Free compact boson on branched coverings of the torus

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.