Relative entropy for compact Riemann surfaces

TL;DR

This paper investigates the relative entropy of massive free bosonic fields on compact Riemann surfaces, providing exact and asymptotic formulas that reveal universal and topological effects relevant to gravitational phenomena.

Contribution

It derives exact expressions for the relative entropy on the sphere and torus, and establishes asymptotic behaviors for large mass on general surfaces, linking geometry and topology.

Findings

Exact expression for the sphere's relative entropy.

Asymptotic series for large mass depending on curvature.

Topological corrections in the entropy's asymptotic behavior.

Abstract

The relative entropy of the massive free bosonic field theory is studied on various compact Riemann surfaces as a universal quantity with physical significance, in particular, for gravitational phenomena. The exact expression for the sphere is obtained, as well as its asymptotic series for large mass and its Taylor series for small mass. One can also derive exact expressions for the torus but not for higher genus. However, the asymptotic behaviour for large mass can always be established-up to a constant-with heat-kernel methods. It consists of an asymptotic series determined only by the curvature, hence common for homogeneous surfaces of genus higher than one, and exponentially vanishing corrections whose form is determined by the concrete topology. The coefficient of the logarithmic term in this series gives the conformal anomaly.