Matrix model of Chern–Simons matter theories beyond the spherical limit

Shuichi Yokoyama

doi:10.1093/ptep/pty028

Abstract

A class of matrix models which arises as partition function in U(N) Chern-Simons matter theories on three sphere is investigated. Employing the standard technique of the 1/N expansion we solve the system beyond the planar limit. In particular we study a case where the matrix model potential has 1/N correction and give a general solution thereof up to the order of 1/N^2. We confirm that the general solution correctly reproduces the past exact result of the free energy up to the order in the case of pure Chern-Simons theory. We also apply to the matrix model of N=2 Chern-Simons theory with arbitrary numbers of fundamental chiral multiplets and anti-fundamental ones, which does not admit the Fermi gas analysis in general.

Highlights

Recent progress in supersymmetric Chern–Simons matter theories has been made on the basis of the exact result by means of supersymmetric localization
Progress in non-supersymmetric Chern–Simons matter theories has been made not relying on the localization technique but on the 1/N expansion technique by restricting the class of matter fields to vector fields
Of matrix models incorporating the standard technique of 1/N expansion developed in the study of ordinary Hermitian matrix models [32] beyond the spherical limit [33,34]

Summary

Introduction

Recent progress in supersymmetric Chern–Simons matter theories has been made on the basis of the exact result by means of supersymmetric localization. The differential equation in Eq (40) will be solved as in the original Hermitian matrix model for the one-cut case [33] and the two-cut case [34], though such explicit solutions of the genus-one free energy do not contain the information about the dependence on other coupling constants such as where we define Kby f (w) − 2ω0, 1 (z)f (z). This equation is of the same form as the one without the hole correction by replacing ω0 , W0 with ω0, 1 , W , respectively. We emphasize that this simplification happens only at the genus-one order, and at higher order one may need to solve Eq (37) in general

Applications

Discussion