April 10, 2018

Paper submitted to arXiv

Motivated by the previous result that a holomorphic zero-mode wave function in abelian Chern-Simons theory on the torus can be considered as a quantum version of a modular form of weight 2, we consider how a Hecke operator acts on such wave functions. We argue that the action of the Hecke operator can be considered as a sum over possible gauge transformations of the wave function. (The gauge transformations are induced by doubly periodic translations.) The resultant expressions suggest that the notion of the level which is inherent to the modular forms naturally arises for the wave function. We also present a speculative idea on the computation of the Hecke eigenvalues. I have tried to find more fruitful results but spent a few months in vain. For details please see the paper.