Free compact boson on branched covering of the torus

We have studied the partition function of a free compact boson on a n-sheeted covering of torus gluing along 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 Rényi 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.