Kac-Wakimoto characters and non-holomorphic Jacobi forms (Q2833014)

In this paper, the authors study automorphic properties of the Kac-Wakimto characters pertaining to the affine Lie superalgebra \(s\ell(m|n)^{\wedge}\). These are of the form NEWLINE\[NEWLINE\text{ch}F = e^{\Lambda_0}\prod_{k\geq 1}\frac{\prod_{r=1}^m\left(1+\zeta\beta_rq^{k-\frac{1}{2}}\right)\left(1+\zeta^{-1}\beta_r^{-1}q^{k-\frac{1}{2}}\right)}{\prod_{j=1}^n\left(1-\zeta\beta_{m+j}q^{k-\frac{1}{2}}\right)\left(1-\zeta^{-1}\beta_{m+j}^{-1}q^{k-\frac{1}{2}}\right)}\tag{1}NEWLINE\]NEWLINE (see [\textit{V. G. Kac} and \textit{M. Wakimoto}, Commun. Math. Phys. 215, No. 3, 631--682 (2001; Zbl 0980.17002)] for details). Special instances of ch\(F\) have been studied in several papers with the same goal. The first author considered together with \textit{K. Ono} the case \(n=1\), see [Math. Ann. 345, No. 3, 547--558 (2009; Zbl 1262.11059)]. They added to (1) a non-holomorphic function to obtain a non-holomorphic modular function. The same idea was applied to more general versions of (1). Along the way new types of automorphic functions were developed to describe the more general characters. In [J. Reine Angew. Math. 694, 179--202 (2014; Zbl 1307.11054)] the first author and \textit{A. Folsom} treated the case of arbitrary \(n\in\mathbb N\) but assumed that the coefficients \(\beta_s\) and \(e^{\Lambda_0}\) in (1) are equal to 1. It turned out that the Kac-Wakimoto characters are the holomorphic part of so called \textit{almost hormonic Maas forms of depth $r$}.NEWLINENEWLINEIn the present paper, the authors drop the above mentioned restriction on the \(\beta_s\) and \(e^{\Lambda_0}\) and show that in this situation ch\(F\) is the holomorphic part of a new type of Jacobi form, a \textit{mixed harmonic Maass-Jacobi form}, using the following identity NEWLINE\[NEWLINE\text{ch}F = e^{\Lambda_0}(-1)^{m}i^{-n}\zeta^Mq^{\frac{M}{3}}\eta(\tau)^{-2M}\left(\prod_{r=1}^m\beta_r^{\frac{1}{2}}\right)\left(\prod_{j=1}^n\beta_{m+j}^{-\frac{1}{2}}\right)\Phi\left(z+\frac{\tau}{2},{\mathbf u},\tau\right).NEWLINE\]NEWLINE Here \(\eta\) is the Dedekind eta function and NEWLINE\[NEWLINE\Phi(z,{\mathbf u}, \tau)=\frac{\prod_{r=1}^s\vartheta(z+u_r+\frac{1}{2};\tau)^{m_r}}{\prod_{j=1}^t\vartheta(z-w_j;\tau)^{n_j}}NEWLINE\]NEWLINE is a quotient of the classical Jacobi theta functions \(\vartheta\). In sections 3--6, several properties of \(\Phi\) are derived. Among them, the proof that the Fourier coefficients of this function in \(z\) are the holomorphic parts of almost Maass-Jacobi forms (see Theorem 7.1 in the paper). In section 3, the authors decompose \(\Phi\) into a polar and a finite part by extending results of \textit{A. Dabholkar} et al. [Quantum black holes, wall crossing, and mock modular forms'', Cambridge Monogr. Math. Phys. (to appear); \url{arXiv:1208.4074}]. In section 5, both parts are completed by adding a non-holomorphic term to obtain a non-holomorphic Jacobi form. The paper concludes in section 7 with the proof of Theorem 1.1.