Mock theta functions and indefinite modular forms

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 $S{L}_2(\mathbf{Z})$-invariant.