Classical Theta Functions and Quantum Tori

TL;DR

This paper explores the relationship between classical theta functions and quantum tori, revealing that the multiplication kernel on a quantum torus is a boundary value of a classical theta function and satisfies a Schrödinger-type equation.

Contribution

It demonstrates that the multiplication kernel on a quantum torus is a boundary value of a classical multivariate theta function and satisfies a Schrödinger equation involving the deformation parameter.

Findings

Kernel is the boundary value of a classical theta function

Kernel satisfies a Schrödinger equation with deformation parameter

In some cases, kernel decomposes into products of single-variable theta functions

Abstract

The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schr\"odinger equation in which the role of time is played by the deformation parameter $\hbar$ and the role of the hamiltonian by a Poisson structure. At least in some special cases, the kernel can be written as a sum of products of single-variable theta functions.