Mordell integrals and Giveon-Kutasov duality

TL;DR

This paper computes the partition function of finite $N$ supersymmetric $U(N)$ Chern-Simons theory with hypermultiplets using Mordell integrals, and tests Giveon-Kutasov duality systematically, revealing a modular phase factor.

Contribution

It introduces a novel method to evaluate the matrix model using Mordell integrals and provides a comprehensive test of Giveon-Kutasov duality for various parameters.

Findings

Partition functions expressed as determinants of Hankel matrices.

Explicit phase factor characterized by mod 4 behavior.

Validated duality for up to 12 flavors.

Abstract

We solve, for finite $N$, the matrix model of supersymmetric $U(N)$ Chern-Simons theory coupled to $N_{f}$ massive hypermultiplets of $R$-charge $\frac{1}{2}$, together with a Fayet-Iliopoulos term. We compute the partition function by identifying it with a determinant of a Hankel matrix, whose entries are parametric derivatives (of order $N_{f}-1$) of Mordell integrals. We obtain finite Gauss sums expressions for the partition functions. We also apply these results to obtain an exhaustive test of Giveon-Kutasov (GK) duality in the $\mathcal{N}=3$ setting, by systematic computation of the matrix models involved. The phase factor that arises in the duality is then obtained explicitly. We give an expression characterized by modular arithmetic (mod 4) behavior that holds for all tested values of the parameters (checked up to $N_{f}=12$ flavours).