Fay Identities of Pfaffian Type for Hyperelliptic Curves

Gaëtan Borot^a, Thomas Buc-D''Alche<sup>b

a</sup> Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
^b UMPA UMR 5669, ENS de Lyon, CNRS, 46, allée d’Italie 69007, Lyon, France

Abstract: We establish identities of Pfaffian type for the theta function associated with twice or half the period matrix of a hyperelliptic curve. They are implied by the large size asymptotic analysis of exact Pfaffian identities for expectation values of ratios of characteristic polynomials in ensembles of orthogonal or quaternionic self-dual random matrices. We show that they amount to identities for the theta function with the period matrix of a hyperelliptic curve, and in this form we reprove them by direct geometric methods.

Keywords: random matrix theory, theta function, Fay's identity, hyperelliptic curves.

MSC: 60B20, 14H42

Received: January 30, 2024; in final form June 6, 2024; Published online June 23, 2024

Language: English

DOI: 10.3842/SIGMA.2024.054

ArXiv: 2312.12229