Abstract
We study an analog of the AGT (Alday-Gaiotto-Tachikawa) relation in five dimensions. We conjecture that the instanton partition function of 5D \( \mathcal{N} = 1 \) pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four-dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.
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L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, arXiv:0906.3219 [SPIRES].
D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [SPIRES].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [SPIRES].
N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].
A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [SPIRES].
G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, arXiv:0909.4031 [SPIRES].
L.F. Alday, F. Benini and Y. Tachikawa, Liouville/Toda central charges from M5-branes, arXiv:0909.4776 [SPIRES].
R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [SPIRES].
A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [SPIRES].
N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP 09 (2009) 031 [arXiv:0907.2593] [SPIRES].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, arXiv:0909.0945 [SPIRES].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, arXiv:0909.1105 [SPIRES].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [SPIRES].
S. Cecotti and C. Vafa, BPS Wall Crossing and Topological Strings, arXiv:0910.2615 [SPIRES].
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
K. Takasaki, Integrable structure of melting crystal model with two q-parameters, J. Geom. Phys. 59 (2009) 1244 [arXiv:0903.2607] [SPIRES].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, arXiv:0908.4052 [SPIRES].
N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].
R. Flume and R. Poghossian, An algorithm for the microscopic evaluation of the coefficients of the Seiberg-Witten prepotential, Int. J. Mod. Phys. A 18 (2003) 2541 [hep-th/0208176] [SPIRES].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multi-instanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [SPIRES].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math. 162 (2005) 313 [math/0306198].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
H. Nakajima and K. Yoshioka, Instanton counting on blowup. II: K-theoretic partition function, Transform. Groups 10 (2005) 489 [math/0505553].
H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [SPIRES].
A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [SPIRES].
H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [SPIRES].
I.G. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford University Press, New York U.S.A. (1995).
J. Shiraishi, H. Kubo, H. Awata and S. Odake, A Quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996) 33 [q-alg/9507034] [SPIRES].
H. Awata, H. Kubo, S. Odake and J. Shiraishi, Virasoro-type symmetries in solvable models, in Proceedings of the Canada-China meeting on theoretical physics, L. Lapointe et.al. eds., Les Publications CRM, Montreal Canada (2003) hep-th/9612233 [SPIRES].
B. Feigin and E. Frenkel, Quantum W algebras and elliptic algebras, Commun. Math. Phys. 178 (1996) 653 [q-alg/9508009] [SPIRES].
H. Awata, H. Kubo, S. Odake and J. Shiraishi, Quantum W(N) algebras and Macdonald polynomials, Commun. Math. Phys. 179 (1996) 401 [q-alg/9508011] [SPIRES].
A. Gorsky and N. Nekrasov, Relativistic Calogero-Moser model as gauged WZW theory, Nucl. Phys. B 436 (1995) 582 [hep-th/9401017] [SPIRES].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [SPIRES].
A. Braverman, Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors, math/0401409.
A. Braverman and P. Etingof, Instanton counting via affine Lie algebras. II: From Whittaker vectors to the Seiberg-Witten prepotential, math/0409441.
P. Bouwknegt and K. Pilch, The deformed Virasoro algebra at roots of unity, Commun. Math. Phys. 196 (1998) 249 [q-alg/9710026].
T. Maeda, T. Nakatsu, K. Takasaki and T. Tamakoshi, Five-dimensional supersymmetric Yang-Mills theories and random plane partitions, JHEP 03 (2005) 056 [hep-th/0412327] [SPIRES].
E. Carlsson, Vertex Operators and Moduli Spaces of Sheaves, arXiv:0906.1825 [SPIRES].
M. Jimbo, H. Nagoya and J. Sun, Remarks on the confluent KZ equation for sl 2 and quantum Painlevé equations, J. Phys. A: Math. Theor. 41 (2008) 175205.
K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, Cubic Pencils and Painlevé Hamiltonians, Funkcial. Ekvac. 48 (2005) 147, [nlin/0403009].
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ArXiv ePrint: 0910.4431
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Awata, H., Yamada, Y. Five-dimensional AGT conjecture and the deformed Virasoro algebra. J. High Energ. Phys. 2010, 125 (2010). https://doi.org/10.1007/JHEP01(2010)125
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DOI: https://doi.org/10.1007/JHEP01(2010)125