Entanglement entropy for even spheres

TL;DR

This paper re-computes the entanglement entropy's logarithmic coefficient on even spheres using a local thermodynamic approach in de Sitter space, confirming it matches the conformal anomaly and deriving related formulas.

Contribution

It introduces a local technique for calculating entanglement entropy on even spheres and derives new formulas involving Bernoulli polynomials and higher GJMS Laplacians.

Findings

Re-derivation of the known conformal anomaly coefficient.

Development of a local thermodynamic method for entropy calculation.

Derivation of formulas for conformal anomalies of higher GJMS Laplacians.

Abstract

The coefficient of the logarithmic term in the entropy on even spheres is re-computed by the local technique of integrating the finite temperature energy density up to the horizon on static d--dimensional de Sitter space and thence finding the entropy by thermodynamics. Numeric evaluation yields the known answer i.e. (minus) the conformal anomaly on the d-sphere. The de Sitter quantities are obtained by conformal transformation of the Rindler ones, themselves obtained, for convenience, from those around a cosmic string. The expressions are given in terms of generalised Bernoulli polynomials for which an identity is derived. The arising spherical conformal anomaly is discussed and a formula is given for it for Branson's higher GJMS Laplacian, P_2k, as an oscillating polynomial in the level, k.