Topological R\'enyi and Entanglement Entropy for a 2d q-deformed $U(N)$ Yang-Mills theory and its Chern-Simons dual

TL;DR

This paper explores the calculation of Re9nyi and entanglement entropies in 2d q-deformed topological Yang-Mills theories and their dual 3d Chern-Simons theories, revealing dualities and expressing observables via modular matrices.

Contribution

It introduces a method to compute entanglement measures in q-deformed 2d Yang-Mills and dual 3d Chern-Simons theories, highlighting level-rank dualities and modular matrix relations.

Findings

Re9nyi and entanglement entropies are expressed in terms of WZW modular matrices.

Level-rank duality relates theories with odd K and N.

Wilson line observables are connected to the modular transformation matrices.

Abstract

R\'enyi and entanglement entropies are constructed for 2d q-deformed topological Yang-Mills theories with gauge group $U(N)$, as well as the dual 3d Chern-Simons (CS) theory on Seifert manifolds. When $q=\exp[2\pi i/(N+K)]$, and $K$ is odd, the topological R\'enyi entropy and Wilson line observables of the CS theory can be expressed in terms of the modular transformation matrices of the WZW theory, $\rm{\hat{U}(N)}_{K,N(K+N)}$. If both $K$ and $N$ are odd, there is a level-rank duality of the 2d qYM theory and of the associated CS theory, as well as that of the R\'enyi and entanglement entropies, and Wilson line observables.