (n2 m1 +n1 m2 m1 +n1 m2 ) . 14.3. The composition. We are now going to describe the composition inside the category H(qv , qu ) in the case where u = ξ · z1t1 z2t2 and v = η · z1s1 z2s2 are invertible monomials with s2 , t2 ≥ 0. We moreover restrict our attention to the composition of morphisms in H(qv , qu )((k, σ), (h, σ)) and in H(qv , qu )((h, σ), (g, σ)) where g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and k = κ · z1l1 z2l2 are invertible monomials with n1 ≥ m1 ≥ l1 ≥ 0. In this situation we know that the morphisms H(qv , qu )((k, σ), (h, σ)) agree with the Z-graded complex line



Lqv (k, σ), (h, σ) ⊗ L†qu (k, σ), (h, σ) = F (k, σ), (h, σ) (qv , qe ) ⊗ F (k, σ), (h, σ) (qe , qu ) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 57 Moreover, from Lemma 14.1 we have that  

∗ F (k, σ), (h, σ) (qv , qe ) ∼ = Λtop Im(Γm1 ,l1 ,s2 ,0 ) ⊕ {0} , s2 · (l1 − m1 )  

F (k, σ), (h, σ) (qe , qu ) = Λtop Im(Γm1 ,l1 ,t2 ,0 ) ⊕ {0} , t2 · (m1 − l1 ) and we may single out the explicit morphism:

∗ ⊗ ωm1 ,l1 ,t2 ,0 ∈ Lqv (k, σ), (h, σ) ⊗ L†qu (k, σ), (h, σ) , ωm 1 ,l1 ,s2 ,0 see Equation (14.3). A similar description applies to the morphisms from (h, σ) to (g, σ) and from (k, σ) to (g, σ). The definition of the composition isomorphism appearing in the lemma here below can be found in Definition 10.1. Lemma 14.4. Let u = ξ · z1t1 z2t2 , v = η · z1s1 z2s2 and g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and k = κ · z1l1 z2l2 be invertible monomials with s2 , t2 ≥ 0 and n1 ≥ m1 ≥ l1 ≥ 0. The composition isomorphism

Mqv ,qu : H(qv , qu ) (k, σ), (h, σ) ⊗ H(qv , qu ) (h, σ), (g, σ) is given explicitly by
/ H(qv , qu ) (k, σ), (g, σ) ∗ ⊗ ωm1 ,l1 ,t2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 ⊗ ωn1 ,m1 ,t2 ,0 Mqv ,qu : ωm 1 ,l1 ,s2 ,0 7→ (−1)(n1 −m1 )(m1 −l1 )t2 (t2 −s2 ) · (ωm1 ,l1 ,s2 ,0 ∧ ωn1 ,m1 ,s2 ,0 )∗ ⊗ (ωm1 ,l1 ,t2 ,0 ∧ ωn1 ,m1 ,t2 ,0 ). Proof. The sign (−1)(n1 −m1 )(m1 −l1 )t2 (t2 −s2 ) comes from the symmetry constraint when passing from the Z-graded complex line



Lqv (k, σ), (h, σ) ⊗ L†qu (k, σ), (h, σ) ⊗ Lqv (h, σ), (g, σ) ⊗ L†qu (h, σ), (g, σ) to the Z-graded complex line



Lqv (k, σ), (h, σ) ⊗ Lqv (h, σ), (g, σ) ⊗ L†qu (h, σ), (g, σ) ⊗ L†qu (k, σ), (h, σ) . We are thus claiming that ∗ ⊗ ωn∗ 1 ,m1 ,s2 ,0 ) = (ωm1 ,l1 ,s2 ,0 ∧ ωn1 ,m1 ,s2 ,0 )∗ Mqv (ωm 1 ,l1 ,s2 ,0 M†qu (ωn1 ,m1 ,t2 ,0 and ⊗ ωm1 ,l1 ,t2 ,0 ) = ωm1 ,l1 ,t2 ,0 ∧ ωn1 ,m1 ,t2 ,0 . We shall only establish this claim for the case of the isomorphism

Mqv : Lqv (k, σ), (h, σ) ⊗ Lqv (h, σ), (g, σ)
/ Lqv (k, σ), (g, σ) since the proof in the case of M†qu follows a similar pattern. Alternatively, it is possible to derive the formula for M†qu by applying the duality relation from Proposition 9.12. For more details on the isomorphism Mqv we refer to Definition 9.10. In view of the description of the coproduct from Lemma 14.2, sending the unit 1 ∈ C to the element

ωm1 ,l1 ,s2 ,0 ∧ ωn1 ,m1 ,s2 ,0 ⊗ (ωm1 ,l1 ,s2 ,0 ∧ ωn1 ,m1 ,s2 ,0 )∗ ∈ L†qv (k, σ), (g, σ) ⊗ Lqv (k, σ), (g, σ) it suffices to show that
∗ µqv ωm ⊗ ωn∗ 1 ,m1 ,s2 ,0 ⊗ ωm1 ,l1 ,s2 ,0 ∧ ωn1 ,m1 ,s2 ,0 = 1 . 1 ,l1 ,s2 ,0 58 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We introduce the Fredholm operators F 12 F 23 F†13  (Pzl1 − Pz1m1 )Qz2s2 1  Pz1m1 Qz2s2 :=  0 Pz1m1 Q 0 0 0 0 Pz1n1 Q : Pzl1 Qz2s2 H ⊕ Pz1m1 QH ⊕ Pz1n1 QH  1 Pzl1 Q 0 0 1  0 (Pz1m1 − Pz1n1 )Qz2s2 Pz1n1 Q :=  0 Pz1n1 Qz2s2 0    / P l1 QH ⊕ P m1 Q s2 H ⊕ P n1 QH z1 z2 z1 z 1   / P l1 QH ⊕ P m1 QH ⊕ P n1 Q s2 H : Pzl1 QH ⊕ Pz1m1 Qz2s2 H ⊕ Pz1n1 QH z2 z1 z1 z1   1 0 Pz1n1 Qz2s2 (Pzl1 − Pz1n1 )Qz2s2 1   0 Pz1m1 Q 0 :=   0 0 Pz1n1 Q / P l1 Q s2 H ⊕ P m1 QH ⊕ P n1 QH . : P l1 QH ⊕ P m1 QH ⊕ P n1 Q s2 H z2 z1 z1 z1 z1 z1 z1 z2 and
These Fredholm operators are stabilised versions of the Fredholm operators F (k, σ), (h, σ) (qv , qe ),

F (h, σ), (g, σ) (qv , qe ) and F (k, σ), (g, σ) (qe , qv ), respectively. Our first task is to compute the torsion isomorphism T = T ◦ (id ⊗ T) : |F 12 | ⊗ |F 23 | ⊗ |F†13 | / |F 12 | ⊗ |F†13 F 23 | / |F†13 F 23 F 12 | . The various products of Fredholm operators appearing can be computed and are given by F†13 F 23   Pz1n1 Qz2s2 0  (Pz1m1 − Pz1n1 )Qz2s2 Pz1n1 Q  0 0 / n s m Pzl1 QH ⊕ Pz1 1 Qz22 H ⊕ Pz1 1 QH Pzl1 Qz2s2 H ⊕ Pz1m1 QH ⊕ Pz1n1 QH 1 1  0 (Pzl1 − Pz1m1 )Qz2s2 + Pz1n1 Qz2s2 (Pz1m1 − Pz1n1 )Qz2s2 1  (Pz1m1 − Pz1n1 )Qz2s2 0 Pz1n1 Q  0 0 Pz1n1 Q / P l1 Q s2 H ⊕ P m1 QH ⊕ P n1 QH . P l1 Q s2 H ⊕ P m1 QH ⊕ P n1 QH (Pzl1 − Pz1n1 )Qz2s2 1  0 = n Pz1 1 Q :   F†13 F 23 F 12 =  : z1 z2 z1 z1 z1 z1 z1 z2 and Moreover, the respective kernels and cokernels are given by Ker(F 12 ) = {0} = Ker(F 23 ) = Coker(F†13 ) Coker(F 12 ) ∼ = Im(Γm ,l ,s ,0 ) ⊕ {0} ⊕ {0} 1 1 2 Coker(F 23 ) ∼ = {0} ⊕ Im(Γn1 ,m1 ,s2 ,0 ) ⊕ {0} ∼ = Coker(F†13 F 23 ) Ker(F†13 ) = Im(Γn1 ,l1 ,s2 ,0 ) ⊕ {0} ⊕ {0} = Ker(F†13 F 23 ) Ker(F 13 F 23 F 12 ) = {0} ⊕ Im(Γn ,m ,s ,0 ) ⊕ {0} ∼ = Coker(F 13 F 23 F 12 ) † 1 The torsion isomorphism T : |F 23 | ⊗ |F†13 | {0} 1 † 2 / |F 13 F 23 | therefore comes from the six term exact sequence † / Im(Γn1 ,l1 ,s2 ,0 ) Im(Γn1 ,l1 ,s2 ,0 ) 0 {0} o Im(Γn1 ,m1 ,s2 ,0 )  Im(Γn1 ,m1 ,s2 ,0 ) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS whereas the torsion isomorphism T : |F 12 |⊗|F†13 F 23 | {0} O / |F 13 F 23 F 12 | comes from the six term exact sequence † / Im(Γn1 ,m1 ,s2 ,0 ) ι / Im(Γn1 ,l1 ,s2 ,0 ) P l z11 Im(Γn1 ,m1 ,s2 ,0 ) 59 Im(Γn1 ,m1 ,s2 ,0 ) o −Pz m1 1  0 Im(Γm1 ,l1 ,s2 ,0 ) where the map ι is the inclusion, see Definition 3.2. The isomorphism T : |F 23 | ⊗ |F†13 | therefore given explicitly by / |F 13 F 23 | is † T : ωn∗ 1 ,m1 ,s2 ,0 ⊗ ωn1 ,l1 ,s2 ,0 7→ (−1)(n1 −l1 )(n1 −m1 )s2 · ωn1 ,l1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 / |F 13 F 23 F 12 | is given explicitly by whereas the isomorphism T : |F 12 | ⊗ |F†13 F 23 | † ∗ T : ωm ⊗ ωn1 ,m1 ,s2 ,0 ∧ ωm1 ,l1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 7→ ωn1 ,m1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 . 1 ,l1 ,s2 ,0 For details on these torsion isomorphisms we refer to Definition 2.6. Combining these computations we / |F 13 F 23 F 12 | is described by conclude that the torsion isomorphism T : |F 12 | ⊗ |F 23 | ⊗ |F†13 | † ∗ T : ωm ⊗ ωn∗ 1 ,m1 ,s2 ,0 ⊗ ωn1 ,m1 ,s2 ,0 ∧ ωm1 ,l1 ,s2 ,0 7→ (−1)(n1 −l1 )(n1 −m1 )s2 · ωn1 ,m1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 . 1 ,l1 ,s2 ,0 Comparing with Equation (9.4) and Equation (14.2) we now introduce the idempotent f := Pzl1 (Q − Qz2s2 ) ⊕ 0 ⊕ 0 . 1 We are going to disregard the extra subspace given by the idempotent Pzl1 (1−Q)⊕Pz1m1 (1−Q)⊕Pz1n1 (1−Q) 1 since it appears twice in the stabilisation procedure present in Equation (9.4) and therefore does not influence the final result. We apply f to stabilise the Fredholm operator F†13 F 23 F 12 and we thereby obtain the Fredholm operator   0 (Pz1n1 − Pz1m1 )Qz2s2 + Pzl1 Q (Pz1m1 − Pz1n1 )Qz2s2 1   0 Pz1n1 Q  (Pz1m1 − Pz1n1 )Qz2s2 F†13 F 23 F 12 + f =  0 0 Pz1n1 Q which acts as an endomorphism of the Hilbert space Pzl1 QH ⊕ Pz1m1 QH ⊕ Pz1n1 QH. We are going to 1 trivialise the Z-graded complex line associated to the above Fredholm operator and to this end we introduce the invertible operators   (Pzl1 − Pz1m1 )Q Pz1m1 Q 0 1   Pz1m1 Q 0 0 Ω12 :=   0 0 Pz1n1 Q   Pzl1 Q 0 0 1   0 (Pz1m1 − Pz1n1 )Q Pz1n1 Q  and Ω23 :=  0 0 Pz1n1 Q   0 Pz1n1 Q (Pzl1 − Pz1n1 )Q 1   0 Pz1m1 Q 0 Ω13 :=   0 0 Pz1n1 Q all acting as automorphisms of the Hilbert space Pzl1 QH ⊕ Pz1m1 QH ⊕ Pz1n1 QH. 1 invertible operators is then given by  0 (Pzl1 − Pz1m1 + Pz1n1 )Q (Pz1m1 − Pz1n1 )Q 1  13 23 12 (Pz1m1 − Pz1n1 )Q 0 Pz1n1 Q Ω Ω Ω = 0 0 Pz1n1 Q We remark that the difference The product of these   .  (Pz1n1 − Pz1m1 )(Q − Qz2s2 ) (Pz1m1 − Pz1n1 )(Q − Qz2s2 ) 0 0 0  Ω13 Ω23 Ω12 − (F†13 F 23 F 12 + f ) =  (Pz1m1 − Pz1n1 )(Q − Qz2s2 ) 0 0 0  60 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON makes sense (the operators in question act on the same Hilbert space) and is of trace class (in fact of finite rank). Our task is now to compute the perturbation isomorphism / Ω13 Ω23 Ω12 = (C, 0) . P : F†13 F 23 F 12 + f We consider the finite rank operator N := 0 ⊕ Γn1 ,m1 ,s2 ,0 ⊕ 0 which induces an isomorphism / Coker(F 13 F 23 F 12 + f ) , N : Ker(F 13 F 23 F 12 + f ) † † F†13 F 23 F 12 + f makes + N invertible, and has Ker(N ) as a vector space complement of Ker(F†13 F 23 F 12 + f ). It thus follows from Example 4.4 that P(ωn1 ,m1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 ) = det(Σ) where Σ := (Ω13 Ω23 Ω12 )(F†13 F 23 F 12 + f + N )−1 . We record that  (Pzl1 − Pz1m1 + Pz1n1 )Q (Pz1m1 − Pz1n1 )Q 1  13 23 12 −1 (Pz1m1 − Pz1n1 )Q 0 (Ω Ω Ω ) =  0 Pz1n1 Q and hence that Σ−1 = (F†13 F 23 F 12 + f + N )(Ω13 Ω23 Ω12 )−1  (Pz1n1 − Pz1m1 )Qz2s2 + Pzl1 Q (Pz1m1 − Pz1n1 )Qz2s2 1  Γn1 ,m1 ,s2 ,0 (Pz1m1 − Pz1n1 )Qz2s2 = 0 Pz1n1 Q  (Pzl1 − Pz1m1 + Pz1n1 )Q (Pz1m1 − Pz1n1 )Q 1  (Pz1m1 − Pz1n1 )Q 0 · n 0 Pz1 1 Q  Pzl1 Q − Γn1 ,m1 ,s2 ,0 Γn1 ,m1 ,s2 ,0 0 1  m Γn1 ,m1 ,s2 ,0 Pz1 1 Q − Γn1 ,m1 ,s2 ,0 0 = 0 0 Pz1n1 Q 0   Pz1n1 Q  0 0   Pz1n1 Q  0  0  Pz1n1 Q  0   . The Fredholm determinant of this determinant class operator can be computed and is given by det(Σ−1 ) = (−1)(n1 −m1 )s2 . We thus obtain that P(ωn1 ,m1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 ) = (−1)(n1 −m1 )s2 . / F 13 F 23 F 12 and Combining the above computation of the torsion isomorphism T : |F 12 |⊗|F 23 |⊗|F†13 | † / Ω13 Ω23 Ω12 = (C, 0) we obtain that the trivialisation the perturbation isomorphism P : F†13 F 23 F 12 + f


/ (C, 0) µqv = (PS) ◦ (TS) : F (k, σ), (h, σ) (qv , qe ) ⊗ F (h, σ), (g, σ) (qv , qe ) ⊗ F (k, σ), (g, σ) (qe , qv ) is described by the assignment ∗ ⊗ ωn∗ 1 ,m1 ,s2 ,0 ⊗ ωm1 ,l1 ,s2 ,0 ∧ ωn1 ,m1 ,s2 ,0 ) µqv (ωm 1 ,l1 ,s2 ,0 = (−1)(n1 −l1 )(n1 −m1 )s2 +(m1 −l1 )(n1 −m1 )s2 · P(ωn1 ,m1 ,s2 ,0 ⊗ ωn∗ 1 ,m1 ,s2 ,0 ) = (−1)(n1 −l1 )(n1 −m1 )s2 +(m1 −l1 )(n1 −m1 )s2 +(n1 −m1 )s2 = 1 . This ends the proof of the present lemma.  µ · z1n1 z2n2 , · z1m1 z2m2 κ · z1l1 z2l2 14.4. Non-triviality. We now return to the setting where g = h=ν and k = are invertible monomials with exponents n1 , n2 , m1 , m2 , l1 , l2 in N ∪ {0}. According to Equation (14.10) and Lemma 14.4, the automorphism yielding the number c(g, h, k) ∈ C∗ is described by the composition (14.11) (gh, σ) b−1 gh,k / (e, σ) µm1 l2 ·(ωn1 ,0,n2 +m2 ,0 ∧ωn1 +m1 ,n1 ,n2 +m2 ,0 )∗ ⊗(ωn1 ,0,n2 +m2 +l2 ,0 ∧ωn1 +m1 ,n1 ,n2 +m2 +l2 ,0 ) / (gh, σ) up to the sign (−1)l2 (n2 m1 +n1 m2 m1 +n1 m2 )+n1 ·m1 ·(n2 +m2 +l2 )·l2 = (−1)(n1 m2 +n1 m1 +n2 m1 )l2 +n1 n2 m1 l2 . The result of the next proposition relies on the explicit choices made in the beginning of Subsection 14 near the statement of Lemma 14.1. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 61 Proposition 14.5. Let g = µ · z1n1 z2n2 , h = ν · z1m1 z2m2 and k = κ · z1l1 z2l2 be invertible monomials with exponents n1 , n2 , m1 , m2 , l1 , l2 in N ∪ {0}. It holds that c(g, h, k) = c(P, Q, α)(g, h, k) = µm1 l2 · (−1)ε(g,h,k) , where the sign is given by ε(g, h, k) := (n1 m2 + n1 m1 + n2 m1 )l2 + n1 n2 m1 l2 + n1 m1 · (n2 + m2 − 1)(n2 + m2 )/2 + n1 m1 · (n2 + m2 + l2 − 1)(n2 + m2 + l2 )/2 . Proof. We recall from Equation (14.4) that bgh,k = ωn∗ 1 +m1 ,0,n2 +m2 ,0 ⊗ ωn1 +m1 ,0,n2 +m2 +l2 ,0 , thus to compute the remaining composition in Equation (14.11), we consult Equation (14.3) and obtain that ^ ωn1 ,0,n2 +m2 ,0 ωn1 +m1 ,n1 ,n2 +m2 ,0 = ([1] ∧ . . . ∧ [z1n1 −1 ]) ∧ ([z2 ] ∧ . . . ∧ [z1n1 −1 z2 ]) ∧ . . . ∧ ([z2n2 +m2 −1 ] ∧ . . . ∧ [z1n1 −1 z2n2 +m2 −1 ]) ^ ([z1n1 ] ∧ . . . ∧ [z1n1 +m1 −1 ]) ∧ ([z1n1 z2 ] ∧ . . . ∧ [z1n1 +m1 −1 z2 ]) ∧ . . . ∧ ([z1n1 z2n2 +m2 −1 ] ∧ . . . ∧ [z1n1 +m1 −1 z2n2 +m2 −1 ]) = (−1)n1 ·m1 · Pn2 +m2 −1 j=1 j · ([1] ∧ . . . ∧ [z1n1 +m1 −1 ]) ∧ ([z2 ] ∧ . . . ∧ [z1n1 +m1 −1 z2 ]) ∧ . . . ∧ ([z2n2 +m2 −1 ] ∧ . . . ∧ [z1n1 +m1 −1 z2n2 +m2 −1 ]) = (−1)n1 ·m1 ·(n2 +m2 −1)(n2 +m2 )/2 · ωn1 +m1 ,0,n2 +m2 ,0 . Similarly, we have that ωn1 ,0,n2 +m2 +l2 ,0 ∧ ωn1 +m1 ,n1 ,n2 +m2 +l2 ,0 = (−1)n1 ·m1 ·(n2 +m2 +l2 −1)(n2 +m2 +l2 )/2 · ωn1 +m1 ,0,n2 +m2 +l2 ,0 . These computations prove the proposition.  As a consequence of the above Proposition 14.5 we may
show that our group 3-cocycle yields a nontrivial cohomology class [c(P, Q, α)] ∈ H 3 C ∞ (T2 )∗ , C∗ /{±1} . We recall the definition of the class in group 3-homology {z1 , z2 , λ} ∈ H3 (C ∞ (T2 )∗ , Z) from Equation (14.1). Corollary 14.6. We have the identity [c(P, Q, α)], {z1 , z2 , λ} = [λ] ∈ C∗ /{±1} for the pairing between the group 3-cohomology class [c(P, Q, α)] and the group 3-homology class {z1 , z2 , λ}. Proof. The result of Proposition 14.5 implies that c(λ, z1 , z2 ) = λ whereas c(λ, z2 , z1 ) = c(z1 , λ, z2 ) = c(z1 , z2 , λ) = c(z2 , λ, z1 ) = c(z2 , z1 , λ) = 1 .  We immediately obtain the following: Corollary 14.7. Let V be a finite dimensional Hilbert space, and let Γ be a Lie group with a smooth linear ∗ / Γ. Then / L (V ) factors through a homomorphism C∗ action on V such that scalar multiplication
C 3 ∞ 2 ∗ the class [c] ∈ H C (T , Γ), C /{±1} associated to the representation of Γ on V as in Theorem 13.9 is non-trivial. 15. Proofs of properties of the composition in Lp and its dual L†p We return to the setup described in Section 9. We will thus work under the conditions in Assumption 9.1 and p0 ∈ R will be a fixed idempotent. Moreover, we will fix an idempotent p ∈ R such that p − p0 ∈ I. We are going to prove that Lp is indeed a category, thus that the composition is associative and satisfies both left and right unitality conditions. We shall moreover show that the composition L†p can be obtained / L† (λ, µ)⊗Lp (λ, µ) and ψ : Lp (λ, µ)⊗ from the composition in Lp using the duality isomorphisms ϕ : (C, 0) p / (C, 0), for λ, µ ∈ Λ. This result ensures that L†p is a category as well. L†p (λ, µ) Since both of the idempotents p and p0 are fixed in this section we apply the following notation, see Notation 9.2 for further information: 62 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Notation 15.1. For any pair of indices λ, µ in the non-empty index set Λ we define / πλ (p0 )H ⊕ πµ (p)H F (λ, µ) := F (λ, µ)(p, p0 ) : πλ (p)H ⊕ πµ (p0 )H F† (λ, µ) := F (λ, µ)(p0 , p) : πλ (p0 )H ⊕ πµ (p)H and / πλ (p)H ⊕ πµ (p0 )H . 15.1. Associativity. Throughout this subsection, we fix a quadruple (λ, µ, ν, τ ) of indices in Λ. This quadruple will therefore often be suppressed. We introduce a couple of abbreviations (which can be compared with the notation applied in Section 9): For 1 ≤ i < j ≤ 4 we use the notation F ij , F†ij , meaning that the corresponding Fredholm operator relates to the representations sitting in position i and j in the direct sum πλ ⊕ πµ ⊕ πν ⊕ πτ and that the base point p0 ∈ Idem(R) is used to stabilise. For example, we have that F†12 = F† (λ, µ) ⊕ πν (p0 ) ⊕ πτ (p0 ) : πλ (p0 )H ⊕ πµ (p)H ⊕ πν (p0 )H ⊕ πτ (p0 )H / πλ (p)H ⊕ πµ (p0 )H ⊕ πν (p0 )H ⊕ πτ (p0 )H .
For an extra idempotent q ∈ R we also apply the notation Ωij (q) ∈ L πλ (q)H⊕πµ (q)H⊕πν (q)H⊕πτ (q)H , which means that the Ω(q)-operator from Notation 9.3 relates to the representations in position i and j and that we have stabilised with the idempotent q. For 1 ≤ i < j < k ≤ 4, we also define F ijk := F†ik F jk F ij and Ωijk (q) := Ωik (q)Ωjk (q)Ωij (q) . We consider the following idempotent operators on H⊕4 : e0 := πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) e := πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) and f := πλ (1 − p0 ) ⊕ πµ (1 − p) ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) . Using the analogue of Lemma 9.9 for (4 × 4)-matrices we obtain the following isomorphism: µp : |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| S / F 14 F 34 F 23 F 12 + e † P T◦S / F 14 F 34 F 23 F 12 † / Ω14 (p0 )Ω34 (p0 )Ω23 (p0 )Ω12 (p0 ) + e0 = (C, 0) . This isomorphism allows us to define a ternary version of the composition in Lp . Definition 15.2. The isomorphism of Z-graded complex lines Mp : Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ Lp (ν, τ ) / Lp (λ, τ ) is defined as the composition |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| id⊗3 ⊗ϕ µp ⊗id / |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| ⊗ |F (λ, τ )| / |F (λ, τ )| . We are going to use the ternary multiplication operator to prove the associativity of the composition in Lp . More precisely, we are going to prove Theorem 9.14 by showing that each of the two triangles in the following diagram commutes: (15.1) Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ Lp (ν, τ ) ❱❱❱❱ ✐✐ ❱❱❱id⊗M ❱❱❱❱ p ✐ ❱❱❱❱ ✐✐✐ ✐ ✐ ✐ ❱❱* t✐✐✐ Mp Lp (λ, ν) ⊗ Lp (ν, τ ) Lp (λ, µ) ⊗ Lp (µ, τ ) ❯❯❯❯ ❯❯❯❯ ✐✐✐✐ ❯❯❯❯ ✐✐✐✐ ✐ ✐ ✐ ❯❯❯❯ Mp ✐✐✐✐ Mp ❯❯❯*  t✐✐✐✐ Lp (λ, τ ) Mp ⊗id✐✐✐✐✐ A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 63 It turns out that the commutativity of the left hand side and the right hand side can not be established by similar methods even though one could believe this to be the case at a first glance. Establishing the commutativity of the diagram on the right is more involved so we start with the diagram on the left: Proposition 15.3. The following diagram of isomorphisms of Z-graded complex lines is commutative: Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ Lp (ν, τ ) ❙❙❙❙ ❙❙❙M ❙❙p❙❙ Mp ⊗id ❙❙❙❙  ) / Lp (λ, τ ) Lp (λ, ν) ⊗ Lp (ν, τ ) Mp Proof. From the definition of the binary and ternary multiplications, the commutativity of our diagram is equivalent to proving that the trivialisation |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| id⊗2 ⊗ϕ−1 ⊗id⊗2 (15.2) S / |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| / F†14 F 34 F 23 F 12 + e P T◦S / F†14 F 34 F 23 F 12 / Ω14 (p0 )Ω34 (p0 )Ω23 (p0 )Ω12 (p0 ) + e0 = (C, 0) agrees with the trivialisation |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| (15.3) (T◦S)⊗(T◦S) P⊗P / |F†13 F 23 F 12 ⊗ F†14 F 34 F 13 S⊗S / |F†13 F 23 F 12 + e ⊗ F†14 F 34 F 13 + e / Ω13 (p0 )Ω23 (p0 )Ω12 (p0 ) + e0 ⊗ Ω14 (p0 )Ω34 (p0 )Ω13 (p0 ) + e0 ∼ = (C, 0) . Using that torsion commutes with perturbation and stabilisation (Theorem 4.9 and Proposition 5.2), the trivialisation in Equation (15.3) becomes equal to |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| T◦S (15.4) P / F†14 F 34 F 13 F†13 F 23 F 12 S / F†14 F 34 F 13 F†13 F 23 F 12 + e / Ω14 (p0 )Ω34 (p0 )Ω23 (p0 )Ω12 (p0 ) + e0 = (C, 0) . Using the associativity of perturbation (Theorem 4.1) and that perturbation commutes with stabilisation (Proposition 5.3) the trivialisation in Equation (15.4) can be rewritten as |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| T◦S (15.5) S / F 14 F 34 F 13 F 13 F 23 F 12 † † / F 14 F 34 F 23 F 12 + e † P P / F 14 F 34 F 23 F 12 † / Ω14 (p0 )Ω34 (p0 )Ω23 (p0 )Ω12 (p0 ) + e0 = (C, 0) . Since the last half of the trivialisation in Equation (15.2) agrees with the last half of the trivialisation in Equation (15.5) we focus on proving that the following two isomorphisms agree: |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| id⊗2 ⊗ϕ−1 ⊗id⊗2 / |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| (15.6) |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| T◦S / F 14 F 34 F 13 F 13 F 23 F 12 † † P / F 14 F 34 F 23 F 12 . † T◦S / F 14 F 34 F 23 F 12 † and 64 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Commuting perturbation past torsion (Theorem 4.9) and applying the associativity of torsion (Proposition 3.3) yield that the second of the isomorphisms in Equation (15.6) is equal to the composition |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F† (λ, ν)| ⊗ |F (λ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| S⊗(T◦S)⊗S / |F 12 | ⊗ |F 23 | ⊗ |F 13 F†13 | ⊗ |F 34 | ⊗ |F†14 | id⊗2 ⊗P⊗id⊗2 / |F 12 | ⊗ |F 23 | ⊗ |F 34 | ⊗ |F†14 | T / F†14 F 34 F 23 F 12 . But this isomorphism agrees with the first isomorphism in Equation (15.6) and we have proved the proposition.  15.1.1. Commutativity of the right hand side of Equation (15.1). We start with a small lemma whose proof is almost identical to the proof of Lemma 9.9 (and it will therefore not be repeated here). The notation which we apply was introduced in the beginning of Subsection 15.1. Lemma 15.4. The two bounded operators F†12 F 234 F 12 + e and Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0
agree up to an element in the trace ideal L 1 πλ (1)H ⊕ πµ (1)H ⊕ πν (1)H ⊕ πτ (1)H . The next proposition is in a way the analogue of Proposition 15.3 and it shows why proving the commutativity of the right hand side of Equation (15.1) is more complicated than proving the commutativity of the left hand side. We notice that Lemma 9.9 and Lemma 15.4 imply that the perturbation isomorphisms appearing in the statement make sense. Proposition 15.5. The right hand side of the diagram in Equation (15.1) commutes if and only if the following two trivialisations coincide: (15.7) F 12 ⊗ F 234 ⊗ F†12 P / F†12 F 234 F 12 S / F†12 F 234 F 12 + e / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0 = (C, 0) F 12 ⊗ F 234 ⊗ F†12 = T / F 12 ⊗ F†12 id⊗S⊗id ψ◦S−1 and / F 12 ⊗ F 234 + f ⊗ F†12 id⊗P⊗id / F 12 ⊗ Ω234 (p0 ) + e0 ⊗ F†12 / (C, 0) . Proof. First of all, an argument which is similar to the proof of Proposition 15.3 shows that the ternary trivialisation µp : |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| S / F†14 F 34 F 23 F 12 + e P T◦S / F†14 F 34 F 23 F 12 / Ω14 (p0 ) · Ω34 (p0 ) · Ω23 (p0 ) · Ω12 (p0 ) + e0 = (C, 0) agrees with the following trivialisation: |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| (id⊗4 ⊗ϕ⊗id)(id⊗3 ⊗ϕ⊗id) / |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (µ, τ )| ⊗ |F† (λ, µ)| ⊗ |F (λ, µ)| ⊗ |F (µ, τ )| ⊗ |F† (λ, τ )| (15.8) (T◦S)⊗µp P / F†12 · F 234 · F 12 S / F†12 · F 234 · F 12 + e / Ω12 (p0 ) · Ω234 (p0 ) · Ω12 (p0 ) + e0 = (C, 0) . On the other hand, the trivialisation corresponding to the composition Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ Lp (ν, τ ) id⊗Mp / Lp (λ, µ) ⊗ Lp (µ, τ ) Mp / Lp (λ, τ ) A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 65 appearing in Equation (15.1) is given by |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| (15.9) id⊗3 ⊗ϕ⊗id id⊗µp ⊗id⊗2 / |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (µ, τ )| ⊗ |F (µ, τ )| ⊗ |F† (λ, τ )| / |F (λ, µ)| ⊗ |F (µ, τ )| ⊗ |F† (λ, τ )| µp / (C, 0) . Now, using Lemma 9.7 the trivialisation in Equation (15.9) agrees with the trivialisation |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (λ, τ )| (id⊗4 ⊗ϕ⊗id)(id⊗3 ⊗ϕ⊗id) / |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (µ, τ )| ⊗ |F† (λ, µ)| ⊗ |F (λ, µ)| ⊗ |F (µ, τ )| ⊗ |F† (λ, τ )| id⊗µp ⊗id⊗µp ψ / |F (λ, µ)| ⊗ |F† (λ, µ)| / (C, 0) . Comparing with Equation (15.8) we see that the right hand side of Equation (15.1) commutes if and only if the trivialisation T◦S |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (µ, τ )| ⊗ |F† (λ, µ)| S / F†12 · F 234 · F 12 + e P / F†12 · F 234 · F 12 / Ω12 (p0 ) · Ω234 (p0 ) · Ω12 (p0 ) + e0 = (C, 0) agrees with the trivialisation |F (λ, µ)| ⊗ |F (µ, ν)| ⊗ |F (ν, τ )| ⊗ |F† (µ, τ )| ⊗ |F† (λ, µ)| id⊗µp ⊗id / |F (λ, µ)| ⊗ |F† (λ, µ)| ψ / (C, 0) . The proposition is now proved since it follows by an application of the associativity of torsion (Proposition 3.3) and the definition of the binary trivialisation µp that this latter identity of trivialisations is equivalent to the identity of the two trivialisations in Equation (15.7).  We are now going to prove that the two trivialisations appearing in Proposition 15.5 do in fact agree and this will be accomplished in a few steps. We start by establishing some extra notation together with a crucial lemma. Define the Fredholm operator

G(λ, µ) := πλ (1 − p0 ) ⊕ πµ (1 − p) Ω(λ, µ)(1) πλ (1 − p) ⊕ πµ (1 − p0 ) / πλ (1 − p0 )H ⊕ πµ (1 − p)H . : πλ (1 − p)H ⊕ πµ (1 − p0 )H Lemma 15.6. The Fredholm operator F (λ, µ) + G(λ, µ) : πλ (1)H ⊕ πµ (1)H / πλ (1)H ⊕ πµ (1)H is a trace class perturbation of the invertible operator Ω(λ, µ)(1) ∈ L (πλ (1)H ⊕ πµ (1)H). Proof. This is a consequence of Lemma 9.4. Indeed, F (λ, µ) + G(λ, µ) = (πλ (p0 ) ⊕ πµ (p))Ω(λ, µ)(1)(πλ (p) ⊕ πµ (p0 )) + (πλ (1 − p0 ) ⊕ πµ (1 − p))Ω(λ, µ)(1)(πλ (1 − p) ⊕ πµ (1 − p0 )) ∼1 Ω(λ, µ)(1)(πλ (p) ⊕ πµ (p0 ) + πλ (1 − p) ⊕ πµ (1 − p0 )) = Ω(λ, µ)(1) . Let us choose n, m ∈ N ∪ {0} such that Index(F (λ, µ)) = n − m . It follows from the above lemma that Index(G(λ, µ)) = −Index(F (λ, µ)) and we may thus find a bounded finite rank operator L : πλ (1 − p)H ⊕ πµ (1 − p0 )H ⊕ Cn / πλ (1 − p0 )H ⊕ πµ (1 − p)H ⊕ Cm  66 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON such that the bounded operator Γ := G(λ, µ) + L : πλ (1 − p)H ⊕ πµ (1 − p0 )H ⊕ Cn / πλ (1 − p0 )H ⊕ πµ (1 − p)H ⊕ Cm is invertible. We introduce the Hilbert spaces H1 := πλ (1)H ⊕ πµ (1)H ⊕ Cn ⊕ πν (1)H ⊕ πτ (1)H and m H2 := πλ (1)H ⊕ πµ (1)H ⊕ C ⊕ πν (1)H ⊕ πτ (1)H together with the Fredholm operators Fb 12 := (F (λ, µ) + Γ) ⊕ πν (1) ⊕ πτ (1) : H1 / H2 Fb†12 := (F† (λ, µ) + Γ−1 ) ⊕ πν (1) ⊕ πτ (1) : H2 and / H1 . We record from Definition 5.1 that we have the stabilisation isomorphisms S : |F 12 | / |Fb12 | and S : |F†12 | We also introduce a couple of idempotent operators: / |Fb12 | . † e := πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ 1n ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) : H1 b eb0 := πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ 1n ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) : H1 fb := πλ (1 − p0 ) ⊕ πµ (1 − p) ⊕ 1m ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) : H2 fb0 := πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ 1m ⊕ πν (1 − p0 ) ⊕ πτ (1 − p0 ) : H2 / H1 / H1 / H2 / H2 . A straightforward computation shows that Fb†12 (F 234 + fb)Fb 12 = F†12 F 234 F 12 + eb: H1 / H1 and hence we obtain from Lemma 15.4 that Fb†12 (F 234 + fb)Fb12 is a trace class perturbation of Ω12 (p0 ) · Ω234 (p0 ) · Ω12 (p0 ) + eb0 : H1 / H1 . The next lemma is now a consequence of Proposition 5.2 and Proposition 5.3. Lemma 15.7. The following two trivialisations agree: F 12 ⊗ F 234 ⊗ F†12 T / Fb†12 · (F 234 + fb) · Fb 12 F 12 ⊗ F 234 ⊗ F†12 S S⊗S⊗S T / F†12 F 234 F 12 + e / Fb12 ⊗ F 234 + fb ⊗ Fb†12 P / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + b e0 = (C, 0) and / F†12 F 234 F 12 P / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0 = (C, 0) . To continue, we introduce the trivialisation (15.10) ψb : |Fb12 | ⊗ |Fb†12 | T together with the invertible operator / |Fb†12 · Fb 12 | P / |idH1 | = (C, 0). ∆ := Ω(λ, µ)(p0 ) + Ω(λ, µ)(1 − p0 ) : πλ (1)H ⊕ πµ (1)H / πλ (1)H ⊕ πµ (1)H We remark that Lemma 15.6 and Lemma 9.5 show that F (λ, µ) + Γ = F (λ, µ) + G(λ, µ) + L is a trace class / πλ (1)H ⊕ πµ (1)H ⊕ Cm and that F† (λ, µ) + Γ−1 is a perturbation of ∆ + 0m,n : πλ (1)H ⊕ πµ (1)H ⊕ Cn / πλ (1)H ⊕ πµ (1)H ⊕ Cn . It is also convenient trace class perturbation of ∆ + 0n,m : πλ (1)H ⊕ πµ (1)H ⊕ Cm to define the idempotent operators f1 := πλ (1) ⊕ πµ (1) and f2 := πν (1) ⊕ πτ (1). A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 67 Lemma 15.8. The following two trivialisations coincide: Fb 12 ⊗ F 234 + fb ⊗ Fb†12 T Fb 12 ⊗ F 234 + fb ⊗ Fb†12 id⊗P⊗id P (15.11) = / Fb 12 · (F 234 + fb) · Fb 12 † / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + eb0 = (C, 0) / Fb12 ⊗ Fb†12 b ψ and / Fb 12 ⊗ Ω234 (p0 ) + fb0 ⊗ Fb†12 / (C, 0) . Proof. Using that perturbation commutes with torsion (Theorem 4.9) together with the transitivity of perturbation (Theorem 4.1) the second of the two trivialisations in Equation (15.11) becomes equal to Fb 12 ⊗ F 234 + fb ⊗ Fb†12 = P⊗P⊗P / ∆ + 0m,n + f2 ⊗ Ω234 (p0 ) + fb0 ⊗ ∆ + 0n,m + f2 / ∆ + 0m,n + f2 ⊗ ∆ + 0n,m + f2 T P / f1 + 0n + f2 / idH1 = (C, 0) . Next, a direct computation reveals that we have the identities of operators on H1 : (∆ + 0n,m + f2 )(Ω234 (p0 ) + fb0 )(∆ + 0m,n + f2 ) = Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0 + 0n . Hence, using again that perturbation commutes with torsion and the transitivity of perturbation, we can replace the first of the two trivialisations in Equation (15.11) with Fb 12 ⊗ F 234 + fb ⊗ Fb†12 T P⊗P⊗P / ∆ + 0m,n + f2 ⊗ Ω234 (p0 ) + fb0 ⊗ ∆ + 0n,m + f2 / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0 + 0n P / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + eb0 = (C, 0) . We thus see that proving the identity of trivialisations claimed in the lemma amounts to verifying that the following two trivialisations agree: ∆ + 0m,n + f2 ⊗ Ω234 (p0 ) + fb0 ⊗ ∆ + 0n,m + f2 P (15.12) / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + eb0 = (C, 0) ∆ + 0m,n + f2 ⊗ Ω234 (p0 ) + fb0 ⊗ ∆ + 0n,m + f2 T / f1 + 0n + f2 P T / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0 + 0n and = / ∆ + 0m,n + f2 ⊗ ∆ + 0n,m + f2 / idH1 = (C, 0) . That these two latter trivialisations coincide can now be checked by hand using the definition of the torsion isomorphism and the perturbation isomorphism. In fact, one may explicitly compute the kernels and cokernels of all the involved operators and the two trivialisations then both take the form |Cn | ⊗ |Cm |∗ ⊗ |Cm | ⊗ |Cn |∗ T / |Cn | ⊗ |Cn |∗ P / (C, 0) , Moreover, it can be seen from Definition 3.2, Definition 2.6 and Example 4.4 that both of the trivialisations are given explicitly by the map λ · (s ⊗ t∗ ⊗ t ⊗ s∗ ) 7→ λ, for all non-trivial vectors s ∈ Λtop (Cn ) and t ∈ Λtop (Cm ) and all λ ∈ C. Remark however that the torsion and perturbation isomorphisms appearing are not a priori the same since they do in fact depend on different operators as reflected in Equation (15.12). This ends the proof of the lemma.  We observe that the identity Fb†12 · Fb12 = F†12 · F 12 + eb of operators on H1 holds. The following lemma is then a consequence of Proposition 5.2 and Proposition 5.3 where we recall the definitions of the trivialisations ψb and ψ from Equation (15.10) and Equation (9.3). Lemma 15.9. The diagram here below is commutative: −1 S Fb12 ⊗ Fb†12 ▲▲▲ ▲▲▲ ▲ b ▲▲▲ ψ & ⊗S−1 (C, 0) / F 12 ⊗ F 12 † r r r r rrr −1 −1 xrrrψ◦(S ⊗S ) 68 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Combining the results obtained so far we may give a full proof of the associativity of the composition in Lp : Proof of Theorem 9.14. We need to show that the diagram in Equation (15.1) is commutative. The left hand side of this diagram commutes by Proposition 15.3 so we only need to establish that the right hand side also commutes. By Proposition 15.5 this amounts to verifying that the following two trivialisations agree: (15.13) F 12 ⊗ F 234 ⊗ F†12 P / F†12 F 234 F 12 S / F†12 F 234 F 12 + e / Ω12 (p0 )Ω234 (p0 )Ω12 (p0 ) + e0 = (C, 0) F 12 ⊗ F 234 ⊗ F†12 = T / F 12 ⊗ F†12 id⊗S⊗id ψ◦S−1 and / F 12 ⊗ F 234 + f ⊗ F†12 id⊗P⊗id / F 12 ⊗ Ω234 (p0 ) + e0 ⊗ F†12 / (C, 0) . However, using Lemma 15.7, Lemma 15.8 and Lemma 15.9 we see that the first of the two trivialisations in Equation (15.13) coincides with the trivialisation F 12 ⊗ F 234 ⊗ F†12 = / F 12 ⊗ F†12 S / F 12 ⊗ F 234 + fb ⊗ F 12 † ψ◦S−1 P / (C, 0) . / F 12 ⊗ Ω234 (p0 ) + eb0 ⊗ F 12 † The fact that the two trivialisations in Equation (15.13) agree is now a consequence of Proposition 5.3. This proves the theorem.  15.2. Unitality. We continue by proving the unitality condition in Lp . Recall that for λ ∈ Λ, the unit / λ in Lp (λ, λ) is given by the unit in C under the identification idλ : λ   0 πλ (p0 ) (C, 0) = = |F (λ, λ)| = Lp (λ, λ) . πλ (p) 0 Proof of Theorem 9.13. Since Lp (λ, µ) is a Z-graded one-dimensional vector space over C and since both of the degree 0 maps x 7→ Mp (x ⊗ idµ ) and x 7→ Mp (idλ ⊗ x) are automorphisms of Lp (λ, µ), it suffices to show that they are both idempotents. However, by the associativity of the composition in Lp (Theorem 9.14) this amounts to showing that Mp (idλ ⊗ idλ ) = idλ for all λ ∈ Λ. Thus, let λ ∈ Λ be given. It suffices to show that the two perturbation isomorphisms F (λ, λ) · F† (λ, λ) P / πλ (p0 ) ⊕ πλ (p) and F†13 (λ, λ, λ)F 23 (λ, λ, λ)F 12 (λ, λ, λ) + πλ (1 − p) ⊕ πλ (1 − p0 ) ⊕ πλ (1 − p0 ) P / Ω13 (p0 )Ω23 (p0 )Ω12 (p0 ) + πλ (1 − p0 ) ⊕ πλ (1 − p0 ) ⊕ πλ (1 − p0 ) both agree with the identity automorphism when identifying all the graded lines with (C, 0). However, a straightforward computation shows that F (λ, λ) · F† (λ, λ) = πλ (p0 ) ⊕ πλ (p) and F†13 (λ, λ, λ)F 23 (λ, λ, λ)F 12 (λ, λ, λ) + πλ (1 − p) ⊕ πλ (1 − p0 ) ⊕ πλ (1 − p0 ) = Ω13 (p0 )Ω23 (p0 )Ω12 (p0 ) + πλ (1 − p0 ) ⊕ πλ (1 − p0 ) ⊕ πλ (1 − p0 ) = πλ (1) ⊕ πλ (1) ⊕ πλ (1) . The two relevant perturbation isomorphisms are therefore indeed both equal to the identity automorphism and the result of the theorem follows.  15.3. Duality. We end this section by giving a proof of Proposition 9.12 regarding the duality relation between the compositions in Lp and L†p . We are working with respect to a fixed triple of indices (λ, µ, ν) from the index set Λ and this triple will be suppressed from the notation throughout this subsection. More precisely, we apply the notation introduced in the beginning of Subsection 9.2. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 69 Proof of Proposition 9.12. Using Lemma 9.7, we notice that it suffices to verify that the trivialisation F (λ, µ) ⊗ F (µ, ν) ⊗ F† (λ, ν) ⊗ F (λ, ν) ⊗ F† (µ, ν) ⊗ F† (λ, µ) µp ⊗µ†p / (C, 0) agrees with the trivialisation F (λ, µ) ⊗ F (µ, ν) ⊗ F† (λ, ν) ⊗ F (λ, ν) ⊗ F† (µ, ν) ⊗ F† (λ, µ) id⊗2 ⊗ϕ−1 ⊗id⊗2 (15.14) id⊗ψ⊗id / F (λ, µ) ⊗ F (µ, ν) ⊗ F† (µ, ν) ⊗ F† (λ, µ) / F (λ, µ) ⊗ F† (λ, µ) ψ / (C, 0) . Using that torsion and perturbation commute with stabilisation, see Proposition 5.2 and Proposition 5.3, / (C, 0) as the we may alternatively describe the isomorphism of graded lines ϕ−1 : F† (λ, ν) ⊗ F (λ, ν) composition S F† (λ, ν) ⊗ F (λ, ν) / F†13 ⊗ F 13 T / F 13 · F†13 P / πλ (p0 ) ⊕ πµ (p0 ) ⊕ πν (p) = (C, 0) . / (C, 0) and ψ : F (λ, µ) ⊗ F† (λ, µ) Since similar results apply to ψ : F (µ, ν) ⊗ F† (µ, ν) may rewrite the trivialisation in Equation (15.14) as the composition / (C, 0) we F (λ, µ) ⊗ F (µ, ν) ⊗ F† (λ, ν) ⊗ F (λ, ν) ⊗ F† (µ, ν) ⊗ F† (λ, µ) S (15.15) / F 12 ⊗ F 23 ⊗ F†13 ⊗ F 13 ⊗ F†23 ⊗ F†12 id⊗2 ⊗(P◦T)⊗id⊗2 id⊗(P◦T)⊗id / F 12 ⊗ F 23 ⊗ F†23 ⊗ F†12 P◦T / F 12 ⊗ F†12 / (C, 0) . Repeated use of the transitivity of perturbation and the fact that torsion commutes with perturbation (Theorem 4.1 and Theorem 4.9) shows that the trivialisation in Equation (15.15) agrees with the composition F (λ, µ) ⊗ F (µ, ν) ⊗ F† (λ, ν) ⊗ F (λ, ν) ⊗ F† (µ, ν) ⊗ F† (λ, µ) S / F 12 ⊗ F 23 ⊗ F†13 ⊗ F 13 ⊗ F†23 ⊗ F†12 T / F†12 F†23 F 13 F†13 F 23 F 12 P / πλ (p) ⊕ πµ (p0 ) ⊕ πν (p0 ) = (C, 0) . Let us define the idempotents e := πλ (1 − p) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) and e0 := πλ (1 − p0 ) ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) together with the Fredholm operators F 123 := F†13 F 23 F 12 , F†123 := F†12 F†23 F 13 and Ω123 (p0 ) := Ω13 (p0 )Ω23 (p0 )Ω12 (p0 ). Using the associativity of torsion (Proposition 3.3), the result of the present proposition is then a consequence of the commutative diagram here below: F 123 ⊗ F†123 T / F 123 F 123 † S⊗S  F†123 + e ⊗ F 123 + e T P⊗P 
−1 Ω123 (p0 ) + e0 ⊗ Ω123 (p0 ) + e0 T ❯❯❯❯ ❯❯❯❯ ❯❯❯❯ S ❯❯❯❯ P ❯❯*  / F 123 F 123 + e π (p) ⊕ πµ (p0 ) ⊕ πν (p0 ) λ † ✐ ✐ ✐ ✐ ✐✐✐✐ P ✐✐✐✐S ✐ ✐ ✐  t✐✐ / πλ (1) ⊕ πµ (1) ⊕ πν (1) which in turn follows from Theorem 4.9, Proposition 5.2 and Proposition 5.3.  16. Proofs of properties of the change of base point In this section we give full proofs of the main properties of the change-of-base point isomorphism which we introduced in Section 11. Thus, we shall see that the change-of-base-point isomorphism is an isomorphism of coproduct categories. The main properties needed for establishing this result are the cofunctoriality and the functoriality of the change-of-base-point isomorphism. These properties are stated as Proposition 11.4 and Proposition 11.5 in Section 11. The proof of cofunctoriality is rather straightforward and we take care 70 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON of it in Subsection 16.1, but the proof of functoriality is involved and occupies Subsection 16.2-16.5. The technical core of the proof of functoriality can be found as Lemma 16.9 in Subsection 16.5, where we show that a certain Fredholm determinant is equal to one and this allows us to relate the compositions in the coproduct categories to a sort of higher (ternary) change-of-base-point isomorphism in Proposition 16.10. The rest of the proof (Subsection 16.2-16.4) develops alternative descriptions of the change-of-base-point isomorphism with the aim of expressing the ternary change-of-base-point isomorphism as a tensor product of change-of-base-point isomorphisms, see Proposition 16.8. The overall setting is as follows: we are given a unital ring R and an ideal I ⊆ R together with a family of representations {πλ }λ∈Λ satisfying Assumption 9.1. We moreover fix two different base points, thus two idempotents p0 and p′0 in R and we assume that they agree modulo the ideal I. We will apply the results on the commutativity and associativity relations for torsion, perturbation and stabilisation from Section 3, 4 and 5 many times, and we will do so without further notice. 16.1. Cofunctoriality. We let p, e and q be idempotents in the unital ring R, all agreeing with the base points p0 and p′0 modulo the ideal I. Moreover, we fix two indices λ, µ ∈ Λ. For an element x ∈ R and indices i, j ∈ {2, 3, 4} we let / πλ (1)H ⊕ (πµ (1)H)⊕3 eij (πµ (x)) : πλ (1)H ⊕ (πµ (1)H)⊕3 denote the bounded operator represented by the (4 × 4)-matrix with πµ (x) in position (i, j) and zeroes elsewhere. We now present the proof of the cofunctoriality of the change-of-base-point isomorphism: Proof of Proposition 11.4. We suppress the tuples of indices (λ, µ) and (λ, µ, µ, µ) throughout the proof. We first notice that the isomorphism

B(p0 , p′0 ) ⊗ B(p0 , p′0 ) : |F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)|

/ |F (p, p′0 )| ⊗ |F (p′0 , e)| ⊗ |F (e, p′0 )| ⊗ |F (p′0 , q)| agrees with the composition of isomorphisms:

|F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)| (TS)⊗(TS) (16.1) / F 13 (p0 , q, e, p) · F 14 (p, q, e, p0 ) ⊗ F 12 (p0 , q, e, p) · F 13 (e, q, p0 , p) (S−1 PS)⊗(S−1 PS) / F 13 (p′ , q, e, p) · F 14 (p, q, e, p′ ) ⊗ F 12 (p′ , q, e, p) · F 13 (e, q, p′ , p) 0 0 0 0

/ |F (p, p′ )| ⊗ |F (p′ , e)| ⊗ |F (e, p′ )| ⊗ |F (p′ , q)| , 0 0 0 0 (TS)−1 ⊗(TS)−1 where the stabilisations in the middle are using the operators e34 (πµ (1 − p0 )) and e23 (πµ (1 − p0 )) as well as their primed versions (replacing p0 by p′0 ). Likewise, we notice that the isomorphism / |F (p, p′ )| ⊗ |F (p′ , q)| B(p0 , p′ ) : |F (p, p0 )| ⊗ |F (p0 , q)| 0 0 0 agrees with the composition of isomorphisms: (16.2) |F (p, p0 )| ⊗ |F (p0 , q)| S−1 PS TS / |F 12 (p0 , q, e, p) · F 14 (p, q, e, p0 )| / |F 12 (p′0 , q, e, p) · F 14 (p, q, e, p′0 )| (TS)−1 / |F (p, p′0 )| ⊗ |F (p′0 , q)| , where the stabilisations in the middle use the operators e24 (πµ (1 − p0 )) and e24 (πµ (1 − p′0 )). Recall now (from Definition 10.1) that the isomorphism / |F (p, p′0 )| ⊗ |F (p′0 , q)| (∆′e )−1 : |F (p, p′0 )| ⊗ |F (p′0 , e)| ⊗ |F (e, p′0 )| ⊗ |F (p′0 , q)| is given by the composition (∆′e )−1 = (id ⊗ P ⊗ id) ◦ (id ⊗ T ⊗ id) = id ⊗ (ϕ′ )−1 ⊗ id. Using the description of B(p0 , p′0 ) ⊗ B(p0 , p′0 ) from Equation (16.1), we thus obtain that


(∆′e )−1 ◦ B(p0 , p′0 ) ⊗ B(p0 , p′0 ) : |F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)| / |F (p, p′ )| ⊗ |F (p′ , q)| 0 0 A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 71 agrees with the composition of isomorphisms: |F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)| TS / |F 12 (p0 , q, e, p) · F 13 (e, q, p0 , p) · F 13 (p0 , q, e, p) · F 14 (p, q, e, p0 )| S−1 PS / |F 12 (p′0 , q, e, p) · F 13 (e, q, p′0 , p) · F 13 (p′0 , q, e, p) · F 14 (p, q, e, p′0 )| (TS)−1 / |F (p, p′0 )| ⊗ |F (p′0 , e) · F (e, p′0 )| ⊗ |F (p′0 , q)| id⊗P⊗id / |F (p, p′ )| ⊗ |F (p′ , q)| , 0 0 where the stabilisations in the middle use the invertible operators e24 (πµ (1 − p0 )) and e24 (πµ (1 − p′0 )). But this latter composition agrees with the composition: |F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)| (16.3) TS / |F 12 (p0 , q, e, p) · F 14 (p, q, e, p0 )| (TS) −1 (id⊗P⊗id)◦(id⊗T⊗id) S−1 PS / |F (p, p0 )| ⊗ |F (p0 , q)| / |F 12 (p′ , q, e, p) · F 14 (p, q, e, p′ )| 0 0 / |F (p, p′ )| ⊗ |F (p′ , q)| . 0 0 However, using our alternative description of the change-of-base-point isomorphism B(p0 , p′0 ) : |F (p, p0 )| ⊗ |F (p0 , q)| / |F (p, p′0 )| ⊗ |F (p′0 , q)| from Equation (16.2) we see that the composition of isomorphisms in Equation (16.3) agrees with the composition B(p0 , p′0 ) ◦ ∆−1 e : |F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)| / |F (p, p′0 )| ⊗ |F (p′0 , q)| . We thus conclude that
(∆′e )−1 ◦ B(p0 , p′0 ) ⊗ B(p0 , p′0 ) = B(p0 , p′0 ) ◦ ∆−1 e : |F (p, p0 )| ⊗ |F (p0 , e)| ⊗ |F (e, p0 )| ⊗ |F (p0 , q)| / |F (p, p′0 )| ⊗ |F (p′0 , q)| , and this ends the proof of the proposition.  16.2. The dual change of base point. Throughout this subsection we fix two indices λ, µ ∈ Λ together with two idempotents p and q in the unital ring R, both agreeing with the base points p0 and p′0 modulo the ideal I ⊆ R. We develop an alternative version of the change-of-base-point isomorphism which we refer to as the dual change-of-base-point isomorphism. It is described by the isomorphism B† (p0 , p′0 ) : L†q (λ, µ) ⊗ Lp (λ, µ) / (L′q )† (λ, µ) ⊗ L′p (λ, µ) , defined as the composition B† (p0 , p′0 ) : |F (p0 , q)| ⊗ |F (p, p0 )| S−1 PS TS / F 23 (q, p, p0 )F 13 (p0 , p, q) / F 23 (q, p, p′0 )F 13 (p′0 , p, q) (TS)−1 / |F (p′0 , q)| ⊗ |F (p, p′0 )| , where we are suppressing the tuples of indices (λ, µ) and (λ, λ, µ). The stabilisations appearing in the middle are adding the matrices e21 (πλ (1 − p0 )) and e21 (πλ (1 − p′0 )), respectively (see the beginning of Subsection 16.1 for an explanation of the notation). Remark that the perturbation isomorphism in the middle exists by the argument given in the proof of Lemma 11.1. We will often suppress the pair of idempotents (p0 , p′0 ) and simply denote the dual change og base point isomorphism by B† . On many occasions we are going to diverge a bit from the notation introduced in Notation 9.3 and only refer to the idempotents which are directly involved in the Fredholm operators. So instead of writing F ij (λ1 , λ2 , . . . , λn )(p1 , p2 , . . . , pn ) we shall apply the notation F ij (λ1 , λ2 , . . . , λn )(pi , pj ) leaving it to the reader to guess the remaining idempotents (they are hopefully going to be clear from the context). 72 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Lemma 16.1. The automorphisms cp and c†p : (C, 0) cp : (C, 0) c†p : (C, 0) ψ −1 ϕ B(p0 ,p′0 ) / Lp ⊗ L†p / L†p ⊗ Lp / (C, 0) defined by B † (p0 ,p′0 ) ψ′ / L′p ⊗ (L′p )† / (L′p )† ⊗ L′p / (C, 0) and ′ −1 (ϕ ) / (C, 0) are both equal to the identity. Proof. The case of cp follows by combining Lemma 9.7 and Proposition 11.4. Indeed, it suffices to show that c2p = cp and an application of Lemma 9.7 implies that this amounts to verifying the commutativity of the diagram / L′p ⊗ (L′p )† ⊗ L′p ⊗ (L′p )† Lp ⊗ L†p ⊗ Lp ⊗ L†p B(p0 ,p′0 )⊗B(p0 ,p′0 ) id⊗(ϕ′ )−1 ⊗id id⊗ϕ−1 ⊗id  Lp ⊗ L†p  / L′p ⊗ (L′p )† B(p0 ,p′0 ) But the commutativity of this diagram follows immediately from Proposition 11.4. We therefore focus on / (C, 0). c†p : (C, 0) We show that (c†p )2 = c†p . By Lemma 9.7, this amounts to verifying that the diagram (16.4) L†p ⊗ Lp ⊗ L†p ⊗ Lp / (L′p )† ⊗ L′p ⊗ (L′p )† ⊗ L′p B† ⊗B† id⊗ψ ′ ⊗id id⊗ψ⊗id  L†p ⊗ Lp  / (L′ )† ⊗ L′ p p B† is commutative. To prove that this holds, we notice that the diagram (16.5) L†p ⊗ Lp ⊗ L†p ⊗ Lp / (L′p )† ⊗ L′p ⊗ (L′p )† ⊗ L′p B† ⊗B† TS TS  F 34 (p, p0 )F 24 (p0 , p)F 24 (p, p0 )F 14 (p0 , p) S−1 PS  / F 34 (p, p′0 )F 24 (p′0 , p)F 24 (p, p′0 )F 14 (p′0 , p) commutes, where we are suppressing the tuple of indices (λ, λ, λ, µ). Remark that the stabilisations appearing in the lower row are adding the matrices e31 (πλ (1 − p0 )) and e31 (πλ (1 − p′0 )). But the commutativity of the diagram in Equation (16.5) implies that the diagram L†p ⊗ Lp ⊗ L†p ⊗ Lp / (L′p )† ⊗ L′p ⊗ (L′p )† ⊗ L′p B† ⊗B† (TS)◦(id⊗ψ ′ ⊗id) (TS)◦(id⊗ψ⊗id)  F 34 (p, p0 )F 14 (p0 , p) −1 S  / F 34 (p, p′0 )F 14 (p′0 , p) PS is commutative as well. The commutativity of the diagram in Equation (16.4) now follows.  The next result does to some extent explain the relationship between the dual change-of-base-point isomorphism and the change-of-base-point isomorphism introduced in Section 11. As usual we suppress the pair of indices (λ, µ) and the pair of idempotents (p0 , p′0 ) from the notation. We recall that ǫ denotes the commutativity constraint in the Picard category of Z-graded complex lines, see Notation 2.2. Proposition 16.2. The following diagram is commutative: (16.6) B† ⊗B L†q ⊗ Lp ⊗ Lq ⊗ L†p / (L′ )† ⊗ L′ ⊗ L′ ⊗ (L′ )† q p q p id⊗ǫ  L†q ⊗ Lq ⊗ L†p ⊗ Lp id⊗ǫ ′ ϕϕ −1 ′ ⊗ϕ ϕ  / (L′q )† ⊗ L′q ⊗ (L′p )† ⊗ L′p −1 A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 73 Proof. We start by noticing that the diagram L†q ⊗ Lp ⊗ Lq ⊗ L†p B† ⊗B / (L′q )† ⊗ L′p ⊗ L′q ⊗ (L′p )† TS TS  F 13 (p0 , p)F 14 (q, p0 )F 23 (p, p0 )F 13 (p0 , q)  / F 13 (p′0 , p)F 14 (q, p′0 )F 23 (p, p′0 )F 13 (p′0 , q) S−1 PS is commutative, where we suppress the tuple of indices (λ, λ, µ, µ) and where the stabilisations appearing in the lower row are adding the matrices e21 (πλ (1 − p0 )) + e34 (πµ (1 − p0 )) and e21 (πλ (1 − p′0 )) + e34 (πµ (1 − p′0 )), respectively. Using now that F 14 (q, p0 ) and F 23 (p, p0 ) commute since they operate on different direct summands (and similarly with p0 replaced by p′0 ) we achieve from Proposition 5.4 that the diagram L†q ⊗ Lp ⊗ Lq ⊗ L†p (16.7) B† ⊗B / (L′q )† ⊗ L′p ⊗ L′q ⊗ (L′p )† TS(id⊗ǫ⊗id) TS(id⊗ǫ⊗id)  F 13 (p0 , p)F 23 (p, p0 )F 14 (q, p0 )F 13 (p0 , q) −1 S PS  / F 13 (p′ , p)F 23 (p, p′ )F 14 (q, p′ )F 13 (p′ , q) 0 0 0 0 is commutative as well. In particular, when p = q we obtain from Lemma 16.1 that the diagram L†p ⊗ Lp ⊗ Lp ⊗ L†p (16.8) ϕ′ ϕ−1 ⊗(ψ ′ )−1 ψ / (L′p )† ⊗ L′p ⊗ L′p ⊗ (L′p )† TS(id⊗ǫ⊗id) TS(id⊗ǫ⊗id)  F 13 (p0 , p)F 23 (p, p0 )F 14 (p, p0 )F 13 (p0 , p) −1 S PS  / F 13 (p′0 , p)F 23 (p, p′0 )F 14 (p, p′0 )F 13 (p′0 , p) is commutative. To continue, we remark that the diagram (16.9) F 13 (p0 , p)F 23 (p, p0 )F 14 (q, p0 )F 13 (p0 , q) S−1 PS / F 13 (p′0 , p)F 23 (p, p′0 )F 14 (q, p′0 )F 13 (p′0 , q) S−1 PS  F 13 (p0 , p)F 23 (p, p0 )F 14 (p, p0 )F 13 (p0 , p) S−1 PS −1 S PS  / F 13 (p′0 , p)F 23 (p, p′0 )F 14 (p, p′0 )F 13 (p′0 , p) is commutative, where the two stabilisations appearing in each of the two columns are adding the matrices e43 (πµ (1 − q)) and e43 (πµ (1 − p)), respectively. By combining the commutative diagrams in Equation (16.7)(16.9) we obtain the commutative diagram L†q ⊗ Lp ⊗ Lq ⊗ L†p (16.10) B† ⊗B / (L′q )† ⊗ L′p ⊗ L′q ⊗ (L′p )† ǫ(TS)−1 (S−1 PS)(TS)ǫ  L†p ⊗ Lp ⊗ Lp ⊗ L†p ǫ(TS)−1 (S−1 PS)(TS)ǫ ϕ′ ϕ−1 ⊗(ψ ′ )−1 ψ  / (L′p )† ⊗ L′p ⊗ L′p ⊗ (L′p )† where we clarify that the vertical commutativity constraints are really of the form id⊗ǫ⊗id. We now observe that the left vertical isomorphism in the above diagram agrees with the composition of isomorphisms |F (p0 , q)| ⊗ |F (p, p0 )| ⊗ |F (q, p0 )| ⊗ |F (p0 , p)| B(q,p)⊗id⊗2 id⊗ǫ⊗id id⊗ǫ⊗id / |F (p0 , q)| ⊗ |F (q, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| / |F (p0 , p)| ⊗ |F (p, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| / |F (p0 , p)| ⊗ |F (p, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| 74 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON and hence, using Lemma 16.1, it does in fact agree with the composition |F (p0 , q)| ⊗ |F (p, p0 )| ⊗ |F (q, p0 )| ⊗ |F (p0 , p)| ψq ⊗id⊗2 id⊗ǫ⊗id / |F (p, p0 )| ⊗ |F (p0 , p)| id⊗ǫ⊗id ψp−1 ⊗id⊗2 / |F (p0 , q)| ⊗ |F (q, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| / |F (p0 , p)| ⊗ |F (p, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| / |F (p0 , p)| ⊗ |F (p, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| , where the subscripts p and q on the duality isomorphism ψ indicate that these idempotents function as base points. Remark that, changing the point of view and using p0 as a base point, we have that ψq = / (C, 0). Using Lemma 9.7 we then notice that / (C, 0) and similarly ψp = ϕ−1 : L†p ⊗ Lp ϕ−1 : L†q ⊗ Lq the composition of isomorphisms |F (p, p0 )| ⊗ |F (p0 , p)| id⊗ǫ⊗id ψp−1 ⊗id⊗2 / |F (p0 , p)| ⊗ |F (p, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| / |F (p0 , p)| ⊗ |F (p, p0 )| ⊗ |F (p, p0 )| ⊗ |F (p0 , p)| ϕ−1 ⊗ψ / (C, 0) agrees with the composition of isomorphisms ǫ |F (p, p0 )| ⊗ |F (p0 , p)| / |F (p0 , p)| ⊗ |F (p, p0 )| ϕ−1 / (C, 0) . Combining these results we obtain that the composition of the left vertical isomorphism in Equation (16.10) / (C, 0) agrees with the trivialisation with the trivialisation ϕ−1 ⊗ ψ : L†p ⊗ Lp ⊗ Lp ⊗ L†p L†q ⊗ Lp ⊗ Lq ⊗ L†p id⊗ǫ / L†q ⊗ Lq ⊗ L†p ⊗ Lp ϕ−1 ⊗ϕ−1 / (C, 0) . Since a similar description applies when p0 is replaced by p′0 , the result of the present proposition follows from the commutativity of the diagram in Equation (16.10).  16.3. The symmetrised change of base point. Let us fix two indices λ and µ in the index set Λ. We now derive one more alternative version of the change-of-base-point isomorphism. This alternative version is more symmetric in the indices λ and µ and we refer to it as the symmetrised change-of-base-point isomorphism. We are applying notational conventions similar to those described in the beginning of Subsection 16.1 and 16.2. We need a small lemma (which can be compared with Lemma 11.1). Remark that the invertible operator Ω14 (p0 ) appearing in the statement agrees with Ω(λ, µ)(p0 ) up to stabilisation with the idempotent πλ (p) ⊕ πµ (q) (and similarly with p0 replaced by p′0 ), see Notation 9.3. We are also omitting the tuples of indices (λ, λ, µ, µ) and (λ, µ) from the notation. Lemma 16.3. The difference of the two Fredholm operators F 13 (p0 , q)F 24 (p, p0 )Ω14 (p0 ) + e21 (πλ (1 − p0 )) + e34 (πµ (1 − p0 )) F 13 (p′0 , q)F 24 (p, p′0 )Ω14 (p′0 ) + e21 (πλ (1 − p′0 )) + e34 (πµ (1 − and p′0 )) , with domain πλ (1)H ⊕ πλ (p)H ⊕ πµ (q)H ⊕ πµ (1)H and codomain πλ (q)H ⊕ πλ (1)H ⊕ πµ (1)H ⊕ πµ (p)H, is of trace class. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 75 Proof. This is a consequence of Lemma 9.4 and Lemma 9.5 together with the following computation modulo L 1 (H⊕4 ): Ω13 (1)Ω24 (1)Ω14 (1) · (πλ (p0 − p′0 ) ⊕ 0 ⊕ 0 ⊕ πµ (p0 − p′0 ))     πλ (1) 0 0 0 0 0 0 0   0 0 πλ (1)πµ (1)  0 πλ (1) 0 0   · 0 ∼1      πµ (1)πλ (1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   0 0 0 πλ (1)πµ (1)   0 0 0 0  · (πλ (p0 − p′0 ) ⊕ 0 ⊕ 0 ⊕ πµ (p0 − p′0 )) ·   0 0 0 0 πµ (1)πλ (1) 0 0 0     0 0 0 πµ (p0 − p′0 ) ∼1 ⊕ . πλ (p0 − p′0 ) 0 0 0  We recall the notation L(·) and R(·) from Example 3.4 and apply Lemma 16.3 to define the symmetrised change-of-base-point isomorphism e 0 , p′ ) : Lp ⊗ L† ⊗ Lq ⊗ L† B(p q 0 q as the composition / L′ ⊗ (L′ )† ⊗ L′ ⊗ (L′ )† p q q q (16.11) e 0 , p′ ) : |F (p, p0 )| ⊗ |F (p0 , q)| ⊗ |F (q, p0 )| ⊗ |F (p0 , q)| B(p 0 R(Ω14 (p0 ))(TS)⊗L(Ω14 (p0 ))(TS) S−1 PS⊗S−1 PS / F 13 (p0 , q)F 24 (p, p0 )Ω14 (p0 ) ⊗ Ω14 (p0 )F 24 (p0 , q)F 13 (q, p0 ) / F 13 (p′0 , q)F 24 (p, p′0 )Ω14 (p′0 ) ⊗ Ω14 (p′0 )F 24 (p′0 , q)F 13 (q, p′0 ) (TS)−1 R(Ω14 (p′0 ))⊗(TS)−1 L(Ω14 (p′0 )) / |F (p, p′0 )| ⊗ |F (p′0 , q)| ⊗ |F (q, p′0 )| ⊗ |F (p′0 , q)| . The stabilisations surrounding the two perturbation isomorphisms are adding the matrices e21 (πλ (1 − p0)) + e34 (πµ (1 − p0 )) and e12 (πλ (1 − p0 )) + e43 (πµ (1 − p0 )) and similarly with p0 replaced by p′0 . The next result explains the relationship between the symmetrised change-of-base-point isomorphism and the change-of-base-point isomorphism introduced in Section 11. Proposition 16.4. The following diagram is commutative: Lp ⊗ L†q ⊗ Lq ⊗ L†q e 0 ,p′ ) B(p 0 / L′p ⊗ (L′q )† ⊗ L′q ⊗ (L′q )† id⊗(ϕ′ )−1 ⊗id id⊗ϕ−1 ⊗id  Lp ⊗ L†q B(p0 ,p′0 )  / L′ ⊗ (L′ )† p q Proof. We start by noticing that the symmetrised change-of-base-point isomorphism agrees with the following composition of isomorphisms: (16.12) Lp ⊗ L†q ⊗ Lq ⊗ L†q S−1 PS L(Ω15 (p0 ))R(Ω14 (p0 ))TS / Ω15 (p0 )F 25 (p0 , q)F 13 (q, p0 )F 13 (p0 , q)F 24 (p, p0 )Ω14 (p0 ) / Ω15 (p′0 )F 25 (p′0 , q)F 13 (q, p′0 )F 13 (p′0 , q)F 24 (p, p′0 )Ω14 (p′0 ) (TS)−1 L(Ω15 (p′0 ))R(Ω14 (p′0 )) / L′p ⊗ (L′q )† ⊗ L′q ⊗ (L′q )† , where the stabilisations surrounding the perturbation isomorphism are adding the matrix e11 (πλ (1 − p0 )) + e54 (πµ (1 − p0 )) (and similarly with p0 replaced by p′0 ) and where the tuple of indices (λ, λ, µ, µ, µ) has been 76 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON suppressed from the notation. We then remark that the composition of isomorphisms in Equation (16.12) agrees with the following composition of isomorphisms: id⊗ϕ−1 id Lp ⊗ L†q ⊗ Lq ⊗ L†q S−1 PS / Lp ⊗ L†q L(Ω14 (p0 ))R(Ω13 (p0 ))TS / Ω14 (p0 )F 24 (p0 , q)F 23 (p, p0 )Ω13 (p0 ) / Ω14 (p′ )F 24 (p′ , q)F 23 (p, p′ )Ω13 (p′ ) 0 0 0 0 (TS)−1 L(Ω14 (p′0 ))R(Ω13 (p′0 )) / L′p ⊗ (L′q )† id⊗ϕ′ ⊗id / L′p ⊗ (L′q )† ⊗ L′q ⊗ (L′q )† , where the non-obvious stabilisations are adding the matrices e11 (πλ (1 − p0 )) + e43 (πµ (1 − p0 )) and e11 (πλ (1 − p′0 )) + e43 (πµ (1 − p′0 )), respectively. Combining these results, we see that it suffices to show that the change/ L′p ⊗ (L′q )† agrees with the composition of isomorphisms of-base-point isomorphism B : Lp ⊗ L†q (16.13) Lp ⊗ L†q L(Ω14 (p0 ))R(Ω13 (p0 ))TS S−1 PS / Ω14 (p0 )F 24 (p0 , q)F 23 (p, p0 )Ω13 (p0 ) / Ω14 (p′0 )F 24 (p′0 , q)F 23 (p, p′0 )Ω13 (p′0 ) (TS)−1 L(Ω14 (p′0 ))R(Ω13 (p′0 )) / L′p ⊗ (L′q )† . To prove this result we let
Ω14 (1 − p0 ) ∈ L πλ (1)H ⊕ πλ (q)H ⊕ πµ (p)H ⊕ πµ (1)H
Ω13 (1 − p0 ) ∈ L πλ (1)H ⊕ πλ (p)H ⊕ πµ (1)H ⊕ πµ (q)H and be stabilized by 0 and remark that
Ω14 (1 − p0 ) · F 24 (1, p0 , p, q)F 23 (1, p, p0 , q) + e43 (πµ (1 − p0 )) Ω13 (1 − p0 )
= Ω14 (1 − p0 ) e11 (πλ (1 − p0 )) + e43 (πµ (1 − p0 )) Ω13 (1 − p0 ) = e11 (πλ (1 − p0 )) + e43 (πµ (1 − p0 )) , where the second identity follows from Lemma 9.5. We thus obtain that Ω14 (p0 )F 24 (p0 , q)F 23 (p, p0 )Ω13 (p0 ) + e11 (πλ (1 − p0 )) + e43 (πµ (1 − p0 ))
= (Ω14 (p0 ) + Ω14 (1 − p0 )) · F 24 (1, p0 , p, q)F 23 (1, p, p0 , q) + e43 (πµ (1 − p0 )) · (Ω13 (p0 ) + Ω13 (1 − p0 )) . A similar result applies to the case where p0 has been replaced by p′0 . Using this computation one may verify that the composition of isomorphisms in Equation (16.13) agrees with the following composition of isomorphisms: Lp ⊗ L†q STS / F 24 (1, p0 , p, q)F 23 (1, p, p0 , q) + e43 (πµ (1 − p0 )) L(Ω14 (p0 )+Ω14 (1−p0 ))R(Ω13 (p0 )+Ω13 (1−p0 )) / Ω14 (p0 )F 24 (p0 , q)F 23 (p, p0 )Ω13 (p0 ) + e11 (πλ (1 − p0 )) + e43 (πµ (1 − p0 )) (16.14) P / Ω14 (p′0 )F 24 (p′0 , q)F 23 (p, p′0 )Ω13 (p′0 ) + e11 (πλ (1 − p′0 )) + e43 (πµ (1 − p′0 )) L(Ω14 (p′0 )+Ω14 (1−p′0 ))R(Ω13 (p′0 )+Ω13 (1−p′0 )) (STS)−1 / F 24 (1, p′0 , p, q)F 23 (1, p, p′0 , q) + e43 (πµ (1 − p′0 )) / L′p ⊗ (L′q )† . This latter composition of isomorphisms can be seen to coincide with the change-of-base-point isomorphism / L′ ⊗ (L′q )† multiplied with the square of the Fredholm determinant det Ω(λ, µ)(p′0 ) + B : Lp ⊗ L†q
p

′ Ω(λ, µ)(1 − p0 ) Ω(λ, µ)(p0 ) + Ω(λ, µ)(1 − p0 ) . The result of the proposition therefore follows since


2 det Ω(λ, µ)(p′0 ) + Ω(λ, µ)(1 − p′0 ) Ω(λ, µ)(p0 ) + Ω(λ, µ)(1 − p0 ) = 1.  A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 77 16.4. Binary and ternary versions of the change of base point. Throughout this subsection we fix three elements in the index set λ, µ, ν ∈ Λ together with two idempotents p and q in R, both agreeing with the base points p0 and p′0 modulo the ideal I ⊆ R. As an extra technical step towards establishing the functoriality of the change-of-base-point isomorphism, we now develop binary and ternary versions of this isomorphism. Moreover, these higher analogues of the change-of-base-point isomorphism will be related to tensor products of the original change-of-base-point isomorphisms. We begin with two lemmas asserting the existence of perturbation isomorphisms. In the first lemma we are suppressing the tuple of indices (λ, λ, µ, ν) and in the second lemma we are suppressing the tuple of indices (λ, λ, λ, µ, ν). Lemma 16.5. The difference of the following two Fredholm operators F 34 (p0 , q) · F 14 (q, p0 ) · F 24 (p0 , p) · F 34 (p, p0 ) + e12 (πλ (1 − p0 )) + e44 (πν (1 − p0 )) F 34 (p′0 , q) ·F 14 (q, p′0 ) ·F 24 (p′0 , p) + e12 (πλ (1 − p′0 )) + and 34 · F (p, p′0 ) e44 (πν (1 − p′0 )) , both acting from the Hilbert space πλ (q)H⊕πλ (1)H⊕πµ (p)H⊕πν (1)H to the Hilbert space πλ (1)H⊕πλ (p)H⊕ πµ (q)H ⊕ πν (1)H, is of trace class. Proof. We compute modulo L 1 (H⊕4 ), using Lemma 9.4 and Assumption 9.1 together with the fact that p0 − p′0 ∈ I: F 34 (p0 , q) · F 14 (q, p0 ) · F 24 (p0 , p) · F 34 (p, p0 ) ∼1 − F 34 (p′0 , q) · F 14 (q, p′0 ) · F 24 (p′0 , p) · F 34 (p, p′0 )  πλ (1) 0 0 
0 0 0 πλ (p0 − p′0 ) ⊕ 0⊕2 ⊕ πν (p0 − p′0 ) ·   0 0 0 0 0 πν (1)πµ (1)    0 0 0 πλ (1)πν (1) 0 0 0  0 0   0 0 0 0 0 · ·  0 0 πµ (1)   0 0 0 πµ (1) 0 0 0 0 0 πν (1)πλ (1) 0   0 0 0 0   0 πλ (1) 0 0  ·  0 0 0 πµ (1)πν (1)  0 0 0 0  0 0   0  0  0 0   0  0 ∼1 e12 (πλ (p0 − p′0 )) + e44 (πν (p0 − p′0 )) .  Lemma 16.6. The difference of the following two Fredholm operators Ω14 (p0 ) · F 24 (p0 , q)F 45 (p0 , q)F 25 (q, p0 ) · F 35 (p0 , p)F 45 (p, p0 )F 34 (p, p0 ) · Ω14 (p0 ) + πλ (1 − p0 ) ⊕ 0 ⊕ 0 ⊕ πµ (1 − p0 ) ⊕ πν (1 − p0 ) and Ω14 (p′0 ) · F 24 (p′0 , q)F 45 (p′0 , q)F 25 (q, p′0 ) · F 35 (p′0 , p)F 45 (p, p′0 )F 34 (p, p′0 ) · Ω14 (p′0 ) + πλ (1 − p′0 ) ⊕ 0 ⊕ 0 ⊕ πµ (1 − p′0 ) ⊕ πν (1 − p′0 ) , both acting on the Hilbert space πλ (1)H ⊕ πλ (q)H ⊕ πλ (p)H ⊕ πµ (1)H ⊕ πν (1)H, is of trace class. Proof. The argument is similar to the argument given in the proof of Lemma 16.5 and will therefore not be repeated here.  78 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON As a consequence of Lemma 16.5, we may define the following binary change-of-base-point isomorphism N(p0 , p′0 ) : Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) / L′ (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) ⊗ (L′ )† (µ, ν) p p q q as the composition |F (µ, ν)(p, p0 )| ⊗ |F (λ, ν)(p0 , p)| ⊗ |F (λ, ν)(q, p0 )| ⊗ |F (µ, ν)(p0 , q)| TS (16.15) / F 34 (p0 , q) · F 14 (q, p0 ) · F 24 (p0 , p) · F 34 (p, p0 ) S−1 PS (TS)−1 / F 34 (p′0 , q) · F 14 (q, p′0 ) · F 24 (p′0 , p) · F 34 (p, p′0 ) / |F (µ, ν)(p, p′0 )| ⊗ |F (λ, ν)(p′0 , p)| ⊗ |F (λ, ν)(q, p′0 )| ⊗ |F (µ, ν)(p′0 , q)| , where we are suppressing the tuple of indices (λ, λ, µ, ν) and where the stabilisations in the middle add the invertible operators e12 (πλ (1 − p0 )) + e44 (πν (1 − p0 )) and e12 (πλ (1 − p′0 )) + e44 (πν (1 − p′0 )), respectively. Similarly, using Lemma 16.6, we define the ternary change-of-base-point isomorphism M(p0 , p′0 ) : Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ L†q (λ, µ) / L′p (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′q )† (λ, µ) as the composition M(p0 , p′0 ) : |F (λ, µ)(p, p0 )| ⊗ |F (µ, ν)(p, p0 )| ⊗ |F (λ, ν)(p0 , p)| ⊗ |F (λ, ν)(q, p0 )| ⊗ |F (µ, ν)(p0 , q)| ⊗ |F (λ, µ)(p0 , q)| R(Ω14 (p0 ))L(Ω14 (p0 ))TS / Ω14 (p0 ) · F 24 (p0 , q)F 45 (p0 , q)F 25 (q, p0 ) · F 35 (p0 , p)F 45 (p, p0 )F 34 (p, p0 ) · Ω14 (p0 ) (16.16) S−1 PS / Ω14 (p′0 ) · F 24 (p′0 , q)F 45 (p′0 , q)F 25 (q, p′0 ) · F 35 (p′0 , p)F 45 (p, p′0 )F 34 (p, p′0 ) · Ω14 (p′0 ) (TS)−1 R(Ω14 (p′0 ))L(Ω14 (p′0 )) / |F (λ, µ)(p, p′0 )| ⊗ |F (µ, ν)(p, p′0 )| ⊗ |F (λ, ν)(p′0 , p)| ⊗ |F (λ, ν)(q, p′0 )| ⊗ |F (µ, ν)(p′0 , q)| ⊗ |F (λ, µ)(p′0 , q)| , where we are suppressing the tuple of indices (λ, λ, λ, µ, ν) from the notation and where the stabilisations in the middle are now adding the invertible operators e11 (πλ (1 − p0 )) + e44 (πµ (1 − p0 )) + e55 (πν (1 − p0 )) and e11 (πλ (1 − p′0 )) + e44 (πµ (1 − p′0 )) + e55 (πν (1 − p′0 )), respectively. In the next proposition we express the binary change-of-base-point isomorphism in terms of the changeof-base-point isomorphism and the dual change-of-base-point isomorphism (see Definition 11.2 and the beginning of Subsection 16.2): Proposition 16.7. The isomorphism of Z-graded complex lines B(p0 , p′0 ) ⊗ B† (p0 , p′0 ) : Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) / L′p (µ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) agrees with the composition of isomorphisms of Z-graded complex lines: Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) (16.17) N(p0 ,p′0 ) ǫ ǫ / Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) / L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) / L′ (µ, ν) ⊗ (L′ )† (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) . p q p q A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 79 Proof. We suppress the tuple of indices (λ, λ, µ, ν, ν) from the notation. Our first claim is that the binary change-of-base-point isomorphism N(p0 , p′0 ) agrees with the composition of isomorphisms F (µ, ν)(p, p0 ) ⊗ F (λ, ν)(p0 , p) ⊗ F (λ, ν)(q, p0 ) ⊗ F (µ, ν)(p0 , q) TS (16.18) / F 34 (p0 , q) · F 15 (q, p0 ) · F 25 (p0 , p) · F 35 (p, p0 ) S−1 PS / F 34 (p′0 , q) · F 15 (q, p′0 ) · F 25 (p′0 , p) · F 35 (p, p′0 ) (TS)−1 / F (µ, ν)(p, p′0 ) ⊗ F (λ, ν)(p′0 , p) ⊗ F (λ, ν)(q, p′0 ) ⊗ F (µ, ν)(p′0 , q) , where the stabilisations in the middle use the invertible operators e12 (πλ (1 − p0 )) + e45 (πν (1 − p0 )) and e12 (πλ (1 − p′0 )) + e45 (πν (1 − p′0 )), respectively. / (C, 0), To prove the above claim, we recall from Lemma 16.1 that the automorphism cq (µ, ν) : (C, 0) defined as the composition (C, 0) ψ −1 / |F (µ, ν)(q, p0 )| ⊗ |F (µ, ν)(p0 , q)| S−1 PS (TS)−1 TS / F 34 (p0 , q) · F 35 (q, p0 ) / F 34 (p′0 , q) · F 35 (q, p′0 ) / |F (µ, ν)(q, p′ )| ⊗ |F (µ, ν)(p′ , q)| 0 0 ψ′ / (C, 0) , is equal to the identity map. We are here considering F 34 (p0 , q) · F 35 (q, p0 ) as a Fredholm operator acting on the Hilbert space πλ (p0 )H ⊕ πλ (p)H ⊕ πµ (q)H ⊕ πν (q)H ⊕ πν (p0 )H and we apply the invertible operator e11 (πλ (1 − p0 )) + e45 (πν (1 − p0 )) for the (non-obvious) stabilisation procedure (a similar comment applies to the primed version). This observation allows us to replace N(p0 , p′0 ) with N(p0 , p′0 ) · cq (µ, ν). However, this latter isomorphism can now be described as the composition of isomorphisms |F (µ, ν)(p, p0 )| ⊗ |F (λ, ν)(p0 , p)| ⊗ |F (λ, ν)(q, p0 )| ⊗ |F (µ, ν)(p0 , q)| id⊗4 ⊗ψ −1 / |F (µ, ν)(p, p0 )| ⊗ |F (λ, ν)(p0 , p)| ⊗ |F (λ, ν)(q, p0 )| ⊗ |F (µ, ν)(p0 , q)| ⊗ |F (µ, ν)(q, p0 )| ⊗ |F (µ, ν)(p0 , q)| TS / F 34 (p0 , q) · F 35 (q, p0 ) · F 35 (p0 , q) · F 15 (q, p0 ) · F 25 (p0 , p) · F 35 (p, p0 ) S−1 PS / F 34 (p′0 , q) · F 35 (q, p′0 ) · F 35 (p′0 , q) · F 15 (q, p′0 ) · F 25 (p′0 , p) · F 35 (p, p′0 ) (TS)−1 / |F (µ, ν)(p, p′ )| ⊗ |F (λ, ν)(p′ , p)| ⊗ |F (λ, ν)(q, p′ )| 0 0 0 ⊗ |F (µ, ν)(p′0 , q)| ⊗ |F (µ, ν)(q, p′0 )| ⊗ |F (µ, ν)(p′0 , q)| id⊗4 ⊗ψ ′ / |F (µ, ν)(p, p′0 )| ⊗ |F (λ, ν)(p′0 , p)| ⊗ |F (λ, ν)(q, p′0 )| ⊗ |F (µ, ν)(p′0 , q)| , where the stabilisation appearing in the middle adds the invertible operator e12 (πλ (1 − p0 )) + e45 (πν (1 − p0 )) (and similarly for the primed case). Using Lemma 9.7 we may cancel out F 35 (q, p0 ) · F 35 (p0 , q) and F 35 (q, p′0 ) · F 35 (p′0 , q) from the above expression and we arrive exactly at the composition of isomorphisms given in Equation (16.18). This establishes our first claim. The point of the alternative description of the binary change-of-base-point isomorphism given in Equation (16.18) is that the Fredholm operators F 34 (p0 , q) and F 15 (q, p0 ) · F 25 (p0 , p) operate on different direct summands (and similarly for the primed versions). Hence, by the commutativity property of the torsion 80 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON isomorphism (Proposition 5.4) we obtain that the isomorphism Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) N(p0 ,p′0 ) ǫ ǫ / Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) / L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) / L′ (µ, ν) ⊗ (L′ )† (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) p q p q agrees with the composition |F (µ, ν)(p, p0 )| ⊗ |F (µ, ν)(p0 , q)| ⊗ |F (λ, ν)(p0 , p)| ⊗ |F (λ, ν)(q, p0 )| TS / F 15 (q, p0 ) · F 25 (p0 , p) · F 34 (p0 , q) · F 35 (p, p0 ) S−1 PS (TS)−1 / F 15 (q, p′0 ) · F 25 (p′0 , p) · F 34 (p′0 , q) · F 35 (p, p′0 ) / |F (µ, ν)(p, p′0 )| ⊗ |F (µ, ν)(p′0 , q)| ⊗ |F (λ, ν)(p′0 , p)| ⊗ |F (λ, ν)(q, p′0 )| , where the stabilisations appearing in the middle add the invertible operators e12 (πλ (1 − p0 ))+ e45 (πν (1 − p0 )) and e12 (πλ (1 − p′0 )) + e45 (πν (1 − p′0 )), respectively. But this latter isomorphism can be seen to agree with the isomorphism B(p0 , p′0 ) ⊗ B† (p0 , p′0 ) and the proposition is therefore proved.  We end this subsection by providing a description of the ternary change-of-base-point isomorphism in terms of tensor products of the change-of-base-point isomorphism and the dual change-of-base-point isomorphism: Proposition 16.8. The isomorphism of Z-graded complex lines B(p0 , p′0 ) ⊗ B(p0 , p′0 ) ⊗ B† (p0 , p′0 ) : Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) agrees with the composition of isomorphisms of Z-graded complex lines Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ǫ / Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L† (λ, ν) ⊗ Lq (λ, ν) ⊗ L† (µ, ν) ⊗ L† (λ, µ) p M(p0 ,p′0 ) ǫ q q / L′p (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′q )† (λ, µ) / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) . Proof. We suppress the tuple of indices (λ, λ, λ, µ, µ, ν) from the notation. A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 81 Observe first that M(p0 , p′0 ) agrees with the composition of isomorphisms Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ L†q (λ, µ) id⊗5 ⊗ϕ⊗id / Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ L†q (λ, µ) ⊗ Lq (λ, µ) ⊗ L†q (λ, µ) L(Ω15 (p0 ))R(Ω15 (p0 ))TS / Ω15 (p0 ) · F 25 (p0 , q)F 14 (q, p0 )F 14 (p0 , q)F 56 (p0 , q)F 26 (q, p0 ) · F 36 (p0 , p)F 56 (p, p0 )F 35 (p, p0 ) · Ω15 (p0 ) (16.19) S−1 PS / Ω15 (p′0 ) · F 25 (p′0 , q)F 14 (q, p′0 )F 14 (p′0 , q)F 56 (p′0 , q)F 26 (q, p′0 ) · F 36 (p′0 , p)F 56 (p, p′0 )F 35 (p, p′0 ) · Ω15 (p′0 ) (TS)−1 L(Ω15 (p′0 ))R(Ω15 (p′0 )) / L′p (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′q )† (λ, µ) ⊗ L′q (λ, µ) ⊗ (L′q )† (λ, µ) id⊗5 ⊗(ϕ′ )−1 ⊗id / L′p (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′q )† (λ, µ) , where the stabilisations in the middle are adding the invertible operators e11 (πλ (1 − p0 )) + e55 (πµ (1 − p0 )) + e66 (πν (1−p0 )) and e11 (πλ (1−p′0 ))+e55 (πµ (1−p′0 ))+e66 (πν (1−p′0 )), respectively. Next, using that F 14 (p0 , q) and F 56 (p0 , q)F 26 (q, p0 ) · F 36 (p0 , p)F 56 (p, p0 ) operate on different direct summands (and similarly for the primed version), we obtain from Proposition 5.4 that the composition of isomorphisms in Equation (16.19) agrees with the composition Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ L†q (λ, µ) id⊗5 ⊗ϕ⊗id / Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L† (λ, ν) ⊗ Lq (λ, ν) ⊗ L† (µ, ν) p ⊗ ǫ L†q (λ, µ) q ⊗ Lq (λ, µ) ⊗ L†q (λ, µ) / Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ Lq (λ, µ) ⊗ L†q (λ, µ) L(Ω15 (p0 ))R(Ω15 (p0 ))TS / Ω15 (p0 ) · F 25 (p0 , q)F 14 (q, p0 )F 56 (p0 , q)F 26 (q, p0 ) · F 36 (p0 , p)F 56 (p, p0 )F 14 (p0 , q)F 35 (p, p0 ) · Ω15 (p0 ) (16.20) S−1 PS / Ω15 (p′0 ) · F 25 (p′0 , q)F 14 (q, p′0 )F 56 (p′0 , q)F 26 (q, p′0 ) · F 36 (p′0 , p)F 56 (p, p′0 )F 14 (p′0 , q)F 35 (p, p′0 ) · Ω15 (p′0 ) (TS)−1 L(Ω15 (p′0 ))R(Ω15 (p′0 )) / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) ⊗ L′q (λ, µ) ⊗ (L′q )† (λ, µ) ǫ / L′ (λ, µ) ⊗ L′ (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) p p p q ⊗ (L′q )† (µ, ν) ⊗ (L′q )† (λ, µ) ⊗ L′q (λ, µ) ⊗ (L′q )† (λ, µ) id⊗5 ⊗(ϕ′ )−1 ⊗id / L′ (λ, µ) ⊗ L′ (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) ⊗ (L′ )† (µ, ν) ⊗ (L′ )† (λ, µ) . p p p q q q 82 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON We then notice that the composition in Equation (16.20) can be rewritten using the symmetrised and the binary change-of-base-point isomorphisms e 0 , p′ ) : Lp (λ, µ) ⊗ L† (λ, µ) ⊗ Lq (λ, µ) ⊗ L† (λ, µ) B(p 0 q q / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′q (λ, µ) ⊗ (L′q )† (λ, µ) N(p0 , p′0 ) : Lp (µ, ν) ⊗ / L′p (µ, ν) ⊗ ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ and L†p (λ, ν) (L′q )† (µ, ν) e + (p0 , p′ ) ⊗ B e 0 , p′ ) = B e − (p0 , p′ ) defined in Equation (16.11) and Equation (16.15). Indeed, using that B(p 0 0 0 already factorises as a tensor product of two isomorphisms, we obtain the following alternative description of the isomorphism in Equation (16.20): Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ L†q (λ, µ) id⊗5 ⊗ϕ⊗id / Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L† (λ, ν) ⊗ Lq (λ, ν) ⊗ L† (µ, ν) p q ⊗ L†q (λ, µ) ⊗ Lq (λ, µ) ⊗ L†q (λ, µ) ǫ / Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lq (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ L†q (µ, ν) e 0 ,p′ )⊗N(p0 ,p′ ) B(p 0 0 (16.21) / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′q (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ (L′q )† (µ, ν) ǫ / L′ (λ, µ) ⊗ L′ (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) ⊗ (L′ )† (µ, ν) p ⊗ p ′ † (Lq ) (λ, µ) id⊗5 ⊗(ϕ′ )−1 ⊗id ⊗ p ′ Lq (λ, µ) ⊗ q ′ † (Lq ) (λ, µ) q / L′ (λ, µ) ⊗ L′ (µ, ν) ⊗ (L′ )† (λ, ν) ⊗ L′ (λ, ν) ⊗ (L′ )† (µ, ν) ⊗ (L′ )† (λ, µ) . p p p q q q The result of the present proposition now follows by an application of Proposition 16.4 and Proposition 16.7.  16.5. Functoriality. Throughout this subsection, we let λ, µ, ν ∈ Λ be three indices and p, q ∈ R be idempotents agreeing with the base points p0 and p′0 modulo the ideal I ⊆ R. /H′ (p, q) respects the We are going to prove that the change-of-base-point isomorphism B(p0 , p′0 ) : H(p, q) ′ composition laws in the categories H(p, q) and H (p, q). The main reason for the validity of this functoriality result is that a certain Fredholm determinant is equal to one and this is the statement of the next lemma. We introduce the representation τ := πλ ⊕ πλ ⊕ πλ ⊕ πµ ⊕ πν : R / L (H⊕5 ) and suppress the tuple of indices (λ, λ, λ, µ, ν) from the notation. Lemma 16.9. The invertible operator   Ω14 (p′0 )Ω24 (p′0 )Ω45 (p′0 )Ω25 (p′0 )Ω35 (p′0 )Ω45 (p′0 )Ω34 (p′0 )Ω14 (p′0 ) + τ (1 − p′0 )  −1 (16.22) · Ω14 (p0 )Ω24 (p0 )Ω45 (p0 )Ω25 (p0 )Ω35 (p0 )Ω45 (p0 )Ω34 (p0 )Ω14 (p0 ) + τ (1 − p0 ) / τ (1)H⊕5 : τ (1)H⊕5 is of determinant class and the determinant is equal to one. Proof. We introduce the invertible operators acting on the Hilbert space τ (1)H⊕5 : Ω1345 (p0 ) := Ω14 (p0 )Ω45 (p0 )Ω35 (p0 )Ω45 (p0 )Ω34 (p0 )Ω14 (p0 ) + τ (1 − p0 ) Ω 1245 14 45 25 45 24 14 (p0 ) := Ω (p0 )Ω (p0 )Ω (p0 )Ω (p0 )Ω (p0 )Ω (p0 ) + τ (1 − p0 ) . and A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 83 A similar notation applies when p0 is replaced by p′0 or by the unit 1 ∈ R. Using Lemma 9.5 we then observe that   Ω14 (p′0 )Ω24 (p′0 )Ω45 (p′0 )Ω25 (p′0 )Ω35 (p′0 )Ω45 (p′0 )Ω34 (p′0 )Ω14 (p′0 ) + τ (1 − p′0 )  −1 · Ω14 (p0 )Ω24 (p0 )Ω45 (p0 )Ω25 (p0 )Ω35 (p0 )Ω45 (p0 )Ω34 (p0 )Ω14 (p0 ) + τ (1 − p0 )
−1 1245
−1 1345 ′ ·Ω (p0 ) . ·Ω (p0 ) · Ω1345 (p0 ) = Ω1245 (p′0 ) To show that the invertible operator in Equation (16.22) is of determinant class it thus suffices to check that
Ω1345 (p0 ) − Ω1345 (p′0 ) and Ω1245 (p0 ) − Ω1245 (p′0 ) ∈ L 1 τ (1)H⊕5 . Since Ω1345 agrees with Ω1245 up to conjugation by the permutation matrix   0 πλ (1) Σ := πλ (1) ⊕ ⊕ πµ (1) ⊕ πν (1) πλ (1) 0 we may focus on the first of these two differences. We compute modulo trace class operators, using Assumption 9.1, Lemma 9.4 and Lemma 9.5 together with the fact that p0 − p′0 ∈ I: Ω1345 (p0 ) − Ω1345 (p′0 ) ∼1 τ (p′0 − p0 ) + τ (p0 − p′0 ) · Ω1345 (1) ∼1 τ (p′0 − p0 )    0 πµ (1)πν (1)
πν (1)πµ (1) 0  0 0 πλ (1)πν (1)
 0 πµ (1) 0 · πλ (1)⊕2 ⊕  πν (1)πλ (1) 0 0   0 πµ (1)πν (1)
· πλ (1)⊕3 ⊕ πν (1)πµ (1) 0   
0 πλ (1)πµ (1) · πλ (1)⊕2 ⊕ ⊕ πν (1) Ω14 (1) πµ (1)πλ (1) 0
′ ′ 14 ⊕3 ∼1 τ (p0 − p0 ) + τ (p0 − p0 ) · Ω (1) · πλ (1) ⊕ πµ (1) ⊕ πν (1) · Ω14 (1) = 0 . + τ (p0 − p′0 ) · Ω14 (1)  πλ (1)⊕3 ⊕ Now, to show that the determinant of the invertible operator in Equation (16.22) is equal to one, we use that the Fredholm determinant is multiplicative and invariant under conjugation, to compute that  
−1 1345 ′
−1 1245 ·Ω (p0 ) · Ω1345 (p0 ) ·Ω (p0 ) det Ω1245 (p′0 )  
−1 
−1  · det Ω1245 (p0 ) · Ω1245 (p′0 ) = det Ω1345 (p′0 ) · Ω1345 (p0 )  
−1 
−1  · det Σ · Ω1345 (p0 ) · Ω1345 (p′0 ) ·Σ = det Ω1345 (p′0 ) · Ω1345 (p0 ) = 1.  To ease the notation, we define the Z-graded complex lines Ap (λ, µ, ν) := Lp (λ, µ) ⊗ Lp (µ, ν) ⊗ L†p (λ, ν) and A†q (λ, µ, ν) := Lq (λ, ν) ⊗ L†q (µ, ν) ⊗ L†q (λ, µ) . A similar notation applies with p0 replaced by p′0 and hence L replaced by L′ and A replaced by A′ . / (C, 0) from / (C, 0) and µ†q : A†q (λ, µ, ν) Recall the construction of the trivialisations µp : Ap (λ, µ, ν) Equation (9.4) and Equation (9.5). Recall also the construction of the ternary change-of-base-point isomor/ A′p (λ, µ, ν) ⊗ (A′q )† (λ, µ, ν) from Equation (16.16). phism M(p0 , p′0 ) : Ap (λ, µ, ν) ⊗ A†q (λ, µ, ν) The next result relates the ternary change-of-base-point isomorphism to the compositions in the categories H(p, q) and H′ (p, q). Proposition 16.10. We have the identity
−1 ◦ (µp ⊗ µ†q ) : Ap (λ, µ, ν) ⊗ A†q (λ, µ, ν) M(p0 , p′0 ) = µ′p ⊗ (µ′q )† of isomorphisms of Z-graded complex lines. / A′p (λ, µ, ν) ⊗ (A′q )† (λ, µ, ν) . 84 JENS KAAD, RYSZARD NEST AND JESSE WOLFSON Proof. We suppress the tuple of indices (λ, λ, λ, µ, ν). Notice first that the trivialisation µp ⊗ µ†q can be rewritten as the composition Ap (λ, µ, ν) ⊗ A†q (λ, µ, ν) L(Ω14 (p0 ))R(Ω14 (p0 ))TS / Ω14 (p0 ) · F 24 (p0 , q)F 45 (p0 , q)F 25 (q, p0 ) · F 35 (p0 , p)F 45 (p, p0 )F 34 (p, p0 ) · Ω14 (p0 ) PS / Ω14 (p0 ) · Ω24 (p0 )Ω45 (p0 )Ω25 (p0 ) · Ω35 (p0 )Ω45 (p0 )Ω34 (p0 ) · Ω14 (p0 ) + τ (1 − p0 ) = (C, 0) , where the last stabilisation uses the invertible operator πλ (1−p0 )⊕πλ (1−q)⊕πλ (1−p)⊕πµ (1−p0 )⊕πν (1−p0 ). Since a similar description holds for the trivialisation µ′p ⊗(µ′q )† , we see that the ternary change-of-base-point isomorphism / A′p (λ, µ, ν) ⊗ (A′q )† (λ, µ, ν) M(p0 , p′0 ) : Ap (λ, µ, ν) ⊗ A†q (λ, µ, ν) agrees with the composition
−1 ◦ (µp ⊗ µ†q ) : Ap (λ, µ, ν) ⊗ A†q (λ, µ, ν) µ′p ⊗ (µ′q )† / A′p (λ, µ, ν) ⊗ (A′q )† (λ, µ, ν) up to the Fredholm determinant of the quotient   Ω14 (p′0 )Ω24 (p′0 )Ω45 (p′0 )Ω25 (p′0 )Ω35 (p′0 )Ω45 (p′0 )Ω34 (p′0 )Ω14 (p′0 ) + τ (1 − p′0 )  −1 · Ω14 (p0 )Ω24 (p0 )Ω45 (p0 )Ω25 (p0 )Ω35 (p0 )Ω45 (p0 )Ω34 (p0 )Ω14 (p0 ) + τ (1 − p0 ) see the description of the perturbation isomorphism in Example 4.3. But this Fredholm determinant is equal to one by Lemma 16.9.  We may now combine the results achieved in Subsection 16.4 with the results of this subsection and obtain a proof of the functoriality of the change-of-base-point isomorphism (notice that we already established the unitality condition in Section 11). Proof of Proposition 11.5. Recall from Definition 10.1 that the multiplication operator Mp,q : Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) / Lp (λ, ν) ⊗ L†q (λ, ν) is defined as the composition Mp,q : Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) id⊗4 ⊗ϕ⊗ϕ ǫ / Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lp (λ, ν) ⊗ L†q (λ, ν) ⊗ Lq (λ, ν) / Ap (λ, µ, ν) ⊗ A†q (λ, µ, ν) ⊗ Lp (λ, ν) ⊗ L†q (λ, ν) µp ⊗µ†q ⊗id⊗2 / Lp (λ, ν) ⊗ L†q (λ, ν) . Using Proposition 16.10 and Proposition 16.8, we may rewrite this multiplication operator as the composition Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) id⊗4 ⊗ϕ⊗ϕ ǫ / Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lp (λ, ν) ⊗ L†q (λ, ν) ⊗ Lq (λ, ν) / Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) ⊗ L†p (λ, ν) ⊗ Lq (λ, ν) ⊗ Lp (λ, ν) ⊗ L†q (λ, ν) B⊗B⊗B† ⊗id⊗2 / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′q (λ, ν) ⊗ Lp (λ, ν) ⊗ L†q (λ, ν) ǫ / A′p (λ, µ, ν) ⊗ (A′q )† (λ, µ, ν) ⊗ Lp (λ, ν) ⊗ L†q (λ, ν) µ′p ⊗(µ′q )† ⊗id⊗2 / Lp (λ, ν) ⊗ L†q (λ, ν) . To continue, we combine the above description of the multiplication operator with Proposition 16.2 to see that the composition B(p0 , p′0 ) ◦ Mp,q : Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) / L′ (λ, ν) ⊗ (L′ )† (λ, ν) p q A HIGHER KAC-MOODY EXTENSION FOR TWO-DIMENSIONAL GAUGE GROUPS 85 can be rewritten as the composition Lp (λ, µ) ⊗ L†q (λ, µ) ⊗ Lp (µ, ν) ⊗ L†q (µ, ν) B(p0 ,p′0 )⊗B(p0 ,p′0 ) id⊗4 ⊗ϕ′ ⊗ϕ′ / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′q )† (µ, ν) / L′p (λ, µ) ⊗ (L′q )† (λ, µ) ⊗ L′p (µ, ν) ⊗ (L′q )† (µ, ν) ⊗ (L′p )† (λ, ν) ⊗ L′p (λ, ν) ⊗ (L′q )† (λ, ν) ⊗ L′q (λ, ν) ǫ / A′p (λ, µ, ν) ⊗ (A†q )′ (λ, µ, ν) ⊗ L′p (λ, ν) ⊗ (L′q )† (λ, ν) µ′p ⊗(µ′q )† ⊗id⊗2 / L′p (λ, ν) ⊗ (L′q )† (λ, ν) . But this is exactly the composition M′p,q ◦ (B(p0 , p′0 ) ⊗ B(p0 , p′0 )) and the theorem is therefore proved.  References [ADCK89] E. Arbarello, C. De Concini, and V. G. 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