Abstract
We give a full list of known \({\mathcal{N}=1}\) supersymmetric quantum field theories related by the Seiberg electric-magnetic duality conjectures for SU(N), SP(2N) and G 2 gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Römelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them were proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen new elliptic beta integrals on root systems extending the univariate elliptic beta integral discovered by the first author.
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Andrews, G. E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Math. Appl. 71, Cambridge: Cambridge Univ. Press, 1999
Barnes E.W.: On the theory of the multiple gamma function. Cambr. Trans. 19, 374–425 (1904)
Baxter R.J.: Partition function of the eight-vertex lattice model. Ann. Phys. (NY) 70, 193–228 (1972)
Benvenuti S., Feng B., Hanany A., He Y.H.: Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics. JHEP 0711, 050 (2007)
Berkooz M.: The dual of supersymmetric SU(2k) with an antisymmetric tensor and composite dualities. Nucl. Phys. B 452, 513–525 (1995)
Bianchi M., Dolan F. A., Heslop P. J., Osborn H.: N = 4 superconformal characters and partition functions. Nucl. Phys. B 767, 163–226 (2007)
Brodie J.H.: Duality in supersymmetric SU(N c ) gauge theory with two adjoint chiral superfields. Nucl. Phys. B 478, 123–140 (1996)
Brodie J.H., Strassler M.J.: Patterns of duality in \({\mathcal{N}=1}\) SUSY gauge theories, or: Seating preferences of theater going nonAbelian dualities. Nucl. Phys. B 524, 224–250 (1998)
van de Bult, F. J.: An elliptic hypergeometric beta integral transformation. http://arXiv.org/abs/0912.3812v1 [math.CA], 2009
van de Bult F.J., Rains E.M.: Basic hypergeometric functions as limits of elliptic hypergeometric functions. SIGMA 5, 59 (2009)
Cho P. L.: Moduli in exceptional SUSY gauge theories. Phys. Rev. D 57, 5214–5223 (1998)
Cho P.L., Kraus P.: Symplectic SUSY gauge theories with antisymmetric matter. Phys. Rev. D 54, 7640–7649 (1996)
Csáki C., Murayama H.: Discrete anomaly matching. Nucl. Phys. B 515, 114–162 (1998)
Csáki C., Murayama H.: New confining \({\mathcal{N}=1}\) supersymmetric gauge theories. Phys. Rev. D 59, 065001 (1999)
Csáki C., Schmaltz M., Skiba W.: A systematic approach to confinement in \({\mathcal{N}=1}\) supersymmetric gauge theories. Phys. Rev. Lett. 78, 799–802 (1997)
Csáki C., Schmaltz M., Skiba W.: Confinement in \({{\mathcal N}=1}\) SUSY gauge theories and model building tools. Phys. Rev. D 55, 7840–7858 (1997)
Csáki C., Schmaltz M., Skiba W., Terning J.: Selfdual \({{\mathcal N}=1}\) SUSY gauge theories. Phys. Rev. D 56, 1228–1238 (1997)
Csáki C., Skiba W., Schmaltz M.: Exact results and duality for SP(2N) SUSY gauge theories with an antisymmetric tensor. Nucl. Phys. B 487, 128–140 (1997)
van Diejen J.F., Spiridonov V.P.: An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums. Math. Res. Lett. 7, 729–746 (2000)
van Diejen, J. F., Spiridonov, V. P.: Elliptic Selberg integrals, Internat. Math. Res. Notices, no. 20, 1083–1110 (2001)
van Diejen J.F., Spiridonov V.P.: Unit circle elliptic beta integrals. Ramanujan J 10, 187–204 (2005)
Distler J., Karch A.: \({\mathcal{N} = 1}\) dualities for exceptional gauge groups and quantum global symmetries. Fortsch. Phys. 45, 517–533 (1997)
Dixon A.L.: On a generalisation of Legendre’s formula \({KE'-(K-E)K'=\frac{1}{2}\pi}\). Proc. London Math. Soc. 2(1), 206–224 (1905)
Dolan F.A.: Counting BPS operators in \({\mathcal{N}=4}\) SYM. Nucl. Phys. B 790, 432–464 (2008)
Dolan F. A., Osborn H.: On short and semi-short representations for four dimensional superconformal symmetry. Ann. Phys. (NY) 307, 41–89 (2003)
Dolan F.A., Osborn H.: Applications of the superconformal index for protected operators and q-hypergeometric identities to \({{\mathcal N}=1}\) dual theories. Nucl. Phys. B 818, 137–178 (2009)
Eichler, M., Zagier, D.: The Theory of Jacobi Forms. Progress in Math. 55, Boston: Birkhäuser, 1985
Felder G., Varchenko A.: The elliptic gamma function and \({SL(3,{\mathbb Z})\ltimes{\mathbb Z}^3}\). Adv. Math. 156, 44–76 (2000)
Feng B., Hanany A., He Y.H.: Counting gauge invariants: the plethystic program. JHEP 0703, 090 (2007)
Forrester P.J., Warnaar S.O.: The importance of the Selberg integral. Bull. Amer. Math. Soc. (N.S.) 45, 489–534 (2008)
Friedman E., Ruijsenaars S.: Shintani-Barnes zeta and gamma functions. Adv. Math. 187, 362–395 (2004)
Gadde A., Pomoni E., Rastelli L., Razamat S.S.: S-duality and 2d Topological QFT. JHEP 03, 032 (2010)
Giddings S.B., Pierre J.M.: Some exact results in supersymmetric theories based on exceptional groups. Phys. Rev. D 52, 6065–6073 (1995)
Gray J., Hanany A., He Y. H., Jejjala V., Mekareeya N.: SQCD: A geometric apercu. JHEP 0805, 099 (2008)
Gustafson R.A.: Some q-beta and Mellin-Barnes integrals with many parameters associated to the classical groups. SIAM J. Math. Anal. 23, 525–551 (1992)
Gustafson R.A.: Some q-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras. Trans. AMS 341(1), 69–119 (1994)
Gustafson R.A.: Some q-beta integrals on SU(n) and Sp(n) that generalize the Askey–Wilson and Nassrallah-Rahman integrals. SIAM J. Math. Anal. 25, 441–449 (1994)
Gustafson R.A., Rakha M.A.: q-Beta integrals and multivariate basic hypergeometric series associated to root systems of type A m . Ann. Comb. 4, 347–373 (2000)
Hanany A., Mekareeya N.: Counting gauge invariant operators in SQCD with classical gauge groups. JHEP 0810, 012 (2008)
’t Hooft, G.: Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking. In: Recent Developments in Gauge Theories (eds. G. ’t Hooft et. al.), New York: Plenum Press, 1980, pp. 135–157
Intriligator K.: New RG fixed points and duality in supersymmetric SP(N c ) and SO(N c ) gauge theories. Nucl. Phys. B 448, 187–198 (1995)
Intriligator K.A., Leigh R.G., Strassler M.J.: New examples of duality in chiral and nonchiral supersymmetric gauge theories. Nucl. Phys. B 456, 567–621 (1995)
Intriligator K.A., Pouliot P.: Exact superpotentials, quantum vacua and duality in supersymmetric SP(N c ) gauge theories. Phys. Lett. B 353, 471–476 (1995)
Intriligator K.A., Seiberg N.: Phases of \({\mathcal{N}=1}\) supersymmetric gauge theories in four-dimensions. Nucl. Phys. B 431, 551–568 (1994)
Intriligator K.A., Seiberg N.: Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(N c ) gauge theories. Nucl. Phys. B 444, 125–160 (1995)
Intriligator, K. A., Seiberg, N.: Phases of \({\mathcal{N}=1}\) supersymmetric gauge theories and electric - magnetic triality. In: Future perspectives in string theory, (Los Angeles 1995), Singapore: World Scientific, 1996 pp. 270–282
Intriligator K.A., Seiberg N.: Lectures on supersymmetric gauge theories and electric - magnetic duality. Nucl. Phys. Proc. Suppl. 45BC, 1–28 (1996)
Karch A.: More on \({\mathcal{N}=1}\) selfdualities and exceptional gauge groups. Phys. Lett. B 405, 280–286 (1997)
Kinney J., Maldacena J.M., Minwalla S., Raju S.: An index for 4 dimensional super conformal theories. Commun. Math. Phys. 275, 209–254 (2007)
Klein M.: More confining \({\mathcal{N}=1}\) SUSY gauge theories from nonAbelian duality. Nucl. Phys. B 553, 155–204 (1999)
Kutasov D.: A comment on duality in \({{\mathcal N}=1}\) supersymmetric non-Abelian gauge theories. Phys. Lett. B 351, 230–234 (1995)
Kutasov D., Schwimmer A.: On duality in supersymmetric Yang-Mills theory. Phys. Lett. B 354, 315–321 (1995)
Kutasov D., Schwimmer A., Seiberg N.: Chiral rings, singularity theory and electric-magnetic duality. Nucl. Phys. B 459, 455–496 (1996)
Leigh R.G., Strassler M.J.: Duality of Sp(2N c ) and SO(N c ) supersymmetric gauge theories with adjoint matter. Phys. Lett. B 356, 492–499 (1995)
Leigh R. G., Strassler M. J.: Accidental symmetries and N = 1 duality in supersymmetric gauge theory. Nucl. Phys. B 496, 132–148 (1997)
Machacek, M. E., Vaughn, M. T.: Two loop renormalization group equations in a general quantum field theory. 1. Wave function renormalization. Nucl. Phys. B 222 83–103(1983); 2. Yukawa couplings, ibid. B 236 221–232 (1984); 3. Scalar quartic couplings, ibid. B 249 70–92(1985)
Nakayama Y.: Index for orbifold quiver gauge theories. Phys. Lett. B 636, 132–136 (2006)
Nekrasov N. A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003)
Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. In: The Unity of Mathematics Eds. P. Etingof, V. Retakh, I. M. Singer, Progr. Math., 244, Boston, MA: Birkhauser, 2006, pp. 525–596
Pesando I.: Exact results for the supersymmetric G 2 gauge theories. Mod. Phys. Lett. A 10, 1871–1886 (1995)
Pouliot P.: Chiral duals of nonchiral SUSY gauge theories. Phys. Lett. B 359, 108–113 (1995)
Pouliot P.: Duality in SUSY SU(N) with an antisymmetric tensor. Phys. Lett. B 367, 151–156 (1996)
Pouliot P.: Spectroscopy of gauge theories based on exceptional Lie groups. J. Phys. A 34, 8631–8658 (2001)
Poppitz E., Trivedi S.P.: Some examples of chiral moduli spaces and dynamical supersymmetry breaking. Phys. Lett. B 365, 125–131 (1996)
Rains E.M.: Transformations of elliptic hypergeometric integrals. Ann. of Math. 171, 169–243 (2010)
Rains E.M.: BC n -symmetric abelian functions. Duke Math. J. 135(1), 99–180 (2006)
Rains E.M.: Limits of elliptic hypergeometric integrals. Ramanujan J. 18(3), 257–306 (2009)
Rains, E. M.: Elliptic Littlewood identities. http://arXiv.org/abs/0806.0871v1 [math.CO], 2008
Rains, E. M., Spiridonov, V. P.: Determinants of elliptic hypergeometric integrals. Funkt. Analiz i ego Pril. 43 (4), 67–86 (2009) (Funct. Anal. Appl. 43 (4), 297–311 (2009))
Ramond P.: Superalgebras in \({{\mathcal N} = 1}\) gauge theories. Phys. Lett. B 390, 179–184 (1997)
Reiman, A. G., Semenov-Tian-Shansky, M. A., Faddeev, L. D.: Quantum anomalies and cocycles on gauge groups. Funkt. Analiz i ego Pril. 18 (4), 64–72 (1984) (Funct. Anal. Appl. 18 (4), 319–326 (1984))
Römelsberger C.: Counting chiral primaries in \({{\mathcal N}=1, d=4}\) superconformal field theories. Nucl. Phys. B 747, 329–353 (2006)
Römelsberger, C.: Calculating the superconformal index and Seiberg duality. http://arXiv.org/abs/0707.3702v1 [hep-th], 2007
Ruijsenaars S.N.M.: First order analytic difference equations and integrable quantum systems. J. Math. Phys. 38, 1069–1146 (1997)
Seiberg N.: Exact results on the space of vacua of four-dimensional SUSY gauge theories. Phys. Rev. D 49, 6857–6863 (1994)
Seiberg N.: Electric–magnetic duality in supersymmetric non-Abelian gauge theories. Nucl. Phys. B 435, 129–146 (1995)
Seiberg, N., Witten, E.: Electric - magnetic duality, monopole condensation, and confinement in \({\mathcal{N}=2}\) supersymmetric Yang-Mills theory. Nucl. Phys. B 426, 19–52 (1994), Erratum-ibid. B 430, 485–486(1994)
Seiberg N., Witten E.: Monopoles, duality and chiral symmetry breaking in \({\mathcal{N}=2}\) supersymmetric QCD. Nucl. Phys. B 431, 484–550 (1994)
Shifman M. A.: Nonperturbative dynamics in supersymmetric gauge theories. Prog. Part. Nucl. Phys. 39, 1–116 (1997)
Skagerstam B.S.: On the large N c limit of the SU(N c ) colour quark-gluon partition function. Z. Phys. C 24, 97–101 (1984)
Spiridonov, V. P.: On the elliptic beta function. Usp. Mat. Nauk 56 (1), 181–182 (2001) (Russ. Math. Surv 56 (1), 185–186 (2001))
Spiridonov, V. P.: Theta hypergeometric series. In: Proceedings of the NATO ASI Asymptotic Combinatorics with Applications to Mathematical Physics (St. Petersburg, Russia, July 2001), Eds. Malyshev, V. A., Vershik, A. M., Amsterdam: Kluwer, 2002, pp. 307–327
Spiridonov, V. P.: Theta hypergeometric integrals. Algebra i Analiz 15 (6), 161–215 (2003) (St. Petersburg Math. J. 15 (6), 929–967(2003))
Spiridonov, V. P.: A Bailey tree for integrals. Teor. Mat. Fiz. 139, 104–111 (2004) (Theor. Math. Phys. 139, 536–541 (2004))
Spiridonov, V. P.: Elliptic hypergeometric functions. Habilitation thesis, Bogoliubov Laboratory of Theoretical Physics, JINR, September 2004, 218 pp. (Russian)
Spiridonov V.P.: Short proofs of the elliptic beta integrals. Ramanujan J 13, 265–283 (2007)
Spiridonov, V. P.: Elliptic hypergeometric functions and Calogero-Sutherland type models. Teor. Mat. Fiz, 150 (2), 311–324 (2007) (Theor. Math. Phys. 150 (2), 266–277 (2007))
Spiridonov V.P.: Continuous biorthogonality of the elliptic hypergeometric function. Algebra i Analiz (St. Petersburg Math. J.) 20(5), 155–185 (2008)
Spiridonov, V. P.: Essays on the theory of elliptic hypergeometric functions. Usp. Mat. Nauk 63(3), 3–72 (2008) (Russ. Math. Surv. 63(3), 405–472 (2008))
Spiridonov, V. P.: Elliptic hypergeometric terms. In: Proc. of the Workshop “Théories galoisiennes et arithmétiques des équations différentielles” (September 2009, CIRM, Luminy, France), to appear, http://arXiv.orb/abs/1003.4491v2 [math.CA], 2010
Spiridonov V.P., Vartanov G.S.: Superconformal indices for \({{\mathcal N}=1}\) theories with multiple duals. Nucl. Phys. B 824, 192–216 (2010)
Spiridonov V.P., Vartanov G.S.: Supersymmetric dualities beyond the conformal window. Phys. Rev. Lett. 105, 061603 (2010)
Spiridonov V.P., Warnaar S.O.: Inversions of integral operators and elliptic beta integrals on root systems. Adv. Math. 207, 91–132 (2006)
Spiridonov, V. P., Warnaar, S. O.: New multiple 6 ψ 6 summation formulas and related conjectures. Preprint August (2009)
Sundborg B.: The Hagedorn transition, deconfinement and \({{\mathcal N}=4}\) SYM theory. Nucl. Phys. B 573, 349–363 (2000)
Takhtadzhyan, L. A., Faddeev, L. D.: The quantum method of the inverse problem and the Heisenberg XYZ model. Usp. Mat. Nauk 34(5), 13–63 (1979) (Russ. Math. Surv. 34(5), 11–68 (1979))
Terning J.: Modern Supersymmetry: Dynamics and Duality. Clarendon, Oxford, UK (2006)
Volkov A.: Noncommutative hypergeometry. Commun. Math. Phys. 258, 257–273 (2005)
Witten E.: Constraints on supersymmetry breaking. Nucl. Phys. B 202, 253–316 (1982)
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Spiridonov, V.P., Vartanov, G.S. Elliptic Hypergeometry of Supersymmetric Dualities. Commun. Math. Phys. 304, 797–874 (2011). https://doi.org/10.1007/s00220-011-1218-9
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DOI: https://doi.org/10.1007/s00220-011-1218-9