### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 043, 30 pages      arXiv:1712.09933      https://doi.org/10.3842/SIGMA.2018.043
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory

Arash Arabi Ardehali
School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran

Received January 09, 2018, in final form April 29, 2018; Published online May 06, 2018

Abstract
The purpose of this article is to demonstrate that $i)$ the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and $ii)$ analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item $i$ was demonstrated by Dolan and Osborn in 2008. Item $ii$ was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elements were borrowed from the 2006 work of Rains on the hyperbolic limit of certain classes of EHIs. This article contains a concise review of these developments, along with minor refinements and clarifying remarks, written mainly for mathematicians interested in EHIs. In particular, we work with a representation-theoretic definition of a supersymmetric gauge theory, so that readers without any background in gauge theory - but familiar with the representation theory of semi-simple Lie algebras - can follow the discussion.

Key words: elliptic hypergeometric integrals; supersymmetric gauge theory; hyperbolic asymptotics.

pdf (804 kb)   tex (190 kb)

References

1. Aharony O., Razamat S.S., Seiberg N., Willett B., $3d$ dualities from $4d$ dualities, J. High Energy Phys. 2013 (2013), no. 7, 149, 70 pages, arXiv:1305.3924.
2. Ardehali A.A., High-temperature asymptotics of supersymmetric partition functions, J. High Energy Phys. 2016 (2016), no. 7, 025, 61 pages, arXiv:1512.03376.
3. Ardehali A.A., High-temperature asymptotics of the $4d$ superconformal index, Ph.D. Thesis, University of Michigan, 2016, arXiv:1605.06100.
4. Ardehali A.A., Liu J.T., Szepietowski P., High-temperature expansion of supersymmetric partition functions, J. High Energy Phys. 2015 (2015), no. 7, 113, 28 pages, arXiv:1502.07737.
5. Assel B., Cassani D., Martelli D., Localization on Hopf surfaces, J. High Energy Phys. 2014 (2014), no. 8, 123, 56 pages, arXiv:1405.5144.
6. Beem C., Gadde A., The $\mathcal{N}=1$ superconformal index for class $\mathcal{S}$ fixed points, J. High Energy Phys. 2014 (2014), no. 4, 036, 28 pages, arXiv:1212.1467.
7. Bourdier J., Drukker N., Felix J., The exact Schur index of ${\mathcal N}=4$ SYM, J. High Energy Phys. 2015 (2015), no. 11, 210, 10 pages, arXiv:1507.08659.
8. Brünner F., Regalado D., Spiridonov V.P., Supersymmetric Casimir energy and ${\rm SL}(3,{\mathbb Z})$ transformations, J. High Energy Phys. 2017 (2017), no. 7, 041, 21 pages, arXiv:1611.03831.
9. Brünner F., Spiridonov V.P., $4d$ $\mathcal{N}=1$ quiver gauge theories and the $A_n$ Bailey lemma, J. High Energy Phys. 2018 (2018), no. 3, 105, 30 pages, arXiv:1712.07018.
10. van de Bult F.J., Hyperbolic hypergeometric functions, Ph.D. Thesis, University of Amsterdam, 2007.
11. Cardy J.L., Operator content of two-dimensional conformally invariant theories, Nuclear Phys. B 270 (1986), 186-204.
12. Di Pietro L., Honda M., Cardy formula for $4d$ SUSY theories and localization, J. High Energy Phys. 2017 (2017), no. 4, 055, 35 pages, arXiv:1611.00380.
13. Di Pietro L., Komargoski Z., Cardy formulae for SUSY theories in $d=4$ and $d=6$, J. High Energy Phys. 2014 (2014), no. 12, 031, 28 pages, arXiv:1407.6061.
14. van Diejen J.F., Spiridonov V.P., An elliptic Macdonald-Morris conjecture and multiple modular hypergeometric sums, Math. Res. Lett. 7 (2000), 729-746.
15. van Diejen J.F., Spiridonov V.P., Elliptic Selberg integrals, Int. Math. Res. Not. 2001 (2001), 1083-1110.
16. van Diejen J.F., Spiridonov V.P., Unit circle elliptic beta integrals, Ramanujan J. 10 (2005), 187-204, math.CA/0309279.
17. Dolan F.A., Osborn H., Applications of the superconformal index for protected operators and $q$-hypergeometric identities to ${\mathcal N}=1$ dual theories, Nuclear Phys. B 818 (2009), 137-178, arXiv:0801.4947.
18. Dolan F.A.H., Spiridonov V.P., Vartanov G.S., From $4{\rm d}$ superconformal indices to $3{\rm d}$ partition functions, Phys. Lett. B 704 (2011), 234-241, arXiv:1104.1787.
19. Faddeev L.D., Kashaev R.M., Volkov A.Yu., Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality, Comm. Math. Phys. 219 (2001), 199-219, hep-th/0006156.
20. Felder G., Varchenko A., The elliptic gamma function and ${\rm SL}(3,{\bf Z})\ltimes{\bf Z}^3$, Adv. Math. 156 (2000), 44-76, math.QA/9907061.
21. Gadde A., Pomoni E., Rastelli L., Razamat S.S., $S$-duality and $2d$ topological QFT, J. High Energy Phys. 2010 (2010), no. 3, 032, 22 pages, arXiv:0910.2225.
22. Gadde A., Yan W., Reducing the $4d$ index to the $S^3$ partition function, J. High Energy Phys. 2012 (2012), no. 12, 003, 11 pages, arXiv:1104.2592.
23. Hwang C., Lee S., Yi P., Holonomy saddles and supersymmetry, arXiv:1801.05460.
24. Imamura Y., Relation between the $4d$ superconformal index and the $S^3$ partition function, J. High Energy Phys. 2011 (2011), no. 9, 133, 19 pages, arXiv:1104.4482.
25. Intriligator K., Wecht B., The exact superconformal $R$-symmetry maximizes $a$, Nuclear Phys. B 667 (2003), 183-200, hep-th/0304128.
26. Kels A.P., Yamazaki M., Elliptic hypergeometric sum/integral transformations and supersymmetric lens index, SIGMA 14 (2018), 013, 29 pages, arXiv:1704.03159.
27. Kinney J., Maldacena J., Minwalla S., Raju S., An index for 4 dimensional super conformal theories, Comm. Math. Phys. 275 (2007), 209-254, hep-th/0510251.
28. Koornwinder T.H., Askey-Wilson polynomials for root systems of type $BC$, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications (Tampa, FL, 1991), Contemp. Math., Vol. 138, Amer. Math. Soc., Providence, RI, 1992, 189-204.
29. Kutasov D., Lin J., ${\mathcal N} = 1$ duality and the superconformal index, arXiv:1402.5411.
30. Maldacena J., The large $N$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998), 231-252, hep-th/9711200.
31. Narukawa A., The modular properties and the integral representations of the multiple elliptic gamma functions, Adv. Math. 189 (2004), 247-267, math.QA/0306164.
32. Niarchos V., Seiberg dualities and the $3d/4d$ connection, J. High Energy Phys. 2012 (2012), no. 7, 075, 19 pages, arXiv:1205.2086.
33. Rains E.M., $BC_n$-symmetric Abelian functions, Duke Math. J. 135 (2006), 99-180, math.CA/0402113.
34. Rains E.M., Elliptic hypergeometric integrals, mPIM, Oberseminar, Bonn, July 24, 2008, available at https://www.hcm.uni-bonn.de/fileadmin/user_upload/Ellipticintegrablesystems/Rains-Ober.pdf.
35. Rains E.M., Limits of elliptic hypergeometric integrals, Ramanujan J. 18 (2009), 257-306, math.CA/0607093.
36. Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. of Math. 171 (2010), 169-243, math.QA/0309252.
37. Rastelli L., Razamat S.S., The supersymmetric index in four dimensions, J. Phys. A: Math. Gen. 50 (2017), 443013, 34 pages, arXiv:1608.02965.
38. Römelsberger C., Counting chiral primaries in ${\mathcal N}=1$, $d=4$ superconformal field theories, Nuclear Phys. B 747 (2006), 329-353, hep-th/0510060.
39. Rosengren H., Warnaar S.O., Elliptic hypergeometric functions associated with root systems, arXiv:1704.08406.
40. Ruijsenaars S.N.M., First order analytic difference equations and integrable quantum systems, J. Math. Phys. 38 (1997), 1069-1146.
41. Spiridonov V.P., On the elliptic beta function, Russian Math. Surveys 56 (2001), 185-186.
42. Spiridonov V.P., Bailey's tree for integrals, Theoret. and Math. Phys. 139 (2004), 536-541, math.CA/0312502.
43. Spiridonov V.P., Theta hypergeometric integrals, St. Petersburg Math. J. 15 (2004), 929-967, math.CA/0303205.
44. Spiridonov V.P., Vartanov G.S., Elliptic hypergeometry of supersymmetric dualities, Comm. Math. Phys. 304 (2011), 797-874, arXiv:0910.5944.
45. Spiridonov V.P., Vartanov G.S., Elliptic hypergeometric integrals and 't Hooft anomaly matching conditions, J. High Energy Phys. 2012 (2012), no. 6, 016, 22 pages, arXiv:1203.5677.
46. Spiridonov V.P., Vartanov G.S., Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices, Comm. Math. Phys. 325 (2014), 421-486, arXiv:1107.5788.
47. Spiridonov V.P., Warnaar S.O., Inversions of integral operators and elliptic beta integrals on root systems, Adv. Math. 207 (2006), 91-132, math.CA/0411044.
48. Stokman J.V., Hyperbolic beta integrals, Adv. Math. 190 (2005), 119-160, math.QA/0303178.
49. Teschner J., Supersymmetric gauge theories, quantisation of moduli spaces of flat connections, and Liouville theory, arXiv:1412.7140.
50. Vartanov G.S., On the ISS model of dynamical SUSY breaking, Phys. Lett. B 696 (2011), 288-290, arXiv:1009.2153.