Mock theta functions and indefinite modular forms

TL;DR

This paper explores the modular properties of mock theta functions derived from indefinite quadratic forms related to the coroot lattice of a specific Lie superalgebra, using Zwegers' theory to establish their invariance under modular transformations.

Contribution

It explicitly computes the modular transformation properties of certain mock theta functions associated with indefinite quadratic forms and demonstrates their invariance under SL_2(Z).

Findings

The functions form an SL_2(Z)-invariant space.

Modular transformation properties are explicitly derived.

The functions are connected to the coroot lattice of D(2,1;a).

Abstract

In the explicit formula for the signed mock theta functions $\Phi^{(-)[m,s]}$ obtained from the coroot lattice of $D(2,1;a)$, functions with indefinite quadratic forms naturally take place. We compute their modular transformation properties by applying the Zwegers' modification theory of mock theta functions and show that the $\mathbf{C}$-linear span of these functions is $SL_2(\mathbf{Z})$-invariant.