Some properties of zero-mode wave functions in abelian Chern-Simons theory on the torus

In geometric quantization a zero-mode wave function in abelian Chern-Simons theory on the torus can be defined as $\Psi [ a, \bar{a} ] = e^{- \frac{K(a, \bar{a})}{2}} f (a)$ where $K(a ,\bar{a} )$ denotes a K\"ahler potential for the zero-mode variable $a \in \mathbb{C}$ on the torus. We first review that the holomorphic wave function $f(a)$ can be described in terms of the Jacobi theta functions by imposing gauge invariance on $\Psi [ a, \bar{a} ]$ where gauge transformations are induced by doubly periodic translations of $a$. We discuss that $f(a)$ is quantum theoretically characterized by ($i$) an operative relation in the $a$-space representation and ($ii$) an inner product of $\Psi [ a, \bar{a} ]$'s including ambiguities in the choice of $K(a ,\bar{a} )$. We then carry out a similar analysis on the gauge invariance of $\Psi [ a , \bar{a} ]$ where the gauge transformations are induced by modular transformations of the zero-mode variable. We observe that$f(a)$ behaves as a modular form of weight 2 under the condition of $|a|^2 = 1$, namely, $\left. f \left( - \frac{1}{a} \right) = a^2 f(a) \right|_{|a|^2 = 1}$. Utilizing specific forms of $f(a)$ in terms of the Jacobi theta functions, we further investigate how exactly $f(a)$ can or cannot be interpreted as the modular form of weight 2; we extract conditions that make such an interpretation possible.