R\'{e}nyi Entropy with Surface Defects in Six Dimensions

TL;DR

This paper calculates the contribution of surface defects to Rényi entropy in six-dimensional theories, developing a heat kernel method and exploring supersymmetric refinements, with results applicable to free and large N theories.

Contribution

It introduces a heat kernel approach for defect contributions to Rényi entropy in arbitrary dimensions and extends calculations to supersymmetric cases in six dimensions.

Findings

Surface defect contribution verified for free fields.

Supersymmetric Rényi entropy scales polynomially with Rényi index at large N.

Method connects free field and large N results.

Abstract

We compute the surface defect contribution to R\'{e}nyi entropy and supersymmetric R\'{e}nyi entropy in six dimensions. We first compute the surface defect contribution to R\'{e}nyi entropy for free fields, which verifies a previous formula about entanglement entropy with surface defect. Using conformal map to $S^1_\beta\times H^{d-1}$ we develop a heat kernel approach to compute the defect contribution to R\'{e}nyi entropy, which is applicable for $p$-dimensional defect in general $d$-dimensional free fields. Using the same geometry $S^1_\beta\times H^5$ with an additional background field, one can construct the supersymmetric refinement of the ordinary R\'{e}nyi entropy for six-dimensional $(2,0)$ theories. We find that the surface defect contribution to supersymmetric R\'{e}nyi entropy has a simple scaling as polynomial of R\'{e}nyi index in the large $N$ limit. We also discuss how to connect the free field results and large $N$ results.