Semiclassical limit of topological R\'enyi entropy in $3d$ Chern-Simons theory

TL;DR

This paper investigates the asymptotic behavior of Rènyi entropies in 3D SU(2) Chern-Simons theory for torus links, revealing universal and linking-dependent parts, and connecting results to 2D topological Yang-Mills theory and moduli space volumes.

Contribution

It provides a detailed analysis of the semiclassical limit of Rènyi entropies for torus links, identifying universal components and their relation to 2D topological theories and moduli spaces.

Findings

Rènyi entropies converge to finite values as the level k approaches infinity.

Universal parts of the entropy relate to Riemann zeta functions and 2D Yang-Mills partition functions.

Entropies in the double scaling limit also converge, indicating a consistent semiclassical behavior.

Abstract

We study the multi-boundary entanglement structure of the state associated with the torus link complement $S^3 \backslash T_{p,q}$ in the set-up of three-dimensional SU(2)$_k$ Chern-Simons theory. The focal point of this work is the asymptotic behavior of the R\'enyi entropies, including the entanglement entropy, in the semiclassical limit of $k \to \infty$. We present a detailed analysis for several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large $k$ limiting value of the R\'enyi entropy of torus links of type $T_{p,pn}$ is the sum of two parts: (i) the universal part which is independent of $n$, and (ii) the non-universal or the linking part which explicitly depends on the linking number $n$. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the R\'enyi entropy of certain states prepared in topological $2d$ Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large $k$ limits of the entanglement entropy and the minimum R\'enyi entropy for torus links $T_{p,pn}$ can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the R\'enyi entropies of $T_{p,pn}$ link in the double scaling limit of $k \to \infty$ and $n \to \infty$ and propose that the entropies converge in the double limit as well.