Comments on Higher Rank Wilson Loops in ${\cal N}=2^*$

TL;DR

This paper evaluates Wilson loop expectation values in ${ m N}=2^*$ theory with a $U(N)$ gauge group, analyzing eigenvalue distributions and comparing results with holographic duals, especially for symmetric representations.

Contribution

It provides new analytic results for Wilson loops in rectangular Young tableau representations and clarifies the limits and genus expansion in ${ m N}=2^*$ theory.

Findings

Analytic expressions for Wilson loops in various regimes.

Reproduction of known results for symmetric and antisymmetric cases.

New contributions improving holographic comparisons.

Abstract

For ${\cal N}=2^*$ theory with $U(N)$ gauge group we evaluate expectation values of Wilson loops in representations described by a rectangular Young tableau with $n$ rows and $k$ columns. The evaluation reduces to a two-matrix model and we explain, using a combination of numerical and analytical techniques, the general properties of the eigenvalue distributions in various regimes of parameters $(N,\lambda,n,k)$ where $\lambda$ is the 't Hooft coupling. In the large $N$ limit we present analytic results for the leading and sub-leading contributions. In the particular cases of only one row or one column we reproduce previously known results for the totally symmetry and totally antisymmetric representations. We also extensively discusss the ${\cal N}=4$ limit of the ${\cal N}=2^*$ theory. While establishing these connections we clarify aspects of various orders of limits and how to relax them; we also find it useful to explicitly address details of the genus expansion. As a result, for the totally symmetric Wilson loop we find new contributions that improve the comparison with the dual holographic computation at one loop order in the appropriate regime.