Chern-Simons Matrix Models and Unoriented Strings

TL;DR

This paper investigates the sub-leading free energy in matrix models related to SO/Sp gauge groups and Chern-Simons theory, deriving new formulas and confirming results through explicit calculations.

Contribution

It derives a quantum loop equation for the Chern-Simons matrix model and expresses the sub-leading free energy as an integral over the spectral curve, extending previous understanding.

Findings

Derived a quantum loop equation for the matrix model.

Expressed F_{1} as an integral over the spectral curve.

Confirmed the formula with explicit calculations for quadratic potential.

Abstract

For matrix models with measure on the Lie algebra of SO/Sp, the sub-leading free energy is given by F_{1}(S)=\pm{1/4}\frac{\del F_{0}(S)}{\del S}. Motivated by the fact that this relationship does not hold for Chern-Simons theory on S^{3}, we calculate the sub-leading free energy in the matrix model for this theory, which is a Gaussian matrix model with Haar measure on the group SO/Sp. We derive a quantum loop equation for this matrix model and then find that F_{1} is an integral of the leading order resolvent over the spectral curve. We explicitly calculate this integral for quadratic potential and find agreement with previous studies of SO/Sp Chern-Simons theory.