Mordell integrals and Giveon-Kutasov duality

Abstract

We solve, for finite N , the matrix model of supersymmetric U(N ) ChernSimons theory coupled to N f massive hypermultiplets of R-charge 1 2 , together with a FayetIliopoulos 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 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).

N = 2 $$ \mathcal{N}=2 $$ Chern-Simons-matter theories without vortices

“…This is in contradistinction with the behavior of the weak coupling perturbation series in more general N = 2 supersymmetric gauge theories, which is asymptotic [20,21,22]. For integer k, in some cases the partition function on S 3 has a finite number of terms (see [23,24,25,26,27,28] for many examples). This can be illustrated by U(1) N = 2 Chern-Simons theory with a FI deformation, coupled to a pair of massless chiral fields of ∆ = 1/2 and opposite gauge charges.…”

“…This is a Mordell integral [29] (see [26,28] for explicit examples in the context of N = 2 CS theories). For integer k, the integral can be computed by choosing an appropriate rectangular contour, leading to a finite sum [26]…”

Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

“…These specific cases precisely contain the one which is physically relevant: g = 2πi/k with k ∈ Z. This works very well for N f = 1 and has been more recently extended to higher flavour in [12] by studying parametric derivatives of Mordell integrals. A large number of analytic results and explicit tests of Giveon-Kutasov duality can be obtained in that way.…”

Wilson loops and free energies in 3D $\mathcal{N}=4$ SYM: exact results, exponential asymptotics, and duality

“…Notice that the G-Barnes function part in (2.7) is symmetric under η → −η. Therefore, due to the imaginary prefactor Z N (−η) = Z N (η), as also happens when there is a Chern-Simons term [33,34]. Note also that (2.7) admits alternative equivalent expressions, for example involving Gamma functions.…”