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.
Highlights
Six dimensional theory on a manifold Mn of dimension n gives a SUSY gauge theory in (6 − n) dimensions, which is a source of otherwise unexpected correspondence between the field theory on Mn and the SUSY gauge theory
In [25] by considering six dimensional N = (2, 0) superconformal field theory on L(κ, 1)×Σ×S1, it is proposed that 2d topological quantum field theory (TQFT) obtained from complex Chern-Simons theory or N = 2 Chern-Simons theory with an adjoint chiral multiplet on Σ × S1 has a dual description in terms of the Coulomb branch limit of the superconformal index on the lens space L(κ, 1)
When the manifold is Σ × S1, we can make the partition function localized on discrete SUSY vacua which coincide with the solutions (Bethe roots) to the Bethe ansatz equation
Summary
Let us see the other side of 6d theory on L(κ, 1) × Σ × S1. The superconformal index of four dimensional N = 2 theory is defined as the partition function on S3×S1, where we take. When the N = 2 superconformal theory is of class S, the superconformal indices give a 2d TQFT on the punctured Riemann surface Σg,n associated with the class S theory [26, 27] This is regarded as a TQFT version of AGT correspondence, where conformal blocks are replaced by topological correlation functions. In [25] the computation of the superconformal indices for T3 theory has been made by invoking the Argyres-Seiberg duality that allows a weak coupling region. This approach cannot be generalized to TN theory for 3 < N. In this paper we compute the partition function of U(4) theory which is expected to match with the superconformal indices of T4 theory
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