In this talk we describe the imminently coming work in collaboration with B. Estienne where we demonstrate a purely mathematical construction which recovers some results of Cardy and Calabrese on the so-called entanglement entropy. Skipping briefly over introducing the latter notion, we start by defining mathematically a Conformal Field Theory over a (smooth) Riemann surface $(\Sigma, g)$ and its partition function as well as correlation functions. Then for a Riemann surface with "conical singularities" we will define the CFT “partition function” (denoted Z) on it using a simple Hadamard renormalization (removing disks) of the Polyakov anomaly integral, a first main ingredient of the work. Then we state the main result namely for a branched cover $f : \Sigma_d \to \Sigma$ of degree d, the ratio $Z(\Sigma_d, f^*g)/Z(\Sigma, g)^d$ of partition functions transforms under conformal changes of g like a two-point function of CFT primary operators of specific conformal weights. If time permits, we talk briefly about the physical motivation around entanglement entropies.
