Sphere Renyi entropies

TL;DR

This paper calculates scalar field entropies on a d-dimensional lune, explores their behavior with mass, and conjectures about effective actions on manifolds with conical singularities, providing new numerical and analytical insights.

Contribution

It introduces a numerical quadrature method for effective action calculations on lune geometries and derives a universal coefficient from large mass expansion.

Findings

Effective action expressed as a simple quadrature for conformal coupling.

Universal coefficient derived from large mass expansion.

Conjecture that effective action extremizes when conical singularity disappears.

Abstract

I give some scalar field theory calculations on a d-dimensional lune of arbitrary angle, evaluating, numerically, the effective action which is expressed as a simple quadrature, for conformal coupling. Using this, the entanglement and Renyi entropies are computed. Massive fields are also considered and a renormalisation to make the (one-loop) effective action vanish for infinite mass is suggested and used, not entirely successfully. However a universal coefficient is derived from the large mass expansion. From the deformation of the corresponding lune result, I conjecture that the effective action on all manifolds with a simple conical singularity has an extremum when the singularity disappears. For the round sphere, I show how to convert the quadrature form of the conformal Laplacian determinant into the more usual sum of Riemann zeta functions (and log2).