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

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.