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Published February 2018 | Supplemental Material + Submitted
Journal Article Open

Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality


In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class SS theory T[Σ,G] on L(k,1)×S^1, the other is the LGLG "equivariant Verlinde formula", or equivalently partition function of LGCLGC complex Chern–Simons theory on Σ×S^1. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual LGLG. When G is not simply-connected, we provide a recipe of computing the index of T[Σ,G] as summation over the indices of T[Σ,G] with non-trivial background 't Hooft fluxes, where G is the universal cover of G. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for G=SU(2) or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU(N) or PSU(N) and show that equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres–Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients.

Additional Information

© 2018 Springer-Verlag GmbH Germany, part of Springer Nature. Received: 24 July 2016; Accepted: 28 November 2017; First Online: 11 January 2018. We thank Jørgen Ellegaard Andersen, Francesco Benini, Martin Fluder, Abhijit Gadde, Tamás Hausel, Murat Koloğlu, Pavel Putrov, Richard Wentworth, Ingmar Saberi, Jaewon Song, Andras Szenes and Masahito Yamazaki for helpful discussions related to this work. We would also like to thank the organizers of the Simons Summer Workshop 2015, where a significant portion of this project was completed. This work is funded by the DOE Grant DE-SC0011632, U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 ("the GEAR Network"), the Walter Burke Institute for Theoretical Physics, the center of excellence grant "Center for Quantum Geometry of Moduli Space" from the Danish National Research Foundation (DNRF95), and the Center of Mathematical Sciences and Applications at Harvard University.

Attached Files

Submitted - 1605.06528v2.pdf

Supplemental Material - 220_2017_3074_MOESM1_ESM.pdf


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