Asymptotic formulas for coefficients of inverse theta functions

TL;DR

This paper derives asymptotic formulas for coefficients of certain negative index Jacobi forms, which relate to partition numbers, algebraic geometry, and theoretical physics, providing new insights into their growth behavior.

Contribution

It introduces asymptotic formulas for coefficients of negative index Jacobi forms, linking partition theory, algebraic geometry, and physics.

Findings

Derived asymptotic formulas for coefficients

Connected coefficients to Betti numbers and BPS state counts

Enhanced understanding of negative index Jacobi forms

Abstract

We determine asymptotic formulas for the coefficients of a natural class of negative index and negative weight Jacobi forms. These coefficients can be viewed as a refinement of the numbers $p_k(n)$ of partitions of n into k colors. Part of the motivation for this work is that they are equal to the Betti numbers of the Hilbert scheme of points on an algebraic surface S and appear also as counts of Bogomolny-Prasad-Sommerfield (BPS) states in physics.