HU-EP-15/48, HU-MATH-15/XX Quantum deformations of the flat space superstring Anna Pachol∗ Dipartimento di Matematica ”Giuseppe Peano”, Università degli Studi di Torino, Via Carlo Alberto, 10 - 10123 Torino, Italy Stijn J. van Tongeren† arXiv:1510.02389v2 [hep-th] 27 Oct 2015 Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, IRIS Gebäude, Zum Grossen Windkanal 6, 12489 Berlin We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of AdS5 × S5 . This contraction limit turns out to be equivalent to a previously studied limit that yields the so-called mirror model - the model obtained from the light cone gauge fixed AdS5 × S5 string by a double Wick rotation. Reversing this logic, the AdS5 × S5 superstring is the double Wick rotation of a quantum deformation of the flat space superstring. This quantum deformed flat space string realizes symmetries of timelike κPoincaré type, and is T dual to dS5 ×H5 , indicating interesting relations between symmetry algebras under T duality. Our results directly extend to AdS2 × S2 × T6 and AdS3 × S3 × T4 , and beyond string theory to many (semi)symmetric space coset sigma models, such as for example a deformation of the four dimensional Minkowski sigma model with timelike κ-Poincaré symmetry. We also discuss possible null and spacelike deformations. PACS numbers: 11.25.Tq, 02.20.Uw, 11.30.Cp, 04.65.+e, 11.30.Ly, 11.30.Pb Understanding the dynamics of a string in a generic background is a complicated problem. The simplest possible background for a string is flat space, and there its dynamics are well understood. In case of a background such as AdS5 × S5 - playing an important role in the AdS/CFT correspondence [1] - the sigma model is considerably more complicated, but can still be tackled thanks to its integrability [2, 3]. Given the computational control that integrability offers, efforts have been made to find other, less symmetric backgrounds that nevertheless correspond to an integrable model. One way to do so is to deform the string in a controlled fashion, an example of which is the Lunin-Maldacena deformation [4– 6], the string dual to β deformed supersymmetric YangMills theory. A more drastic deformation was proposed in [7], corresponding to a quantum (q) deformation of the AdS5 × S5 string sigma model in the sense that the superconformal algebra of the string is deformed to the corresponding quantum group [8, 9].1 Based on experience with the squashed sphere sigma model [13] - which fits this framework [12, 14] - the full symmetry algebra of this model is expected to be the corresponding quantum affine algebra, a deformation of the Yangian of the AdS5 × S5 string. The status of this deformed sigma model in terms of string theory remains mysterious to date [15, 16],2 see also [17, 18]. In this paper we investigate q deformations in string theory in the simplest ∗ † 1 2 apachol@unito.it svantongeren@physik.hu-berlin.de This generalizes the earlier results of [10–12]. The fermions of the deformed model do not appear to be compatible with supergravity [15]. At the same time, doing formal worldsheet T duality in all Cartan isometry directions does give a background compatible with supergravity [16]. While this solution has a nontrivial dilaton that prevents T dualizing back in possible setting, that of flat space. As we will explain, the deformed flat space string is intimately connected to the AdS5 × S5 string in two ways. The first of these is in its construction. Semisimple Lie (super)algebras can be naturally deformed to quantum groups [19], but this procedure cannot be applied to nonsemisimple algebras such as the ten dimensional (super) Poincaré algebra of the flat space string. However, given a suitable semisimple algebra it is possible to get a nontrivial deformation of some nonsemisimple algebras by the analogue of a Wigner-İnönü contraction. The result can be thought of as a quantum deformation of the corresponding nonsemisimple algebra. This was famously done for the four dimensional Poincaré algebra, yielding what is known as the κ-Poincaré algebra from the q deformed AdS4 algebra [20, 21]. Now, since the flat space string arises from a Wigner-İnönü type contraction of the AdS5 ×S5 one, it should be possible to obtain a q deformation of the flat space string by an appropriate contraction of the q deformed AdS5 × S5 sigma model. The resulting symmetry algebra is of timelike (super) κ-Poincaré type. The implementation of this contraction yields the second connection to the AdS5 × S5 string. It turns out that this contraction is nothing but a previously studied limit of the q deformed AdS5 × S5 string [22–24], related to the so-called mirror model [25]. More precisely, this limit gives a sigma model that in a light cone gauge fixed setting is related to the AdS5 × S5 one by a worldsheet double Wick rotation.3 In other words, the AdS5 × S5 3 supergravity, this still means that the original deformed model should at least be scale invariant at one loop [16]. Furthermore, at the level of scattering theory it seems desirable to do a nontrivial change of basis [15], which may have further consequences. Cf. footnote 2, explicitly matching the fermions of this mirror 2 string is the double Wick rotation of the q deformation of the simplest possible string. In this paper we consider the bosonic sector of the model - where we go from Uq (so(2, 4) ⊕ so(6)) to Uκ (iso(1, 4) ⊕ iso(5)) symmetry - leaving a detailed investigation of fermions for the future. We do however match the well known “lattice” or “spin chain” sin P/2 off shell central extension of psu(2|2) [26, 27] that plays an important role in integrability in AdS/CFT. From the present point of view, it is the double Wick rotation of the contraction of a fermionic anticommutator in Uq (psu(2, 2|4)). We tentatively refer to this only implicitly described contraction of Uq (psu(2, 2|4)) as (inhomogeneous) Uκ (iusp(2, 2|4)) (note that usp(2, 2) ≃ so(4, 1) and usp(4) ≃ so(5)). The metric of the q deformed flat space string, also known as mirror AdS5 × S5 , is related to dS5 × H5 by two T dualities, one involving time. Now, the type IIB∗ sigma model on dS5 × H5 has su∗ (4|4) symmetry [28], which should extend to a full Yangian algebra based on su∗ (4|4). Our model on the other hand has Uκ (iusp(2, 2|4)) symmetry, which we expect to extend to the corresponding quantum affine algebra. Double T duality hence appears to relate (two realizations of) these infinite dimensional symmetry algebras.4 At the bosonic level, the timelike T duality apparently relates (the infinite dimensional extensions of) so(1, 5) and Uκ (iso(1, 4)) while the spacelike one relates so(1, 5) and Uκ (iso(5)). Our construction and these comments readily generalize to other dimensions, in particular to superstrings on AdS2 × S2 × T6 and AdS3 × S3 × T4 . These would present different q deformations of the flat space string, with smaller isometry subgroups being deformed. In this sense then, our contraction of the deformed AdS5 × S5 string gives the largest possible deformation in flat space. At the level of bosonic sigma models we can consider separate spaces and many dimensions, which makes it possible to realize null and spacelike κ-Poincaré type symmetry, and to for instance make contact with the four dimensional κ-Poincaré algebra by considering an analogous contraction of the q deformed AdS4 sigma model. This paper is organized as follows. We will begin by briefly introducing contractions of quantum algebras. Then we implement this type of contraction in the sigma model - demonstrating that the standard Drinfeld-Jimbo r matrix for Uq (so(2, 4) ⊕ so(6)) reduces to the expected κ-Poincaré type r matrix - and indicate its relation to the mirror model. Next we discuss the off shell central extension of psu(2|2), and comment on T duality relations. We then comment on bosonic sigma models in 4 background [23, 24] with those of the contracted deformed sigma model is a subtle point: a direct limit does not appear to give the desired answer, however after a suitable change of basis of scattering states the associated S matrices do exactly match [15]. This assumes the subtleties with the fermions of the deformed model indicated in footnotes 2 and 3 can be appropriately resolved. various dimensions and the associated spacelike and null type contractions. In the conclusions we indicate possible generalizations and open questions. Appendices contain a discussion of the relevant Lie algebras and r matrices, as well as comments on two other possible deformations of AdS5 ×S5 , deformations of dS5 ×H5 , and deformations of S2 , AdS2 , dS2 , and H2 . I. QUANTUM ALGEBRA CONTRACTION The sigma models that we are considering have quantum group symmetry. At the bosonic level the relevant undeformed algebras are so(2, d − 1), so(1, d), and so(d + 1), symmetries of anti-de Sitter space, de Sitter space and the hyperboloid, and the sphere, all d dimensional. The essence of the contraction we are interested in is already captured in the simple case of two dimensions, which we would briefly like to recall. A clear pedagogical discussion of this topic can be found in [29]. The quantum algebras Uq (so(3)) ≃ Uq (su(2)) and Uq (soq (2, 1)) ≃ Uq (su(1, 1)) are two relevant real forms of Uq (sl(2, C)) that are naturally defined with q taken real.5 The algebraic sector of Uq (sl(2, C)) is given by [e+ , e− ] = [h]q , [h, e± ] = ±2e± , (1) where [a]q = q a − q −a . q − q −1 (2) In terms of the physical antihermitian generators we use below, for Uq (su(2)) we have i [−2in56 ]q , 2 [n15 , n56 ] = n16 , [n56 , n16 ] = n15 , [n16 , n15 ] = (3) while for Uq (su(1, 1)) instead i [−2im05 ]q , 2 [m10 , m05 ] = −m15 , [m05 , m15 ] = −m10 . [m15 , m10 ] = (4) These are related by the analytic continuation n16 = im10 , n15 = im15 , n56 = m05 .6 The conventional 5 6 By naturally we mean we are dealing with a standard notion of conjugation [29]. While isomorphic at the undeformed level, as nicely explained in [29], there is a third natural deformation, Uq (sl(2)) for |q| = 1. The difference with Uq (su(1, 1)) is that in this case one deforms the commutator with the noncompact direction on the right hand side. 3 Wigner-İnönü contraction of so(3) to the two dimensional Euclidean algebra starts with the splitting so(3) = so(2)⊕n2 (as vector spaces). Similarly we take so(2, 1) = so(1, 1)⊕m2 for the Poincaré group. We choose this so(2) to be generated by n15 , while for so(1, 1) we take m01 (m15 would give an isomorphic algebra in this case). We then rescale the generators in n or m as ni6 = Rli , i = 1, 5, and mj5 = Rpj , j = 0, 1. To keep a nontrivial deformation we also scale q as log q = −α/R [30]. In the limit R → ∞ we then get sin 2αl5 , 2α [n15 , l5 ] = l1 , [l5 , l1 ] = 0, (5) sin 2αp0 , 2α [m10 , p0 ] = −p1 , [p0 , p1 ] = 0, (6) II. In this section we will discuss the implementation of the contraction procedure described above in the sigma model. We will focus on the bosonic sector of the model. A. [p1 , m10 ] = respectively. In the limit α → 0 these clearly reduce to the two dimensional Euclidean and Poincaré algebras iso(2) = so(2) B l2 and iso(1, 1) = so(1, 1) B p2 , with l2 and p2 generated by the ls and ps. For so(2, 1) we could have also chosen the splitting so(2)⊕ m′2 , but in that case we would trivialize the deformation. Though more involved, higher dimensional algebras can be similarly contracted by appropriately splitting (real forms of) so(n + 1) as so(n) ⊕ mn , and contracting the generators in mn together with an appropriate scaling of q at the level of the corresponding quantum algebra. Contracting Uq (so(2, 3)) this way gives the famous κ-Poincaré algebra [20, 21] for example. Higher rank special orthogonal algebras have not been explicitly contracted, but all should yield κ-Poincaré type deformations of the associated flat space isometry groups of appropriate signature, which do exist in higher dimensions [31, 32]. As we will come back to below, the so-called r matrices associated to this type of deformations are of the form r = aµ m̂µν ∧ pν , (8) where T is the string tension, and g and B denote the background metric and B field respectively. The string action for AdS5 × S5 = SO(2, 4)/SO(1, 4) × SO(6)/SO(5) can be written as a (semi)symmetric space coset sigma model [33], which can be deformed based on a so-called R operator as proposed in [7], giving a so-called YangBaxter sigma model [10, 11]. The bosonic action is7 Z √ T S =−2 (9) dτ dσ 12 ( hhαβ − ǫαβ )sTr(Aα P2 Jβ ) where J = (1 − κRg ◦ P2 )−1 (A) with Rg (X) = g −1 R(gXg −1 )g, and κ labels the deformation. At κ = 0 this gives the undeformed AdS5 × S5 string. The operator R is a skew symmetric map from a relevant (super)algebra to itself, here su(2, 2|4), which solves the modified classical Yang-Baxter equation (mCYBE) [R(x), R(y)] − R([R(x), y] + [x, R(y)]) = ±[x, y]. (10) In this case the R operator is of so-called nonsplit type, meaning it solves the mCYBE with a + sign. For completeness, a − sign means a split solution, and dropping the commutator on the right hand side altogether gives the homogeneous classical Yang-Baxter equation (CYBE).8 Including fermions, this deformed model realizes Uq (psu(2, 2|4)) symmetry [9], where the deformation parameter q is (classically) expressed in terms of T , R and κ as [8] log q = −κ/(R2 T ). (7) where the m̂µν generate an appropriate real form of so(n), and in case of indefinite signature aµ can be a timelike, spacelike, or null vector. The contraction we described for Uq (so(2, 1)) results in the two dimensional analogue of the timelike κ-Poincaré algebra. The idea is now to implement this type of contraction in the q deformed AdS5 × S5 (AdSn × Sn × T10−2n ) string sigma model, which will (hopefully) give a nontrivial deformation of the flat space string with κ-Poincaré type symmetry. The sigma model The action for the bosonic string is given by Z T dτ dσ (gmn dxm dxn − Bmn dxm∧ dxn ) , S =−2 [l1 , n15 ] = and CONTRACTING THE SIGMA MODEL (11) The effect of this deformation on the bosonic background was worked out in [8] (for fermions see [15]), and 7 8 Here h is the world sheet metric, ǫτ σ = 1, Aα = g −1 ∂α g with g ∈ PSU(2, 2|4), sTr denotes the supertrace, and the Pi are the projectors onto the ith Z4 graded components of the semi-symmetric space PSU(2, 2|4)/(SO(4, 1) × SO(5)) (super AdS5 × S5 ). See [34] for a recent unified discussion of these three classes of deformations in the present context. 4 the result is9 1 f+ (ρ) 2 dt + dρ2 + ρ2 dΘρ3 f− (κρ) f+ (ρ)f− (κρ) f− (r) 1 + dφ2 + dr2 + r2 dΘr3 , f+ (κr) f− (r)f+ (κr) R−2 ds2 = − in the split relevant for the contraction, m5 is generated by the ma5 and n5 by the nc6 . This shows that the contraction we are interested in can be translated to the coordinates by considering a singular limit on t and ρ, and r and φ, cf. eqs. (12).10 We can physically implement this by reinstating dimensionful fields through the AdS5 radius as where f± (x) = 1 ± x2, dΘ3 is a deformation of the threesphere metric in Hopf coordinates dΘρ3 ≡ dΘr3 1 1+ 2 κ 2 ρ4 (dζ 2 + cos2 ζdψ22 ) + sin2 ζdψ32 , sin ζ 1 (dξ 2 + cos2 ξdχ22 ) + sin2 ξdχ23 , ≡ 1 + κ 2 r4 sin2 ξ and R denotes the AdS5 radius (of curvature). At κ = 0 this is the metric of AdS5 × S5 , whose factors have curvature −20/R2 and 20/R2 respectively. The B field is given by R−2 B = κ  ρ4 sin 2ζ dψ1 ∧ dζ 1 + κ 2 ρ4 sin2 ζ  r4 sin 2ξ − dχ ∧ dξ . 1 1 + κ 2 r4 sin2 ξ i i gs = eχ fi −ζm13 arcsinh ρ m15 e e 13 fi −ξn e 16 earcsin r n , , (12) where t = ψ1 and φ = χ1 , and the fi are given by f1 = m05 = n56 , f2 = m12 = n12 and f3 = m34 = n34 , which span the Cartan subalgebra of psu(2, 2|4). Our conventions for so(2, 4) and so(6) and their generators mij and nij are discussed in appendix A. As will come back later, there are multiple choices of R operator. The above background corresponds to the standard R operator, associated to the standard Drinfeld-Jimbo r matrix as discussed in appendix A. B. The contraction Now we are ready to consider contractions. Analogously to so(2, 1) discussed above, for so(2, 4) we want to split off an so(1, 4) algebra. Importantly, we take this so(1, 4) to be the algebra of the coset denominator, which is generated by ms with indices running from zero through four, so that at the undeformed level we contract SO(2, 4)/SO(1, 4) to ISO(1, 4)/SO(1, 4) ≃ R1,4 . For so(6) we follow the same procedure. In other words, We hope the distinction between the operator R and the physical scale R is clear. r̃ = Rρ, ρ̃ = Rr, and taking the limit R → ∞ keeping the new coordinates fixed. If we do not do anything else, this limit gives flat space, matching the algebraic situation discussed above, where to keep a nontrivial deformation we had to simultaneously scale q. Given eqn. (11), it is actually natural to do so. We can consider the limit R, κ → ∞, keeping κ/R ≡ κ−1 and the string tension fixed.11 In this limit q → 1, while the symmetry algebra should remain partially deformed as is clear from the simple examples discussed in the previous section. In this limit the B field vanishes, while the metric becomes −dt2 + dr2 + r2 dΘ3 1 − r2 /κ2 dφ2 + dρ2 + + ρ2 dΘ3 , 1 + ρ2 /κ2 (13) where we have dropped tildes - note that the range of φ is no longer compact. Similar to the radius of AdS5 , κ simply sets an overall scale, and factoring it out by rescaling coordinates we get a string on ds2 = −dt2 + dr2 + r2 dΘ3 1 − r2 dφ2 + dρ2 + + ρ2 dΘ3 , 1 + ρ2 (14) with effective string tension g ≡ κ2 T . At this stage the model may appear to contain no deformation parameter anymore, and indeed the deformation parameter can be formally scaled out of κ-Poincaré type algebras, like it can be from eqs. (5) and (6).12 However, this is entirely analogous to how the AdS5 × S5 radius can be scaled out of psu(2, 2|4) and absorbed in the string tension. The value of the deformation parameter becomes relevant as soon as a physical scale is fixed.13 10 11 12 9 φ̃ = Rφ, ds2 = These expression arise by deforming the AdS5 × S5 coset sigma model built on g = diag(ga , gs ) with ga = eψ t̃ = Rt, 13 Had we not chosen our splitting compatible with the coset structure, we would formally need to rescale directions corresponding to gauge degrees of freedom. It is not clear to us whether this can be sensibly done. Note the distinction between κ and κ here. These two variables are standard in their respective fields (deformed string sigma models and κ-Poincaré algebras), so we felt it better not to introduce further ones. This is possible in the so-called q analog version of κ-Poincaré type algebras, where the value of κ can be fixed, see e.g. [35]. At the level of the quantum spectrum the distinction is also clear: 5 As indicated in the introduction, this contraction should result in a symmetry algebra of κ-Poincaré type. Such contractions have not been explicitly worked for our rank three cases Uq (so(2, 4)) and Uq (so(6)), let alone Uq (psu(2, 2|4)). In all lower dimensional cases (AdS3 × S3 × T4 , AdS2 × S2 × T6 , and e.g. the sigma model on AdS4 ) however, these contractions are exactly the ones that give κ-Poincaré type algebras [21], leaving little doubt what the outcome should be. Still, to show this more concretely, rather than contracting the algebras in painstaking detail, let us focus on the associated r matrices. As explained in detail in appendix A, if we take the canonical Drinfeld-Jimbo r matrix for the associated quantum groups, express it in terms of physical generators, and take the contraction limit appropriate for the sigma model, we get r = m0j ∧ pj , and r = n5j ∧ lj , (15) for so(2, 4) and so(6) respectively, sums in j running from one through four. Cf. eqn. (7), these are precisely the κPoincaré type r matrices for Uκ (iso(1, 4)) and Uκ (iso(5)) respectively [32, 36], timelike in the case of iso(1, 4).14 In other words, also these cases correspond to standard κ-Poincaré type algebras that can be found in [31]. To extend these results to the supersymmetric case, √ we should include the supercharges and rescale them by R in the contraction procedure. We will not discuss this in detail in the present paper, but will come back to one nice aspect of it below. C. D. Φ = Φ0 − 1 a β a β β a {Qαa , Q†β b } = δb Rα + δα Lb + δb δα H, 2 where L and R generate the two bosonic su(2)s. Note that this is a conventional Lie superalgebra. If we go off shell by relaxing the Virasoro constraint (level matching condition), this algebra picks up a further central element of the form T sin P/2, where P denotes the total worldsheet momentum [27]. Doing a double Wick rotation interchanges energy and momentum, and for the mirror theory we instead have [25] 1 log(1 − r2 )(1 + ρ2 ), 2 and F = 4e −Φ (ωφ − ωt ) . where Φ0 is a constant, and ωt and ωφ denote wouldbe volume forms on the two five submanifolds, except the flat space string has a fixed integer spectrum, up to an overall scale set by the string tension. Our string has a complicated spectrum depending on a dimensionless parameter g, with an overall scale set by either κ or T . This is of course one fewer free parameter than the q deformed AdS5 × S5 model. The indices 0 on m and 5 on n are entirely due to conventions, though it is relevant that the index on m is timelike. Mirror fermions and off shell central extensions As double Wick rotations preserve conservation laws, the mirror string inherits the light cone symmetries of the AdS5 × S5 superstring. The on shell symmetry algebra of the light cone gauge fixed AdS5 × S5 string is centrally extended psu(2|2)⊕2 , where the central element H corresponds to the worldsheet Hamiltonian. Focusing on one copy of psu(2|2), the supercharges Q and Q† satisfy q deformation as the mirror model This space and the limit to get there were already considered from a different angle in [23, 24], where this background including the corresponding dilaton and RamondRamond five form was shown to correspond to the socalled AdS5 × S5 mirror model - a double Wick rotation of the light cone gauge fixed AdS5 × S5 sigma model [25]. For completeness, this dilaton and five form are [23] 14 with t and φ formally interchanged. While the light cone gauge fixed lagrangians of these two theories are related by a double Wick rotation, note that in particular the Virasoro constraints are different due to the interchange of space and time. By this relationship between the contraction limit and the mirror model, the AdS5 × S5 string is the (off shell) double Wick rotation of the q deformation of the flat space superstring.15 It also shows that the symmetry algebra of the mirror AdS5 × S5 string is Uκ (iso(1, 4) ⊕ iso(5)) at the bosonic level. Modulo the subtleties mentioned earlier, upon including fermions this should extend to what we might denote as Uκ (iusp(2, 2|4)), though have not explicitly described this deformed superalgebra here. Let us however discuss one aspect of this deformed superalgebra, where we can nicely make contact with well known aspects of the off-shell symmetry algebra of the AdS5 × S5 string. a β β a a β {Q̃αa , Q̃†β b } = δb Rα + δα Lb + gδb δα sinh H̃ , 2 (16) where tildes denote mirror quantities, g denotes the effective AdS5 × S5 string tension, and we have put the mirror theory on shell by setting P̃ to zero. Here we see a signature of a would-be q deformation: a hyperbolic sine. To match the above with the contraction of Uq (psu(2, 2|4)), let us consider the simpler Uq (psu(1, 1|2)) instead, either as a subalgebra (before deformation), or as the relevant superalgebra for the AdS2 string where 15 Admittedly, the q deformation itself is found through the AdS5 × S5 model to begin with. 6 the same central extension appears, see e.g. [37]. Standard light cone gauge fixing here produces two psu(1|1) subalgebras with appropriate central extensions that couple them. These are given by the centralizer in psu(2|2) with respect to diag(1, −1, 1, −1) - in other words one psu(1|1) involves rows and columns one and three, the other two and four. Focusing on the first of these psu(1|1)s, its supercharges simply anticommute to h̄1 + h̄3 = diag(1, 0, 1, 0), which upon q deformation becomes {Q, Q† } = [h̄1 + h̄3 ]q , (17) where Q and Q† are the supercharges of psu(1|1). Since we are really dealing with Uq (psu(1, 1|2))), at this stage we should set the overall central element of Uq (su(1, 1|2))) to zero. Doing so means 2(h̄1 + h̄3 ) ∼ diag(1, −1, 1, −1) = H - the light cone string Hamiltonian. To do the quantum contraction we should rescale this generator by R (it is 05 the analogue of i(m − n56 )), and as mentioned earlier √ the fermions by R, which gives {Q, Q† } = κT sinh H . 2κT (18) This matches perfectly with the purely diagonal term of eqs. (16), upon noting firstly that to get the mirror background in the standard form of eqn. (14) - which eqs. (16) refer to - we have to rescale t → t̃/κ, √ meaning H → H̃ = κH, and analogously Q → Q̃ = κQ, and secondly that the conventions under which the off shell central extension was computed involve normalizing spatial translations by the string tension - see e.g. section 2.2.3 of [2] - which we implicitly do not do here since everything remains associated to the time direction. Now that we have matched our deformed symmetry algebra with known results (despite the subtleties surrounding fermions in this model), let us comment further on the resulting model, and related ones. E. in one dimension higher. In fact, the dSn and Hn sigma models should have Yangian symmetry, and we expect our q deformed symmetry to extend to a full quantum affine algebra. Including fermions we need both T dualities to get a clean statement. Taking us a bit beyond conventional strings, dS5 × H5 is a solution of type IIB∗ supergravity [28], and the corresponding superalgebra is a different real form of sl(4|4) known as su∗ (4|4) [38]. This double T duality hence appears to relate Uκ (iusp(2, 2|4)) and su∗ (4|4).4 F. Other dimensions, other spaces The q deformation of [7, 12] applies to any G/H (semi)symmetric space sigma model, and many of them are amenable to Wigner-İnönü contraction. Firstly however, we should come back to the option of using differen R operators to deform AdS5 × S5 . By permuting the signature of su(2, 2), two other and apparently inequivalent deformations of AdS5 were constructed in [9]. We briefly discuss these in appendix B. As explained there, they are not (directly) amenable to the contraction procedure we followed above. Next, attempting to deform dS5 × H5 analogously to AdS5 × S5 as in the main text - which would straightforwardly contract to the T dual of AdS5 × S5 - appears to conflict with the real form su∗ (4|4). It is possible to deform dS5 × H5 , but this results in the analog of one of the other deformations of AdS5 × S5 just mentioned, see appendix C for details. Of course we can consider (anti-)de Sitter space, the sphere, or the hyperboloid in any dimension, with metrics corresponding to the obvious analogue of (parts of) eqn. (13). In particular, timelike four dimensional κPoincaré symmetry arises in the four dimensional sigma model obtained by contracting the deformed four dimensional AdS4 sigma model, namely17 T duality ds2 = Upon formally T dualizing in t and φ, our mirror AdS5 × S5 becomes dS5 × H5 , the product of five dimensional de Sitter space and a five dimensional hyperboloid [23].16 In fact, if we keep the dependence on κ as in eqn. (13), it becomes the radius of dS5 and H5 . Forgetting about fermions for a moment, we see that timelike T duality relates the sigma model on de Sitter space to the Lorentzian submanifold of mirror AdS5 × S5 , and at the level of symmetry thus apparently relates q deformed Poincaré symmetry to undeformed Lorentz symmetry in one dimension higher. Similarly, T duality in φ relates H5 to the Euclidean submanifold, and q deformed Euclidean symmetry to undeformed Lorentz symmetry 16 This fact was previously observed by S. Frolov. −dt2 + dr2 + r2 dΘ2 . 1 − r2 /κ2 (19) Analogously, we expect the T dual of four dimensional anti-de Sitter space to come out of contracting the split type deformation of AdS4 , which should realize spacelike κ-Poincaré symmetry, though we have only concretely investigated this in two dimensions, see appendix D for details. Similarly, split deformations exists for AdS3 and AdS5 , which we expect to contract analogously. These 17 J. Lukierski informed us that eqn. (19) as well as its counterpart for the spacelike deformation alluded to below have been independently obtained by A. Borowiec, H. Kyono, J. Lukierski, J. Sakamoto, and K. Yoshida [39]. We should also note that these backgrounds were considered from a different perspective in [40], there obtained from solutions of the CYBE as opposed to the mCYBE, a point deserving further investigation. 7 cannot be lifted to the corresponding superstring coset sigma models however, since the associated spheres do not admit split deformations. At least in two dimensions, we can also find sigma models with null type κ-Poincaré symmetry. As discussed in appendix D this uses a deformation of AdS2 given in [41]. This deformation is based on the (homogeneous) classical Yang-Baxter equation (CYBE) instead of the mCYBE which is also an option in this context [42] - matching the fact that precisely in the null case the κ-Poincaré r matrix satisfies the CYBE instead of the mCYBE (see e.g. [32]). This means it is associated to a Drinfeld twist and not a proper q deformation, but in this case the contraction appears to work nicely as well. Let us also note that since so(2, 4) contains an iso(1, 3) subalgebra, it is possible to directly deform the AdS5 × S5 string by the four dimensional null κ-Poincaré r matrix, as considered in [43].18 Finally, at the bosonic level it might seem nice to try to deform the flat space string to get a sigma model with ten dimensional κ-Poincaré symmetry. This would result in a ten dimensional analogue of the above metric. However, since it contains only one of the factors of mirror AdS5 × S5 , or equivalently since it is T dual to dS10 , it cannot be embedded in standard supergravity. III. OUTLOOK In this paper we discussed a contraction limit of the q deformation of the AdS5 × S5 superstring sigma model, which can be viewed as a q deformation of the flat space string. Interestingly, this contraction limit turns out to be identical to the one used to obtain the so-called mirror background, showing that the light cone AdS5 × S5 string is the double Wick rotation of the light cone gauge fixed q deformed flat space string. Similar stories apply directly to AdS2 × S2 × T6 and AdS3 × S3 × T4 , the only difference being that there the deformation involves smaller algebras. In this sense, the “most q deformed” flat space superstring is obtained starting from (q deformed) AdS5 × S5 , as it has the biggest semisimple superalgebra underlying a particular string. The backgrounds coming out of our contraction procedure are T dual to other (semi)symmetric spaces, indicating interesting relations between (infinite dimensional) symmetry algebras of sigma models under T duality. There are a number of interesting open questions that we did not address in this paper. Firstly, we did not explicitly contract the deformed algebra, or include the fermions beyond matching the central extension. We hope to address these points in the near future, in particular the relevant super κ-Poincaré algebra. Beyond this, it might be interesting to concretely investigate (contractions of) split deformations of AdS3 , AdS4 and AdS5 , and their relation to spacelike κ-Poincaré symmetry. From the perspective of string theory, looking into an r matrix that contracts to the five dimensional null case is perhaps more interesting however. If this exists (it does in two dimensions) it should correspond to a Drinfeld twist, and can be applied to the full string. At the algebraic level this would give a twisted rather than q deformed algebra, with a nontrivial contraction. It would also be interesting to further investigate the deformations of AdS5 × S5 corresponding to the other choices of Drinfeld-Jimbo r matrices given in [9], regardless of their (lack of) contractibility. Furthermore, in general it would be great to get a full grasp on the infinite dimensional symmetry algebras of these models, even already at the bosonic level, and study their relation under T duality in detail. Also, it might be interesting to consider one-sided contractions in the two parameter deformation of AdS3 × S3 × T4 [14]. Finally, it may be interesting to consider the Penrose limit of AdS5 × S5 in this deformed setting. ACKNOWLEDGMENTS We would like to thank B. Hoare, V. Giangreco M. Puletti, and A. Tseytlin for the insightful discussions, and G. Arutyunov, B. Hoare, and J. Lukierski for comments on the draft. ST is supported by LT. The work of ST is supported by the Einstein Foundation Berlin in the framework of the research project ”Gravitation and High Energy Physics” and acknowledges further support from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No 317089. AP acknowledges the funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 609402 - 2020 researchers: Train to Move (T2M). Part of this work was supported by National Science Center project 2014/13/B/ST2/04043. Appendix A: Algebra and canonical r matrix In this paper we are mainly concerned with the bosonic subalgebra su(2, 2) ⊕ su(4) of psu(2, 2|4). For details and the supersymmetric extension of the material here, we refer to the pedagogical review [2] whose conventions we follow. We will only briefly list the facts we need, beginning with the γ matrices γ 0 = iσ3 ⊗ σ0 , γ 1 = σ2 ⊗ σ2 , γ 2 = −σ2 ⊗ σ1 , γ 3 = σ1 ⊗ σ0 , 18 Generically CYBE based deformations of the AdS5 × S5 string are conjectured to give gravity duals to various noncommutative versions of supersymmetric Yang-Mills theory [43]. γ 4 = σ2 ⊗ σ3 , γ 5 = −iγ 0 , (A1) where σ0 = 12×2 and the remaining σi are the Pauli matrices. With these matrices the generators of so(1, 4) in the spinor representation are given by mij = 41 [γ i , γ j ] 8 where the indices run from zero to four, while for so(5) we can give the same construction with indices running from one to five. The algebra su(2, 2) is spanned by these generators of so(1, 4) together with the mi5 = 21 γ i for i = 0, . . . , 4, satisfying:  ij kl  m , m = η jk mil − η ik mjl − η jl mik + η il mjk (A2) where i, j, k, l = 0, ..., 5 and η = diag (−1, 1, 1, 1, 1, −1). While su(4) is spanned by the combination of so(5) and iγ j for j = 1, . . . , 5 with  ij kl  (A3) n , n = δ jk nil − δ ik njl − δ jl nik + δ il njk where i, j = 1, ..., 5 and ni6 = generators satisfy i i 2γ . Concretely, these   2 −1 0 aij =  −1 2 −1  . 0 −1 2 (A4) n† + n = 0 (A5) for n ∈ su(4). This means that we are dealing with the canonical group metric γ 5 = diag(1, 1, −1, −1) for SU(2, 2), and that eαn and eαm give group elements for real α. The Z4 automorphism of psu(2, 2|4) is generated by Ω, which acts on the bosonic subalgebras as t Ω(m) = −Km K, (A6) e4 = i[e1 , e2 ], f4 = i[f1 , f2 ], (A13) e5 = i[e2 , e3 ], f5 = i[f2 , f3 ], e6 = i[e1 , i[e2 , e3 ]], f6 = i[f1 , i[f2 , f3 ]]. The extended Cartan elements are [hi , hj ] = 0 [ei , fj ] = δij hi [hi , ej ] = aij ej , [hi , fj ] = −aij fj (A7) (A8) (A9) 19 where aij is the Cartan matrix. After deformation commutation relation (A8) becomes the m12 m34 n13 n14 n23 n24 n15 n25 n16 n26 n35 n45 n36 and the relations for the non-simple root vectors (A13) become q-commutators. For the algebras su(2, 2) (and su(4)) we take the Cartan subalgebra to be spanned by n46 h1 = diag(1, −1, 0, 0), h3 = diag(0, 0, 1, −1). h2 = diag(0, 1, −1, 0), (A11) To be precise the deformation procedure requires working with the universal enveloping algebra U (g) of a given Lie algebra g and besides relations (A7-A9) one needs to take into account the so-called Serre relations which also become q deformed. There is a unique Hopf algebra structure on the q deformed algebra U (g), see e.g. Sec. 6 in [44]. i (h1 + 2h2 + h3 ) = n56 , 2 i = (h1 + h3 ) = n12 , 2 i = (h1 − h3 ) = n34 , 2 i = (e1 + f1 − e3 − f3 ) = m13 , 2 1 = (e1 − f1 + e3 − f3 ) = m14 , 2 1 = (e1 − f1 − e3 + f3 ) = m23 , 2 i = − (e1 + f1 + e3 + f3 ) = m24 , 2 i = (e2 + f2 − e6 − f6 ) = im01 , 2 1 = (−e2 + f2 − e6 + f6 ) = im02 , 2 1 = (e2 − f2 − e6 + f6 ) = im15 , 2 i = (e2 + f2 + e6 + f6 ) = im25 , 2 i = − (e4 + e5 + f4 + f5 ) = im03 , 2 i = (−e4 + e5 + f4 − f5 ) = im04 , 2 1 = (−e4 − e5 + f4 + f5 ) = im35 , 2 i = − (−e4 + e5 − f4 + f5 ) = im45 . 2 Note that we have h†i = hi and e†i = fi . The standard Drinfeld-Jimbo classical r matrix r ∈ ∧2 g for so (2, 4) ∼ su (2, 2) is rDJ = i 19 h6 = h1 + h2 + h3 . m05 = (A10) [ei , fj ] = δij [hi ]q , h5 = h2 + h3 , We have raising generators ei and lowering generators fi , i = 1, . . . , 6. The relations between the (bosonic) generators of su(2, 2) (and su(4)) and the Cartan basis elements are 2 4 where K = −γ γ . It leaves the above mentioned subalgebras so(1, 4) and so(5) invariant. The deformation of the above algebras is normally introduced in the Cartan-Chevalley basis for the complexified algebras, so let us give this as well. Here hi are Cartan elements and for simple roots αi we have the corresponding single vectors ei = e+αi , fi = e−αi . They satisfy (A12) The non-simple root vectors follow from h4 = h1 + h2 , m† γ 5 + γ 5 m = 0 for m ∈ su(2, 2), and The Cartan matrix is 6 X j=1 ej ∧ f j (A14) = (m0i ∧ mi5 + m23 ∧ m13 + m24 ∧ m14 ), sums in i running from zero to five. Note that here and below the i = 0 and i = 5 terms give zero however. This 9 r matrix satisfies the inhomogeneous (modified) YangBaxter equation [r12 , r13 ] + [r12 , r23 ] + [r13 , r23 ] = Ω (A15) with the invariant Ω = mij ∧ mjk ∧ mki ∈ ∧3 so (2, 4). The corresponding R operator is defined by R(x) ≡ sTr2 (rDJ (1 ⊗ x)), (A16) where now eqn. (A15) is equivalent to the operator form of the mCYBE of eqn. (10) as so(2, 4) has a nondegenerate Killing form. Redefining mi5 = Rpi , eqn. (A14) becomes rDJ = R(m0i ∧ pi + 1 1 m23 ∧ m13 + m24 ∧ m14 ) (A17) R R At this stage it is convenient to factor in the overall deformation parameter κ = Rκ−1 , so that κ rDJ contracts to κ−1 rκ := lim R→∞ κ 1 rDJ ( ) = κ−1 m0i ∧ pi . 2 R R (A18) For su(4) we analogously have rDJ = (n5i ∧ ni6 + n23 ∧ n31 + n14 ∧ n42 ), (A19) which contracts to rκ = n5i ∧ li . (A20) The R operator referred to in the main text corresponds to the standard psu(2, 2|4) r matrix, which at the bosonic level is just the combination of the above so(2, 4) and so(6) r matrices. Appendix B: Alternate q deformations of AdS5 × S5 The deformation of AdS5 × S5 described in the main text is not the only possibility within the framework of [7, 9]. In [9] it was shown that by combining the R operator with permutations, it is possible to obtain two other deformations of AdS5 [9]. This procedure essentially amounts to permuting the signature of su(2, 2), which does nothing for su(4) and hence should not give further deformations of the sphere. We believe that these three inequivalent permutations correspond to the three possible choices for Cartan involutions for the real form Uq (su(2, 2)) with real q described in [45], cf. eqs. (3.5) and table 1 there. In principle it is possible to contract these different forms of Uq (so(2, 4)). In terms of physical generators, the two r matrices corresponding to deformation P1 and P2 in [9] are given by r1 = m1i ∧ mi2 − m03 ∧ m35 − m04 ∧ m45 , respectively. At this stage an important difference to r of eqn. (A14) becomes clear: the obvious analogous contraction should now involve directions one or two for r1 , or three and four for r2 , however these are in the direction of the denominator of the coset SO(2, 4)/SO(1, 4) (gauge directions). It is therefore not clear to us that the corresponding contraction limit can be simply implemented in the sigma model. Appendix C: Deformed dS5 × H5 In this appendix we briefly describe how to find a deformation of dS5 ×H5 = SO(1, 5)/SO(1, 4)×SO(1, 5)/SO(5) in the spirit of [7, 8]. It will however not be the analogue of the deformation of AdS5 × S5 in the main text, rather it is the analogue of the deformation obtained from the R operator associated to r2 above. The reason for this is that the standard Drinfeld-Jimbo r matrix of sl(4|4) (with respect to our basis) does not preserve the real form su∗ (4|4).20 Without loss of generality, let us describe the situation at the level of sl(4) and su∗ (4). Following [9] we consider permutations of {1, 2, 3, 4}, and use them to construct possibly inequivalent R operators. In our case there turns out to be only one inequivalent permutation that yields an R operator compatible with su∗ (4).21 We can take this permutation to be   1 2 3 4 (C1) P= 1 4 2 3 where the corresponding R operator RP is given by AdP −1 ◦ R ◦ AdP with Pij = δiP(j) , while R denotes the canonical Drinfeld-Jimbo R operator of eqn. (A16). The metric and B field corresponding to RP are given by taking those of the deformed AdS5 × S5 corresponding to P2 of appendix D.1 of [9] and analytically continuing t → it, ρ → ir, and φ → iφ and r → iρ. It can be constructed directly by analytically continuing the generators as mi5 → m̂i5 = imi5 and ni6 → n̂i6 = −ini6 (sign choices of course do not affect the outcome), and using group elements of the form gd = eψ gh = e 20 (B1) 21 and r2 = m3i ∧ mi4 + m10 ∧ m02 + m15 ∧ m51 (B2) i fˆi ζ m̂13 arcsin ρ m̂15 e e χi fˆi ξn̂13 arcsinh r n̂16 e e , , We can of course forget about reality and ask what happens if we use this canonical r matrix to deform dS5 × H5 . Naturally this gives the analytic continuation t → it, ρ → iρ, and φ → iφ and r → ir of deformed AdS5 × S5 described above. This appears to be a real model. However, formally the construction also yields an imaginary exact term in the B field, cf. the continuation of the one AdS5 × S5 , see e.g. footnote 17 in [15]. It would be nice to understand this point in full detail. Note that this counting appears to match with the results of [45], cf. eqs. (3.5) and table 1 there, with three different real q deformations of so(2, 4), but only one for so(1, 5). 10 for de Sitter space (d) and the hyperboloid (h) respectively, where the fˆ generators are related to the m̂ and n̂ generators as the f are to m and n. The algebra su∗ (4) as obtained this way is spanned by the matrices y that satisfy K̂ −1 y ∗ K̂ = y, with K̂ = diag(iσ2 , −iσ2 ) (matching the conventions of [9]). Like the alternate r matrix deformations of AdS5 × S5 discussed in the appendix above, the two obvious sensible contractions of the r matrix for this deformed model would use coset denominator (gauge) directions at the level of the sigma model. T1 T2 S c c 2 AdS nc nc dS2 c nc H2 nc c coord r, ρ gauge 2 T3 r (12) r (13) r (23) c ns ns ns c ns s s nc s s ns nc s ns s φ, t TABLE I. Three possible r matrices for each of S2 , AdS2 , dS2 , and H2 . The Ti denote the three generators of the relevant real form of sl(2, C), which can be associated to compact (c) or noncompact (nc) directions. The associated coordinate type is indicated at the bottom of the table. The r matrices are of either split (s) or nonsplit (ns) type. Appendix D: Deforming S2 , AdS2 , dS2 , and H2 The restrictions imposed by the larger algebras involved for AdS5 × S5 and dS5 × H5 are not necessarily present in lower dimensional algebras. Let us therefore briefly consider the two dimensional sphere and its various analytic continuations. When analytically continuing it becomes possible to have split r matrices. Let us begin the discussion with the sphere, i.e. su(2) at the algebraic level. With respect to our conventions, the standard nonsplit R operator in this case is associated to the r matrix r(12) = T1 ∧T2 , where the Ti , i = 1, 2, 3, denote the generators of su(2). The coset denominator U (1) is associated to T2 , while in our conventions the coordinates r and φ of the main text are associated to T1 and T3 respectively. Of course, these preferred directions have no meaning in the algebra, and any group rotation of the r matrix is still a solution of the mCYBE. For our purposes it will be useful to consider the permutations r(13) = T1 ∧T3 and r(23) = T2 ∧ T3 . The associated deformations of S2 are equivalent, but do not manifestly appear so in terms of coordinates that each have relevance after analytic continuation. We have ds212 = 1 dΩ2 , 1 + κ 2 r2 2 2 dΩ2 , 2 + κ 2 (1 + cos 2φ − cos2 φ r2 ) 2 2 = dΩ2 , 2 2 + κ (1 − cos 2φ − sin2 φ r2 ) 2 (D1) ds213 = (D2) ds223 (D3) all up to total derivative B fields. Here dΩ22 denotes the metric on the two sphere in r and φ coordinates analogous to the ones in the main text. Clearly these last two are related by the shifting φ by π/2, the relation between the first and the two others is not as clear. From here we can directly derive the corresponding deformations of AdS2 , dS2 and H2 by appropriately analytically continuing r and φ as well as the deformation parameter, including a factor of i when the corresponding generator becomes noncompact, or the r matrix changes from nonsplit to split type, cf. table I. This table gives the possibilities that are compatible with the real form under consideration, meaning r matrices containing only r (12) r (23) 2 S T (H ) T (H2 ) AdS2 T (dS2 ) T (AdS2 ) dS2 T (dS2 ) T (AdS2 ) H2 T (H2 ) T (H2 ) 2 TABLE II. Nontrivial contractions of the various deformed spaces. All contractions come out as T dual to some space M (T (M )). The T dual spaces in the first column are obtained in coordinates analogous to the ones in the main text, the T duality referring to the isometry coordinate. Those in the second column are the spaces obtained by analytically continuing both coordinates to imaginary values (which simply flips the overall signature), and then T dualizing. E.g. T (AdS2 ) 2 +dρ2 means ds2 = −dt1−t . 2 real combinations of generators. Since formally multiplying r by i takes a solution of nonsplit type, to one of split type, it is easy to determine whether r(ij) is of split or nonsplit type, by comparing to the original nonsplit su(2) r matrix - if both i and j are compact or noncompact, it is of nonsplit type, otherwise it is of split type.22 Due to the analytic continuations involved, the resulting deformed geometries are not all equivalent under real diffeomorphisms, and may moreover involve analytically continuing κ. If we would like to embed these algebras in a superalgebra, it is of course necessary to require that both r matrices are of the same type, though it need not be sufficient. Let us now consider possible interesting contractions of these models. As mentioned before, we want to contract in the non gauge directions, and the r matrix should have a single component in there.23 This leaves us with r(12) and r(23) . The result of the contractions is collected 22 23 The mention in footnote 6 that in order to q deform sl(2) in a ‘standard’ fashion, we need to consider |q| = 1, while for su(1, 1) we would like q real, seems to match the change from split to nonsplit r matrix here. If we were to take two components in there, we would formally 11 in table II. Given the various relations under analytic continuation, the fact that some of the contractions come out the same is not too surprising - the contraction can trivialize the distinguishing effect. Note that at the level of r matrices, the contraction of the alternate deformation of AdS2 - giving its own T dual - gives precisely the spacelike κ-Poincaré type r matrix for iso(1, 1), namely (23) rAdS2 → m01 ∧ p0 . 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