Free compact boson on branched coverings of the torus

TL;DR

This paper analyzes the partition function of a free compact boson on branched coverings of the torus, revealing its relation to Rényi entanglement entropy for multiple intervals at finite temperature.

Contribution

It introduces a canonical homology basis for the covering surface and expresses the partition function using theta functions, linking geometric and entanglement properties.

Findings

Partition function expressed in terms of theta functions.

Relation established between branched coverings and entanglement entropy.

Provides a framework for calculating entanglement in finite systems.

Abstract

We have studied the free compact boson on a $n$-sheeted covering of the torus gluing alone $m$ branch cuts. It is interesting because when the branched cuts are chosen to be real, the partition function is related to the $n$-th R\'enyi entanglement entropy of $m$ disjoint intervals in a finite system at finite temperature. After proposing a canonical homology basis and its dual basis of the covering surface, we find that the partition function can be written in terms of theta functions.