Soft matrix models and Chern-Simons partition functions

TL;DR

The paper analyzes matrix models with soft confining potentials where the weight is not fixed by moments, focusing on a log-normal type weight that naturally maps to Chern-Simons theory on $S^{3}$. By employing $q$-deformed Stieltjes-Wigert orthogonal polynomials, it exacts the CS partition function and identifies the CS parameter as $q=\mathrm{e}^{-g_s}=\mathrm{e}^{2\pi i/(N+k)}$, linking non-perturbative matrix-model techniques to topological invariants. A key finding is the non-uniqueness: there exist infinitely many distinct matrix models with the same CS partition function due to indeterminate moment problems, illustrated by perturbations of the weight that preserve all moments. This suggests a deeper mathematical richness and potential physical interpretations, such as relating different matrix-model setups to triangulations in 3D topological quantum field theory, and motivates extensions to other geometries and gauge groups using the orthogonal-polynomial framework.

Abstract

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials), and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are infinitely many matrix models with this partition function.

Figures (3)

• Figure 1: The oscillatory term in $f_{\theta }(x)$
• Figure 2: $\theta=1/2$
• Figure 3: $\theta=1$ (solid) and $\theta=-1$ (dashed)