Rényi Entropy with Surface Defects in Six Dimensions

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