Fermions on replica geometries and the Θ - θ relation

Abstract

In arXiv: 1706.09426 we conjectured and provided evidence for an identity between Siegel Theta-constants for special Riemann surfaces of genus n and products of Jacobi theta-functions. This arises by comparing two different ways of computing the nth Renyi entropy of free fermions at finite temperature. Here we show that for n = 2 the identity is a consequence of an old result due to Fay for doubly branched Riemann surfaces. For n > 2 we provide a detailed matching of certain zeros on both sides of the identity. This amounts to an elementary proof of the identity for n = 2, while for n >= 3 it gives new evidence for it. We explain why the existence of additional zeros renders the general proof difficult.