Abstract
We discuss some of the analytic properties of lens space indices for 4d \( \mathcal{N}=2 \) theories of class \( \mathcal{S} \). The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects. We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic RuijsenaarsSchneider model. In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.
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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 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].
F. Dolan and H. Osborn, Applications of the superconformal index for protected operators and q-hypergeometric identities to N = 1 dual theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [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 hypergometric integrals, Ann. Math. 171 (2010) 169 [math.QA/0309252].
V. Spiridonov and G. Vartanov, Elliptic hypergeometry of supersymmetric dualities, Commun. Math. Phys. 304 (2011) 797 [arXiv:0910.5944] [INSPIRE].
V. Spiridonov and G. Vartanov, Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots and vortices, arXiv:1107.5788 [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [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 4d superconformal index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, arXiv:1207.3577 [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].
F. Benini, T. Nishioka and M. Yamazaki, 4d index to 3d index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, Adv. Theor. Math. Phys. 5 (2002) 841 [hep-th/0006010] [INSPIRE].
S.S. Razamat and B. Willett, Global properties of supersymmetric theories and the lens space, arXiv:1307.4381 [INSPIRE].
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].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY gauge theories on three-sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
D. Gang, Chern-Simons theory on L(p,q) lens spaces and localization, arXiv:0912.4664 [INSPIRE].
J. Kallen, Cohomological localization of Chern-Simons theory, JHEP 08 (2011) 008 [arXiv:1104.5353] [INSPIRE].
K. Ohta and Y. Yoshida, Non-Abelian localization for supersymmetric Yang-Mills-Chern-Simons theories on Seifert manifold, Phys. Rev. D 86 (2012) 105018 [arXiv:1205.0046] [INSPIRE].
L.F. Alday, M. Fluder and J. Sparks, The large-N limit of M2-branes on lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE].
S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
F. Benini and S. Cremonesi, Partition functions of N = (2,2) gauge theories on S 2 and vortices, arXiv:1206.2356 [INSPIRE].
N. Doroud, J. Gomis, B. Le Floch and S. Lee, Exact results in D = 2 supersymmetric gauge theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE].
M. Yamazaki, Four-dimensional superconformal index reloaded, Theor. Math. Phys. 174 (2013) 154 [Teor. Mat. Fiz. 174 (2013) 177] [INSPIRE].
L.F. Alday, M. Bullimore and M. Fluder, On S-duality of the superconformal index on lens spaces and 2d TQFT, JHEP 05 (2013) 122 [arXiv:1301.7486] [INSPIRE].
A. Gadde and S. Gukov, 2d index and surface operators, arXiv:1305.0266 [INSPIRE].
I. Cherednik, Double affine Hecke algebra, Cambridge University Press, Cambridge U.K. (2005).
I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge University Press, Cambridge U.K. (2003).
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of two-dimensional N = 2 gauge theories with rank-one gauge groups, arXiv:1305.0533 [INSPIRE].
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford U.K. (1979).
D. Marshall, Symmetric and nonsymmetric Macdonald polynomials, Ann. Comb. 3 (1999) 385.
N. Mekareeya, J. Song and Y. Tachikawa, 2d TQFT structure of the superconformal indices with outer-automorphism twists, JHEP 03 (2013) 171 [arXiv:1212.0545] [INSPIRE].
M. Lemos, W. Peelaers and L. Rastelli, The superconformal index of class S theories of type D, arXiv:1212.1271 [INSPIRE].
I. Cherednik, Jones polynomials of torus knots via DAHA, arXiv:1111.6195 [INSPIRE].
M. Aganagic and S. Shakirov, Knot homology from refined Chern-Simons theory, arXiv:1105.5117 [INSPIRE].
F. Benini, Y. Tachikawa and B. Wecht, Sicilian gauge theories and N = 1 dualities, JHEP 01 (2010) 088 [arXiv:0909.1327] [INSPIRE].
D. Xie and M. Yamazaki, Network and Seiberg duality, JHEP 09 (2012) 036 [arXiv:1207.0811] [INSPIRE].
S. Franco, Bipartite field theories: from D-brane probes to scattering amplitudes, JHEP 11 (2012) 141 [arXiv:1207.0807] [INSPIRE].
M. Yamazaki, in progress.
L.F. Alday, M. Bullimore, M. Fluder and L. Hollands, Surface defects, the superconformal index and q-deformed Yang-Mills, arXiv:1303.4460 [INSPIRE].
M. Noumi, Affine Hecke algebras and Macdonald polynomials, Progr. Math. 160 (1998) 365.
A.N. Kirillov and M. Noumi, Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998) 1 [q-alg/9605004].
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ArXiv ePrint: 1306.1543
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Razamat, S.S., Yamazaki, M. S-duality and the \( \mathcal{N}=2 \) lens space index. J. High Energ. Phys. 2013, 48 (2013). https://doi.org/10.1007/JHEP10(2013)048
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DOI: https://doi.org/10.1007/JHEP10(2013)048