On the density of states in a free CFT and finite volume corrections

TL;DR

This paper investigates the density of states in free conformal field theories on various manifolds, deriving new finite volume corrections and analyzing the applicability of the Cardy-Verlinde formula.

Contribution

It introduces a Taylor series expression for the density of states that improves existing results for manifolds with small boundary area relative to volume.

Findings

Derived finite volume corrections to the density of states.

Provided insights into the conditions for the validity of the Cardy-Verlinde formula.

Developed an improved series expansion for spectral density in free CFTs.

Abstract

Results from spectral geometry such as Weyl's formula can be used to relate the thermodynamic properties of a free massless field to the spatial manifold on which it is defined. We begin by calculating the free energy in two cases: manifolds posessing a boundary and spheres. The subextensive contributions allow us to test the Cardy-Verlinde formula and offer a new perspective on why it only holds in a free theory if one allows for a change in the overall coefficient. After this we derive an expression for the density of states that takes the form of a Taylor series. This series leads to an improvement over known results when the area of the manifold's boundary is nonzero but much less than the appropriate power of its volume.