# Chern-Simons Theory on Seifert Manifold and Matrix Model

## TL;DR

This paper explores the partition functions of Chern-Simons theories on Seifert manifolds, expressing them as unitary matrix models, analyzing eigenvalue distributions, and studying large N phase structures and dualities.

## Contribution

It introduces a matrix model representation of Chern-Simons partition functions on Seifert manifolds and analyzes large N behaviors, including phase transitions and dualities.

## Key findings

- Eigenvalues are discrete and proportional to hook lengths of Young diagrams.
- Eigenvalue density develops an upper cap in the large N limit.
- Multiple large N phases with same free energy are identified for CS on S^3.

## Abstract

Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold are discussed. The partition functions of such theories can be written as a function of modular transformation matrices summed over different integrable representations of affine Lie algebra $u(N)_k$ associated with boundary Wess-Zumino-Witten (WZW) model. Using properties of modular transform matrices we express the partition functions of these theories as a unitary matrix model. We show that, the eigenvalues of unitary matrices are discrete and proportional to hook lengths of the corresponding integrable Young diagram. As a result, in the large $N$ limit, the eigenvalue density develops an upper cap. We consider CS theory on $S^2\times S^1$ coupled with fundamental matters and express the partition functions in terms of modular transformation matrices. Solving this model at large $N$ we find the dominant integrable representations and show how large $N$ representations are related to each other by transposition of Young diagrams as a result of level rank duality. Next we consider $U(N)$ CS theory on $S^3$ and observed that in Seifert framing the dominant representation is no longer an integrable representation after a critical value of 't Hooft coupling. We also show that CS on $S^3$ admits multiple (two-gap phase) large $N$ phases with the same free energy.