Abelian Chern-Simons theory on the torus and physical views on the Hecke operators

TL;DR

This paper explores the action of Hecke operators on zero-mode wave functions in abelian Chern-Simons theory on a torus, providing a physical perspective on these number-theoretic operators.

Contribution

It introduces a gauge-theoretic interpretation of Hecke operators acting on wave functions related to modular forms in abelian Chern-Simons theory.

Findings

Hecke operators can be understood as physical transformations in Chern-Simons theory.

The wave functions correspond to modular forms of weight 2.

Provides a novel physical interpretation of number-theoretic Hecke operators.

Abstract

In the previous paper arXiv:1711.07122, we show 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. Motivated by this result, in this paper we consider an action of a Hecke operator on such a wave function from a gauge theoretic perspective. This leads us to obtain some physical views on the Hecke operators in number theory.