Abstract
Superconformal indices of 4d \( \mathcal{N} \) = 1 SYM theories with SU(N) and SP(2N) gauge groups are investigated for N f = N and N f = N + 1 flavors, respectively. These indices vanish for generic values of the flavor fugacities. However, for a singular submanifold of fugacities they behave like the Dirac delta functions and describe the chiral symmetry breaking phenomenon. Similar picture holds for partition functions of 3d supersymmetric field theories with the chiral symmetry breaking.
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References
F.A. Dolan and H. Osborn, Applications of the superconformal index for protected operators and q-hypergeometric identities to \( \mathcal{N} \) = 1 dual theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
V.P. Spiridonov, On the elliptic beta function, Usp. Mat. Nauk 56 (2001) 181 [Russ. Math. Surv. 56 (2001) 185].
V.P. Spiridonov, Theta hypergeometric integrals, Alg. Analiz. 15 (2003) 161 [St. Petersburg Math. J. 15 (2004) 929] [math.CA/0303205].
V.P. Spiridonov, Essays on the theory of elliptic hypergeometric functions, Usp. Mat. Nauk 63 (2008) 3 [Russ. Math. Surv. 63 (2008) 405] [arXiv:0805.3135].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
C. Romelsberger, Counting chiral primaries in \( \mathcal{N} \) = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
C. Romelsberger, Calculating the superconformal index and Seiberg duality, arXiv:0707.3702 [INSPIRE].
G. Festuccia and N. Seiberg, Rigid supersymmetric theories in curved superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
N. Seiberg, Electric-magnetic duality in supersymmetric non-Abelian gauge theories, Nucl. Phys. B 435 (1995) 129 [hep-th/9411149] [INSPIRE].
E.M. Rains, Transformations of elliptic hypergeometric integrals, Ann. Math. 171 (2010) 169.
V.P. Spiridonov, Short proofs of the elliptic beta integrals, Ramanujan J. 13 (2007) 265 [math.CA/0408369].
V.P. Spiridonov and G.S. Vartanov, Superconformal indices for \( \mathcal{N} \) = 1 theories with multiple duals, Nucl. Phys. B 824 (2010) 192 [arXiv:0811.1909] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Supersymmetric dualities beyond the conformal window, Phys. Rev. Lett. 105 (2010) 061603 [arXiv:1003.6109] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometry of supersymmetric dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots and vortices, Commun. Math. Phys. 325 (2014) 421 [arXiv:1107.5788] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Superconformal indices of \( \mathcal{N} \) = 4 SYM field theories, Lett. Math. Phys. 100 (2012) 97 [arXiv:1005.4196] [INSPIRE].
G.S. Vartanov, On the ISS model of dynamical SUSY breaking, Phys. Lett. B 696 (2011) 288 [arXiv:1009.2153] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions, JHEP 06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The superconformal index of the E 6 SCFT, JHEP 08 (2010) 107 [arXiv:1003.4244] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
M. Sudano, The Romelsberger index, Berkooz deconfinement and infinite families of Seiberg duals, JHEP 05 (2012) 051 [arXiv:1112.2996] [INSPIRE].
B.I. Zwiebel, Charging the superconformal index, JHEP 01 (2012) 116 [arXiv:1111.1773] [INSPIRE].
Y. Nakayama, 4D and 2D superconformal index with surface operator, JHEP 08 (2011) 084 [arXiv:1105.4883] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, arXiv:1112.5179 [INSPIRE].
D. Gang, E. Koh and K. Lee, Line operator index on S 1 × S 3, JHEP 05 (2012) 007 [arXiv:1201.5539] [INSPIRE].
D. Gang, E. Koh and K. Lee, Superconformal index with duality domain wall, JHEP 10 (2012) 187 [arXiv:1205.0069] [INSPIRE].
V.P. Spiridonov, Elliptic beta integrals and solvable models of statistical mechanics, Contemp. Math. 563 (2012) 181 [arXiv:1011.3798] [INSPIRE].
T. Dimofte and D. Gaiotto, An E 7 surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [INSPIRE].
I. Gahramanov and G. Vartanov, Extended global symmetries for 4D \( \mathcal{N} \) = 1 SQCD theories, J. Phys. A 46 (2013) 285403 [arXiv:1303.1443] [INSPIRE].
N. Seiberg, Exact results on the space of vacua of four-dimensional SUSY gauge theories, Phys. Rev. D 49 (1994) 6857 [hep-th/9402044] [INSPIRE].
N.I. Muskhelishvili, Singular integral equations, Springer, Germany (1977).
B. Grinstein and D.R. Nolte, Systematic study of theories with quantum modified moduli, Phys. Rev. D 57 (1998) 6471 [hep-th/9710001] [INSPIRE].
B. Grinstein and D.R. Nolte, Systematic study of theories with quantum modified moduli. 2, Phys. Rev. D 58 (1998) 045012 [hep-th/9803139] [INSPIRE].
V.P. Spiridonov, Continuous biorthogonality of the elliptic hypergeometric function, Alg. Analiz. 20 (2008) 155 [St. Petersburg Math. J. 20 (2009) 791] [arXiv:0801.4137].
C. Csáki, M. Schmaltz and W. Skiba, A systematic approach to confinement in \( \mathcal{N} \) = 1 supersymmetric gauge theories, Phys. Rev. Lett. 78 (1997) 799 [hep-th/9610139] [INSPIRE].
C. Csáki, M. Schmaltz and W. Skiba, Confinement in \( \mathcal{N} \) = 1 SUSY gauge theories and model building tools, Phys. Rev. D 55 (1997) 7840 [hep-th/9612207] [INSPIRE].
V.P. Spiridonov, A Bailey tree for integrals, Teor. Mat. Fiz. 139 (2004) 104] [Theor. Math. Phys. 139 (2004) 536] [math.CA/0312502].
V.P. Spiridonov and S.O. Warnaar, Inversions of integral operators and elliptic beta integrals on root systems, Adv. Math. 207 (2006) 91.
S.E. Derkachov and V.P. Spiridonov, Yang-Baxter equation, parameter permutations and the elliptic beta integral, Usp. Mat. Nauk 68 (2013) 59 [Russ. Math. Surv. 68 (2013) 1027] [arXiv:1205.3520] [INSPIRE].
N.M. Davies, T.J. Hollowood, V.V. Khoze and M.P. Mattis, Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nucl. Phys. B 559 (1999) 123 [hep-th/9905015] [INSPIRE].
K. Konishi and A. Ricco, Calculating gluino condensates in \( \mathcal{N} \) = 1 SYM from Seiberg-Witten curves, Phys. Lett. B 570 (2003) 118 [hep-th/0306128] [INSPIRE].
K.A. Intriligator and P. Pouliot, Exact superpotentials, quantum vacua and duality in supersymmetric SP(N c ) gauge theories, Phys. Lett. B 353 (1995) 471 [hep-th/9505006] [INSPIRE].
J.F. van Diejen and V.P. Spiridonov, Elliptic Selberg integrals, Internat. Math. Res. Notices 20 (2001) 1083.
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].
V. Niarchos, Seiberg dualities and the 3d/4d connection, JHEP 07 (2012) 075 [arXiv:1205.2086] [INSPIRE].
F.A.H. Dolan, V.P. Spiridonov and G.S. Vartanov, From 4d superconformal indices to 3d partition functions, Phys. Lett. B 704 (2011) 234 [arXiv:1104.1787] [INSPIRE].
A. Gadde and W. Yan, Reducing the 4d index to the S 3 partition function, JHEP 12 (2012) 003 [arXiv:1104.2592] [INSPIRE].
Y. Imamura, Relation between the 4d superconformal index and the S 3 partition function, JHEP 09 (2011) 133 [arXiv:1104.4482] [INSPIRE].
O. Aharony, S.S. Razamat, N. Seiberg and B. Willett, 3d dualities from 4d dualities, JHEP 07 (2013) 149 [arXiv:1305.3924] [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of \( \mathcal{N} \) = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
T. Morita and V. Niarchos, F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-matter theories, Nucl. Phys. B 858 (2012) 84 [arXiv:1108.4963] [INSPIRE].
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Spiridonov, V.P., Vartanov, G.S. Vanishing superconformal indices and the chiral symmetry breaking. J. High Energ. Phys. 2014, 62 (2014). https://doi.org/10.1007/JHEP06(2014)062
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DOI: https://doi.org/10.1007/JHEP06(2014)062