Abstract
We propose a Nekrasov-type formula for the instanton partition functions of four-dimensional \( \mathcal{N} \) = 2 U(2) gauge theories coupled to (A1, D2n) Argyres-Douglas theories. This is carried out by extending the generalized AGT correspondence to the case of U(2) gauge group, which requires us to define irregular states of the direct sum of Virasoro and Heisenberg algebras. Using our formula, one can evaluate the contribution of the (A1, D2n) theory at each fixed point on the U(2) instanton moduli space. As an application, we evaluate the instanton partition function of the (A3, A3) theory to find it in a peculiar relation to that of SU(2) gauge theory with four fundamental flavors. From this relation, we read off how the S-duality group acts on the UV gauge coupling of the (A3, A3) theory.
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N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
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] [INSPIRE].
U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multiinstanton calculus and equivariant cohomology, JHEP 05 (2003) 054 [hep-th/0211108] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
H. Nakajima and K. Yoshioka, Lectures on instanton counting, in CRM Workshop on Algebraic Structures and Moduli Spaces, (2003) [math/0311058] [INSPIRE].
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas Theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
M. Del Zotto, C. Vafa and D. Xie, Geometric engineering, mirror symmetry and 6d(1,0) → \( 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} \), JHEP 11 (2015) 123 [arXiv:1504.08348] [INSPIRE].
S. Cecotti and M. Del Zotto, Higher S-dualities and Shephard-Todd groups, JHEP 09 (2015) 035 [arXiv:1507.01799] [INSPIRE].
D. Xie and S.-T. Yau, New N = 2 dualities, arXiv:1602.03529 [INSPIRE].
D. Xie and S.-T. Yau, Argyres-Douglas matter and N = 2 dualities, arXiv:1701.01123 [INSPIRE].
M. Buican, Z. Laczko and T. Nishinaka, \( \mathcal{N} \) = 2 S-duality revisited, JHEP 09 (2017) 087 [arXiv:1706.03797] [INSPIRE].
D. Xie and K. Ye, Argyres-Douglas matter and S-duality: Part II, JHEP 03 (2018) 186 [arXiv:1711.06684] [INSPIRE].
M. Buican, Z. Laczko and T. Nishinaka, Flowing from 16 to 32 Supercharges, JHEP 10 (2018) 175 [arXiv:1807.02785] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Study of N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 471 (1996) 430 [hep-th/9603002] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
D. Gaiotto, Asymptotically free \( \mathcal{N} \) = 2 theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
H. Kanno and M. Taki, Generalized Whittaker states for instanton counting with fundamental hypermultiplets, JHEP 05 (2012) 052 [arXiv:1203.1427] [INSPIRE].
T. Nishinaka and C. Rim, Matrix models for irregular conformal blocks and Argyres-Douglas theories, JHEP 10 (2012) 138 [arXiv:1207.4480] [INSPIRE].
C. Rim, Irregular conformal block and its matrix model, arXiv:1210.7925 [INSPIRE].
H. Kanno, K. Maruyoshi, S. Shiba and M. Taki, \( {\mathcal{W}}_3 \) irregular states and isolated \( \mathcal{N} \) = 2 superconformal field theories, JHEP 03 (2013) 147 [arXiv:1301.0721] [INSPIRE].
Y. Matsuo, C. Rim and H. Zhang, Construction of Gaiotto states with fundamental multiplets through Degenerate DAHA, JHEP 09 (2014) 028 [arXiv:1405.3141] [INSPIRE].
S.K. Choi, C. Rim and H. Zhang, Irregular conformal block, spectral curve and flow equations, JHEP 03 (2016) 118 [arXiv:1510.09060] [INSPIRE].
C. Rim and H. Zhang, Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation, Nucl. Phys. B 919 (2017) 182 [arXiv:1608.05027] [INSPIRE].
H. Itoyama, T. Oota and K. Yano, Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory, Phys. Lett. B 789 (2019) 605 [arXiv:1805.05057] [INSPIRE].
H. Itoyama, T. Oota and K. Yano, Discrete Painlevé system for the partition function of Nf = 2 SU(2) supersymmetric gauge theory and its double scaling limit, J. Phys. A 52 (2019) 415401 [arXiv:1812.00811] [INSPIRE].
T. Nishinaka and T. Uetoko, Argyres-Douglas theories and Liouville Irregular States, JHEP 09 (2019) 104 [arXiv:1905.03795] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
C. Beem and W. Peelaers, Argyres-Douglas Theories in Class S Without Irregularity, arXiv:2005.12282 [INSPIRE].
N. Wyllard, AN−1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
V.A. Alba, V.A. Fateev, A.V. Litvinov and G.M. Tarnopolskiy, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 (2011) 33 [arXiv:1012.1312] [INSPIRE].
M. Buican, L. Li and T. Nishinaka, Peculiar Index Relations, 2D TQFT, and Universality of SUSY Enhancement, JHEP 01 (2020) 187 [arXiv:1907.01579] [INSPIRE].
T.W. Grimm, A. Klemm, M. Mariño and M. Weiss, Direct Integration of the Topological String, JHEP 08 (2007) 058 [hep-th/0702187] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in 16th International Congress on Mathematical Physics, (2009) [DOI] [arXiv:0908.4052] [INSPIRE].
K. Ito, S. Kanno and T. Okubo, Quantum periods and prepotential in \( \mathcal{N} \) = 2 SU(2) SQCD, JHEP 08 (2017) 065 [arXiv:1705.09120] [INSPIRE].
K. Ito and T. Okubo, Quantum periods for \( \mathcal{N} \) = 2 SU(2) SQCD around the superconformal point, Nucl. Phys. B 934 (2018) 356 [arXiv:1804.04815] [INSPIRE].
K. Ito, S. Koizumi and T. Okubo, Quantum Seiberg-Witten curve and Universality in Argyres-Douglas theories, Phys. Lett. B 792 (2019) 29 [arXiv:1903.00168] [INSPIRE].
K. Ito, S. Koizumi and T. Okubo, Quantum Seiberg-Witten periods for \( \mathcal{N} \) = 2 SU(Nc) SQCD around the superconformal point, Nucl. Phys. B 954 (2020) 115004 [arXiv:2001.08891] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT Relation and the Deformed beta-ensemble, Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [INSPIRE].
S. Yanagida, Five-dimensional SU(2) AGT conjecture and recursive formula of deformed Gaiotto state, J. Math. Phys. 51 (2010) 123506 [arXiv:1005.0216] [INSPIRE].
M. Taki, On AGT-W Conjecture and q-Deformed W-Algebra, arXiv:1403.7016 [INSPIRE].
V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings, JHEP 06 (2015) 049 [arXiv:1409.6313] [INSPIRE].
M. Isachenkov, V. Mitev and E. Pomoni, Toda 3-Point Functions From Topological Strings II, JHEP 08 (2016) 066 [arXiv:1412.3395] [INSPIRE].
H. Awata, H. Fujino and Y. Ohkubo, Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence, J. Math. Phys. 58 (2017) 071704 [arXiv:1512.08016] [INSPIRE].
J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang and R.-D. Zhu, Coherent states in quantum \( {\mathcal{W}}_{1+\infty } \) algebra and qq-character for 5d Super Yang-Mills, PTEP 2016 (2016) 123B05 [arXiv:1606.08020] [INSPIRE].
S. Pasquetti, Holomorphic blocks and the 5d AGT correspondence, J. Phys. A 50 (2017) 443016 [arXiv:1608.02968] [INSPIRE].
A. Neguţ, The q-AGT-W relations via shuffle algebras, Commun. Math. Phys. 358 (2018) 101 [arXiv:1608.08613] [INSPIRE].
H. Hayashi and K. Ohmori, 5d/6d DE instantons from trivalent gluing of web diagrams, JHEP 06 (2017) 078 [arXiv:1702.07263] [INSPIRE].
S. Kanno, Y. Matsuo and S. Shiba, W(1 + ∞) algebra as a symmetry behind AGT relation, Phys. Rev. D 84 (2011) 026007 [arXiv:1105.1667] [INSPIRE].
H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi and S. Yanagida, Notes on Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE].
S. Kanno, Y. Matsuo and H. Zhang, Extended Conformal Symmetry and Recursion Formulae for Nekrasov Partition Function, JHEP 08 (2013) 028 [arXiv:1306.1523] [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SHc and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].
A. Mironov, A. Morozov and Y. Zenkevich, Ding-Iohara-Miki symmetry of network matrix models, Phys. Lett. B 762 (2016) 196 [arXiv:1603.05467] [INSPIRE].
H. Awata et al., Explicit examples of DIM constraints for network matrix models, JHEP 07 (2016) 103 [arXiv:1604.08366] [INSPIRE].
H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems. ℛ-matrix and \( \mathrm{\mathcal{R}}\mathcal{TT} \) relations, JHEP 10 (2016) 047 [arXiv:1608.05351] [INSPIRE].
H. Awata et al., Anomaly in RTT relation for DIM algebra and network matrix models, Nucl. Phys. B 918 (2017) 358 [arXiv:1611.07304] [INSPIRE].
H. Awata et al., Generalized Knizhnik-Zamolodchikov equation for Ding-Iohara-Miki algebra, Phys. Rev. D 96 (2017) 026021 [arXiv:1703.06084] [INSPIRE].
J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo and R.-D. Zhu, (p, q)-webs of DIM representations, 5d \( \mathcal{N} \) = 1 instanton partition functions and qq-characters, JHEP 11 (2017) 034 [arXiv:1703.10759] [INSPIRE].
J.-E. Bourgine, M. Fukuda, Y. Matsuo and R.-D. Zhu, Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver, JHEP 12 (2017) 015 [arXiv:1709.01954] [INSPIRE].
H. Awata, H. Kanno, A. Mironov, A. Morozov, K. Suetake and Y. Zenkevich, (q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces, JHEP 03 (2018) 192 [arXiv:1712.08016] [INSPIRE].
S. Sasa, A. Watanabe and Y. Matsuo, A note on the S-dual basis in the free fermion system, PTEP 2020 (2020) 023B02 [arXiv:1904.04766] [INSPIRE].
M. Buican and T. Nishinaka, \( \mathcal{N} \) = 4 SYM, Argyres-Douglas Theories, and an Exact Graded Vector Space Isomorphism, arXiv:2012.13209 [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite Chiral Symmetry in Four Dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
K. Ito and H. Shu, ODE/IM correspondence and the Argyres-Douglas theory, JHEP 08 (2017) 071 [arXiv:1707.03596] [INSPIRE].
L. Hollands, C.A. Keller and J. Song, From SO/Sp instantons to W-algebra blocks, JHEP 03 (2011) 053 [arXiv:1012.4468] [INSPIRE].
L. Hollands, C.A. Keller and J. Song, Towards a 4d/2d correspondence for Sicilian quivers, JHEP 10 (2011) 100 [arXiv:1107.0973] [INSPIRE].
F. Fucito, J.F. Morales and R. Poghossian, Instantons on quivers and orientifolds, JHEP 10 (2004) 037 [hep-th/0408090] [INSPIRE].
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Kimura, T., Nishinaka, T., Sugawara, Y. et al. Argyres-Douglas theories, S-duality and AGT correspondence. J. High Energ. Phys. 2021, 205 (2021). https://doi.org/10.1007/JHEP04(2021)205
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DOI: https://doi.org/10.1007/JHEP04(2021)205