Abstract
We study \( \mathcal{N} = {2} \) supersymmetric SU(2) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A1 (2,0) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Ω-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.
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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, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
G. Bonelli and A. Tanzini, Hitchin systems, N = 2 gauge theories and W-gravity, Phys. Lett. B 691 (2010) 111 [arXiv:0909.4031] [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].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E(n) global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
P.C. Argyres and N. Seiberg, S-duality in N = 2 supersymmetric gauge theories, JHEP 12 (2007) 088 [arXiv:0711.0054] [INSPIRE].
D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [INSPIRE].
S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, arXiv:1103.5832 [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [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].
D. Gaiotto, Asymptotically free N = 2 theories and irregular conformal blocks, arXiv:0908.0307 [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4D/2D correspondences, arXiv:1006.3435 [INSPIRE].
P.C. Argyres, M. 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].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485-486] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
P.C. Argyres, M. Plesser and A.D. Shapere, The Coulomb phase of N = 2 supersymmetric QCD, Phys. Rev. Lett. 75 (1995) 1699 [hep-th/9505100] [INSPIRE].
A. Hanany and Y. Oz, On the quantum moduli space of vacua of N = 2 supersymmetric SU(N c) gauge theories, Nucl. Phys. B 452 (1995) 283 [hep-th/9505075] [INSPIRE].
D. Gaiotto, N. Seiberg and Y. Tachikawa, Comments on scaling limits of 4D N = 2 theories, JHEP 01 (2011) 078 [arXiv:1011.4568] [INSPIRE].
O. Aharony and Y. Tachikawa, A holographic computation of the central charges of D = 4, N =2 SCFTs,JHEP 01 (2008) 037 [arXiv:0711.4532] [INSPIRE].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
D. Nanopoulos and D. Xie, Hitchin equation, singularity and N = 2 superconformal field theories, JHEP 03 (2010) 043 [arXiv:0911.1990] [INSPIRE].
N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91 [INSPIRE].
N.J. Hitchin, The selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
C. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990) 713.
E. Markman, Spectral curves and integrable systems, Comp. Math. 93 (1994) 255.
R. Donagi and E. Markman, Spectral curves, algebraically completely integrable Hamiltonian systems and moduli of bundles, alg-geom/9507017 [INSPIRE].
O. Biquard and P. Boalch, Wild nonabelian Hodge theory on curves, math.DG/0111098.
D. Nanopoulos and D. Xie, Hitchin equation, irregular singularity and N = 2 asymptotical free theories, arXiv:1005.1350 [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
J.A. Harvey, G.W. Moore and A. Strominger, Reducing S duality to T duality, Phys. Rev. D 52 (1995) 7161 [hep-th/9501022] [INSPIRE].
M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4 − D SYM to 2 − D σ-models,Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].
A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands program, hep-th/0604151 [INSPIRE].
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
E. Witten, Gauge theory and wild ramification, arXiv:0710.0631 [INSPIRE].
S.A. Cherkis and A. Kapustin, Singular monopoles and supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 525 (1998) 215 [hep-th/9711145] [INSPIRE].
A. Kapustin, Solution of N = 2 gauge theories via compactification to three-dimensions, Nucl. Phys. B 534 (1998) 531 [hep-th/9804069] [INSPIRE].
M. Taki, On AGT conjecture for pure Super Yang-Mills and W-algebra, JHEP 05 (2011) 038 [arXiv:0912.4789] [INSPIRE].
C. Kozcaz, S. Pasquetti, F. Passerini and N. Wyllard, Affine SL(N ) conformal blocks from N =2 SU(N) gauge theories, JHEP 01 (2011) 045 [arXiv:1008.1412] [INSPIRE].
N. Wyllard, W-algebras and surface operators in N = 2 gauge theories, J. Phys. A 44 (2011) 155401 [arXiv:1011.0289] [INSPIRE].
N. Wyllard, Instanton partition functions in N = 2 SU(N ) gauge theories with a general surface operator and their W-algebra duals, JHEP 02 (2011) 114 [arXiv:1012.1355] [INSPIRE].
H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
V. Belavin and B. Feigin, Super Liouville conformal blocks from N = 2 SU(2) quiver gauge theories, JHEP 07 (2011) 079 [arXiv:1105.5800] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Instantons on ALE spaces and super Liouville conformal field theories, JHEP 08 (2011) 056 [arXiv:1106.2505] [INSPIRE].
Y. Ito, Ramond sector of super Liouville theory from instantons on an ALE space, arXiv:1110.2176 [INSPIRE].
N. Wyllard, Coset conformal blocks and N = 2 gauge theories, arXiv:1109.4264 [INSPIRE].
C.A. Keller, N. Mekareeya, J. Song and Y. Tachikawa, The ABCDEFG of instantons and W-algebras, arXiv:1111.5624 [INSPIRE].
A. Braverman and P. Etingof, Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential, math/0409441 [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On non-conformal limit of the AGT relations, Phys. Lett. B 682 (2009) 125 [arXiv:0909.2052] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Proving the AGT relation for N f = 0, 1, 2 antifundamentals, JHEP 06 (2010) 046 [arXiv:1004.1841] [INSPIRE].
A. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
R. Poghossian, Recursion relations in CFT and N = 2 SYM theory, JHEP 12 (2009) 038 [arXiv:0909.3412] [INSPIRE].
L. Hadasz, Z. Jaskolski and P. Suchanek, Recursive representation of the torus 1-point conformal block, JHEP 01 (2010) 063 [arXiv:0911.2353] [INSPIRE].
V. Fateev and A. Litvinov, On AGT conjecture, JHEP 02 (2010) 014 [arXiv:0912.0504] [INSPIRE].
M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102] [INSPIRE].
G. Bonelli, K. Maruyoshi, A. Tanzini and F. Yagi, Generalized matrix models and AGT correspondence at all genera, JHEP 07 (2011) 055 [arXiv:1011.5417] [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].
A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
K. Maruyoshi and M. Taki, Deformed prepotential, quantum integrable system and Liouville field theory, Nucl. Phys. B 841 (2010) 388 [arXiv:1006.4505] [INSPIRE].
H. Awata, H. Fuji, H. Kanno, M. Manabe and Y. Yamada, Localization with a surface operator, irregular conformal blocks and open topological string, arXiv:1008.0574 [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operatorsin N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].
T. Dimofte, S. Gukov and L. Hollands, Vortex counting and lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE].
G. Bonelli, A. Tanzini and J. Zhao, The Liouville side of the vortex, JHEP 09 (2011) 096 [arXiv:1107.2787] [INSPIRE].
J. Teschner, Quantization of the Hitchin moduli spaces, Liouville theory and the geometric Langlands correspondence I, arXiv:1005.2846 [INSPIRE].
A. Marshakov, A. Mironov and A. Morozov, On AGT relations with surface operator insertion and stationary limit of beta-ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Quantum Hitchin systems via beta-deformed matrix models, arXiv:1104.4016 [INSPIRE].
T.-S. Tai, Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants, JHEP 10 (2010) 107 [arXiv:1008.4332] [INSPIRE].
M. Piatek, Classical conformal blocks from TBA for the elliptic Calogero-Moser system, JHEP 06 (2011) 050 [arXiv:1102.5403] [INSPIRE].
B. Feigin, E. Frenkel and N. Reshetikhin, Gaudin model, Bethe ansatz and correlation functions at the critical level, Commun. Math. Phys. 166 (1994) 27 [hep-th/9402022] [INSPIRE].
E. Frenkel, Gaudin model and opers, math/0407524 [INSPIRE].
B. Feigin, E. Frenkel and V. Toledano Laredo, Gaudin models with irregular singularities, Adv. Math. 223 (2010) 873 [math/0612798] [INSPIRE].
H. Itoyama, K. Maruyoshi and T. Oota, The quiver matrix model and 2D-4D conformal connection, Prog. Theor. Phys. 123 (2010) 957 [arXiv:0911.4244] [INSPIRE].
S. Kanno, Y. Matsuo, S. Shiba and Y. Tachikawa, N = 2 gauge theories and degenerate fields of Toda theory, Phys. Rev. D 81 (2010) 046004 [arXiv:0911.4787] [INSPIRE].
D. Nanopoulos and D. Xie, N = 2 generalized superconformal quiver gauge theory, arXiv:1006.3486 [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
N. Drukker and F. Passerini, (De)Tails of Toda CFT, JHEP 04 (2011) 106 [arXiv:1012.1352] [INSPIRE].
Y. Tachikawa, On W-algebras and the symmetries of defects of 6D N = (2, 0) theory, JHEP 03 (2011) 043 [arXiv:1102.0076] [INSPIRE].
R. Dijkgraaf and C. Vafa, Toda theories, matrix models, topological strings and N = 2 gauge systems, arXiv:0909.2453 [INSPIRE].
T. Eguchi and K. Maruyoshi, Penner type matrix model and Seiberg-Witten theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [INSPIRE].
T. Eguchi and K. Maruyoshi, Seiberg-Witten theory, matrix model and AGT relation, JHEP 07 (2010) 081 [arXiv:1006.0828] [INSPIRE].
A. Mironov, A. Morozov and S. Shakirov, Brezin-Gross-Witten model as ’pure gauge’ limit of Selberg integrals, JHEP 03 (2011) 102 [arXiv:1011.3481] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, arXiv:0908.4052 [INSPIRE].
N. Nekrasov, A. Rosly and S. Shatashvili, Darboux coordinates, Yang-Yang functional and gauge theory, Nucl. Phys. Proc. Suppl. 216 (2011) 69 [arXiv:1103.3919] [INSPIRE].
M.-x. Huang, A.-K. Kashani-Poor and A. Klemm, The Omega deformed B-model for rigid N = 2 theories,arXiv:1109.5728 [INSPIRE].
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Bonelli, G., Maruyoshi, K. & Tanzini, A. Wild quiver gauge theories. J. High Energ. Phys. 2012, 31 (2012). https://doi.org/10.1007/JHEP02(2012)031
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DOI: https://doi.org/10.1007/JHEP02(2012)031