Abstract
We compute the topological partition function (twisted index) of \( \mathcal{N} \) = 2 U(N) Chern-Simons theory with an adjoint chiral multiplet on Σg × S1. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by U(1) equivariant parameter t. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials Pλ(xB, t) with a specialization by Bethe roots xB of the q-boson model. We confirm a proposed duality to the Coulomb branch limit of the lens space superconformal index of four dimensional \( \mathcal{N} \) = 2 theories for SU(2) and SU(3) with lower levels. In SU(2) case we also present more direct computation based on Jeffrey-Kirwan residue operation.
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References
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantum integrability and supersymmetric vacua, Prog. Theor. Phys. Suppl. 177 (2009) 105 [arXiv:0901.4748] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in Proceedings, 16th International Congress on Mathematical Physics (ICMP09), Prague, Czech Republic, 3–8 August 2009, World Scientific, Singapore (2010), pg. 265 [arXiv:0908.4052] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
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].
N. Wyllard, A N − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE].
Y. Terashima and M. Yamazaki, SL(2, R) Chern-Simons, Liouville and gauge theory on duality walls, JHEP 08 (2011) 135 [arXiv:1103.5748] [INSPIRE].
Y. Terashima and M. Yamazaki, Semiclassical analysis of the 3d/3d relation, Phys. Rev. D 88 (2013) 026011 [arXiv:1106.3066] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the squashed three-sphere, JHEP 11 (2017) 119 [arXiv:1305.2891] [INSPIRE].
S. Lee and M. Yamazaki, 3d Chern-Simons theory from M5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].
T. Dimofte, Complex Chern-Simons theory at level k via the 3d-3d correspondence, Commun. Math. Phys. 339 (2015) 619 [arXiv:1409.0857] [INSPIRE].
T. Dimofte, Perturbative and nonperturbative aspects of complex Chern-Simons theory, J. Phys. A 50 (2017) 443009 [arXiv:1608.02961] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
D. Gepner, Fusion rings and geometry, Commun. Math. Phys. 141 (1991) 381 [INSPIRE].
K.A. Intriligator, Fusion residues, Mod. Phys. Lett. A 6 (1991) 3543 [hep-th/9108005] [INSPIRE].
E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in Geometry, topology, and physics, Cambridge, U.K. (1993), pg. 357 [hep-th/9312104] [INSPIRE].
S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].
J.E. Andersen, S. Gukov and D. Pei, The Verlinde formula for Higgs bundles, arXiv:1608.01761 [INSPIRE].
S. Gukov, D. Pei, W. Yan and K. Ye, Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality, Commun. Math. Phys. 357 (2018) 1215 [arXiv:1605.06528] [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].
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].
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].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
S. Okuda and Y. Yoshida, G/G gauged WZW model and Bethe ansatz for the phase model, JHEP 11 (2012) 146 [arXiv:1209.3800] [INSPIRE].
S. Okuda and Y. Yoshida, G/G gauged WZW-matter model, Bethe ansatz for q-boson model and commutative Frobenius algebra, JHEP 03 (2014) 003 [arXiv:1308.4608] [INSPIRE].
S. Okuda and Y. Yoshida, Gauge/bethe correspondence on S 1 × Σh and index over moduli space, arXiv:1501.03469 [INSPIRE].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic genera of 2d N = 2 gauge theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
C. Korff, Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra, Commun. Math. Phys. 318 (2013) 173 [arXiv:1110.6356] [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].
S.S. Razamat and M. Yamazaki, S-duality and the N = 2 lens space index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
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].
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].
I.G. Macdonald, Symmetric functions and Hall polynomials, second edition, Oxford University Press, Oxford, U.K. (1995).
T. Nakanishi and A. Tsuchiya, Level rank duality of WZW models in conformal field theory, Commun. Math. Phys. 144 (1992) 351 [INSPIRE].
S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality in WZW models and Chern-Simons theory, Phys. Lett. B 246 (1990) 417 [INSPIRE].
E.J. Mlawer, S.G. Naculich, H.A. Riggs and H.J. Schnitzer, Group level duality of WZW fusion coefficients and Chern-Simons link observables, Nucl. Phys. B 352 (1991) 863 [INSPIRE].
S.G. Naculich and H.J. Schnitzer, Level-rank duality of the U(N) WZW model, Chern-Simons theory and 2D qYM theory, JHEP 06 (2007) 023 [hep-th/0703089] [INSPIRE].
P.-S. Hsin and N. Seiberg, Level/rank duality and Chern-Simons-matter theories, JHEP 09 (2016) 095 [arXiv:1607.07457] [INSPIRE].
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Kanno, H., Sugiyama, K. & Yoshida, Y. Equivariant U(N) Verlinde algebra from Bethe/gauge correspondence. J. High Energ. Phys. 2019, 97 (2019). https://doi.org/10.1007/JHEP02(2019)097
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DOI: https://doi.org/10.1007/JHEP02(2019)097