  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.
