Partition Functions on the Euclidean Plane with Compact Boundaries in Conformal and Non-Conformal Theories

TL;DR

This paper computes exact partition functions for free bosons on a plane with multiple holes, establishing a correspondence between effective field theory coefficients and string excitation weights, and extends methods to complex boundary conditions.

Contribution

It introduces a novel approach linking Wilson coefficients in effective theories to string boundary excitations and calculates partition functions for multiple boundary configurations using both effective and conformal field theory techniques.

Findings

Exact partition function for two holes matches between effective theory and CFT.

Established correspondence between Wilson coefficients and string excitation weights.

Extended calculations to three holes and higher-order kernels, including non-trivial central charges.

Abstract

In this letter we calculate the exact partition function for free bosons on the plane with lacunae. First the partition function for a plane with two spherical holes is calculated by matching exactly for the infinite set of Wilson coefficients in an effective world line theory and then performing the ensuing Gaussian integration. The partition is then re-calculated using conformal field theory techniques, and the equality of the two results is made manifest. It is then demonstrated that there is an exact correspondence between the Wilson coefficients (susceptabilities) in the effective field theory and the weights of the individual excitations of the closed string coherent state on the boundary. We calculate the partition function for the case of three holes where CFT techniques necessitate a closed form for the map from the corresponding closed string pants diagrams. Finally, it is shown that the Wilson coefficients for the case of quartic and higher order kernels, where standard CFT techniques are no longer applicable, can also be completely determined. These techniques can also be applied to the case of non-trivial central charges.