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 $n\ge 3$ it gives new evidence for it. We explain why the existence of additional zeros renders the general proof difficult

Topics:
High Energy Physics - Theory, Mathematics - Algebraic Geometry, Mathematics - Number Theory

Year: 2018

OAI identifier:
oai:arXiv.org:1805.11114

Provided by:
arXiv.org e-Print Archive

Downloaded from
http://arxiv.org/abs/1805.11114

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.