2018/05/28 by Mukhi, Sunil, Murthy, Sameer
#Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1805.11114
In arXiv:1706:09426 we conjectured and provided evidence for an identity between Siegel Θ-constants for special Riemann surfaces of genus n and products of Jacobi θ-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.