Notes on Planar Resolvents of Chern-Simons-matter Matrix Models

TL;DR

This paper derives explicit formulas for planar resolvents and BPS Wilson loop vevs in ${ m U}(N_1) imes{ m U}(N_2)$ Chern-Simons-matter theories with ${ m N} extgreater=3$, enhancing understanding of their matrix model solutions.

Contribution

It provides an explicit expression for the derivative of the planar resolvent and formulas for BPS Wilson loop vevs in these theories.

Findings

Explicit formula for the derivative of the planar resolvent.

Closed-form expressions for Wilson loop vevs.

Enhanced analytical understanding of matrix models in supersymmetric theories.

Abstract

We revisit planar resolvents of matrix models corresponding to ${\cal N}\ge3$ Chern-Simons-matter theories with the gauge groups of the form ${\rm U}(N_1)\times{\rm U}(N_2)$ coupled to any number of bi-fundamental hypermultiplets. We find that the derivative of a suitably defined planar resolvent can be written explicitly. From this resolvent, we derive the explicit formula for (a linear combination of) the vevs of BPS Wilson loops.