Exact results and Schur expansions in quiver Chern-Simons-matter theories

TL;DR

This paper computes exact partition functions and Wilson loop expectations in quiver Chern-Simons-matter theories on the three-sphere, using Mordell's integral and Schur polynomial techniques, and explores dualities and deformations.

Contribution

It introduces a systematic method combining Mordell's integral and Schur polynomial factorizations to obtain exact results in quiver Chern-Simons-matter theories, including Wilson loops and duality checks.

Findings

Exact partition functions expressed as perturbative expansions in mass parameters.

Wilson loop expectation values in terms of factorized pure Chern-Simons observables.

Inclusion of deformations such as real masses, FI parameters, and adjoint matter.

Abstract

We study several quiver Chern-Simons-matter theories on the three-sphere, combining the matrix model formulation with a systematic use of Mordell's integral, computing partition functions and checking dualities. We also consider Wilson loops in ABJ(M) theories, distinguishing between typical (long) and atypical (short) representations and focusing on the former. Using the Berele-Regev factorization of supersymmetric Schur polynomials, we express the expectation value of the Wilson loops in terms of sums of observables of two factorized copies of $U(N)$ pure Chern-Simons theory on the sphere. Then, we use the Cauchy identity to study the partition functions of a number of quiver Chern-Simons-matter models and the result is interpreted as a perturbative expansion in the parameters $t_j = - e^{2 \pi m_j}$, where $m_j$ are the masses. Through the paper, we incorporate different generalizations, such as deformations by real masses and/or Fayet-Iliopoulos parameters, the consideration of a Romans mass in the gravity dual, and adjoint matter.