Denominator identity for the affine Lie superalgebra $\widehat{\mathfrak{spo}}(2m,2m+1)$ and indefinite theta functions

TL;DR

This paper provides a new proof of a denominator identity for the affine Lie superalgebra dee7f(2m,2m+1), demonstrating that a related indefinite theta function is a modular form using modern techniques.

Contribution

It introduces a novel proof of the denominator identity for a specific affine Lie superalgebra, employing indefinite theta functions and modular form theory.

Findings

Proved the modularity of a certain indefinite theta function.

Established the denominator identity for dee7f(2m,2m+1).

Applied recent techniques by Roehrig and Zwegers.

Abstract

In 1994, Kac and Wakimoto found the denominator identity for classical affine Lie superalgebras, generalizing that for affine Lie algebras. As an application, they obtained power series identities for some powers of $\triangle(q)$, where $\triangle(q)$ is the generating function of triangular numbers. In this article, we give a different proof of one of their identities. The main step is to prove that a certain indefinite theta function involving spherical polynomials is a modular form. We use the technique recently developed by Roehrig and Zwegers.