Almost harmonic Maass forms and Kac–Wakimoto characters

Abstract

We resolve a question of Kac and explain the automorphic properties of characters due to Kac-Wakimoto pertaining to s`.mjn/^highest weight modules, for n 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 D 1. We also provide an explicit asymptotic expansion for the characters.

Negative index Jacobi forms and quantum modular forms

“…The relationship between Jacobi forms and modular forms has appeared in many guises and stems back to important work on holomorphic Jacobi forms, which states that they have theta decompositions relating them to half-integral weight modular forms [1]. The situation for meromorphic positive index Jacobi forms also well understood; a meromorphic Jacobi form of positive index has http://www.resmathsci.com/content/1/1 /11 Fourier coefficients which are almost mock modular forms, which in turn are holomorphic parts of almost harmonic Maass forms [2][3][4][5][6]. Loosely speaking, almost harmonic weak Maass forms are sums of harmonic weak Maass functions under iterates of the raising operator multiplied by almost holomorphic modular forms.…”

A proof of the Thompson moonshine conjecture

“…• its Fourier expansion is of the form 2q −3 + χ 2 (g) + (χ 4 (g) + χ 5 (g))q 4 + O(q 5 ), where χ j is the jth irreducible character of Th as given in Tables 1, 2 [25]. In Sect.…”

“…It took until the first decade of the twenty-first century before work by Zwegers [43], Bruinier and Funke [7] and Bringmann and Ono [5,6] established the "right" framework for these enigmatic functions of Ramanujan's, namely that of harmonic Maaß forms. Since then, there have been many applications of harmonic Maaß forms both in various fields of pure mathematics, see for instance [1,4,9,16], among many others, and mathematical physics, especially in regard to quantum black holes and wall crossing [15] as well as Mathieu and Umbral Moonshine [12,13,18,23]. For a general overview on the subject, we refer the reader to [31,42].…”

“…Projecting this function to the Kohnen plus space then yields the function 4) where A N,ψ is given by Table 5 and define the function This is going to be the explicitly given weakly holomorphic modular form (see Proposition 3.1) mentioned in Conjecture 1.1, meaning that we have…”

“…In order to compute the necessary Fourier coefficients exactly without relying on the rather slow convergence of the Fourier coefficients of the Rademacher series, one can construct linear combinations of weight 1 2 weakly holomorphic eta quotients again using the programs 4 written by Rouse and Webb [34] which have the same principal part at ∞ (and the related cusp [35] is easily seen to be 0. 5 The largest bound up to which coefficients need to be checked turns out to be 384 for [g] = 24CD.…”