Almost harmonic Maass forms and Kac-Wakimoto characters

TL;DR

This paper demonstrates that Kac-Wakimoto characters for sl(m|n)^ highest weight modules are essentially holomorphic parts of new 'almost harmonic Maass forms', generalizing previous work and providing explicit asymptotics.

Contribution

It introduces 'almost harmonic Maass forms' and proves that Kac-Wakimoto characters are their holomorphic parts, extending prior results to n ≥ 1.

Findings

Kac-Wakimoto characters are holomorphic parts of almost harmonic Maass forms

Generalization of harmonic Maass forms to 'almost' versions for n ≥ 1

Explicit asymptotic expansion of characters provided

Abstract

We resolve a question of Kac, and explain the automorphic properties of characters due to Kac-Wakimoto pertaining to sl(m|n)^ highest weight modules, for n \geq 1. We prove that the Kac-Wakimoto characters are essentially holomorphic parts of certain generalizations of harmonic weak Maass forms which we call "almost harmonic Maass forms". Using a new approach, this generalizes prior work of the first author and Ono, and the authors, both of which treat only the case n = 1. We also provide an explicit asymptotic expansion for the characters.