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

• URL: http://arxiv.org/abs/2007.07033v2
• Date: Wed, 23 Dec 2020 14:29:07 GMT
• Title: Semiclassical limit of topological R\'enyi entropy in $3d$ Chern-Simons theory
• Authors: Siddharth Dwivedi, Vivek Kumar Singh, Abhishek Roy
• Abstract summary: We study the multi-boundary entanglement structure of the state associated with the torus link $S3 backslash T_p,q$ in the set-up of three-dimensional SU(2)$_k$ Chern-Simons theory. The focal point of this work is the universal behavior of the R'enyi entropies, including the entanglement entropy, in the semiclassical limit of $k to infty$.
• Score: 2.5685922445338223
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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