Generalized theta functions, projectively flat vector bundles and noncommutative tori

TL;DR

This paper explores the deep connections between theta functions, noncommutative geometry, and vector bundles on 2-tori, extending classical theories through algebraic and geometric techniques to reveal new dualities and symmetries.

Contribution

It introduces a novel framework linking Hermitian-Einstein bundles on 2-tori with noncommutative tori representations, extending Fourier-Mukai-Nahm duality and theta function theory.

Findings

Existence of noncommutative torus actions on sections of vector bundles

Extension of theta functions to vector-valued cases

Algebraic reinterpretation of Matsushima's theory

Abstract

In this paper, the well-known relationship between theta functions and Heisenberg group actions thereon is resumed by combining complex algebraic and noncommutative geometric techniques in that we describe Hermitian-Einstein vector bundles on 2-tori via representations of noncommutative tori, thereby reconstructing Matsushima's setup and elucidating the ensuing Fourier-Mukai-Nahm (FMN) aspects. We prove the existence of noncommutative torus actions on the space of smooth sections of Hermitian-Einstein vector bundles on 2-tori preserving the eigenspaces of a natural Laplace operator. Motivated by the Coherent State Transform approach to theta functions, we extend the latter to vector valued thetas and develop an additional algebraic reinterpretation of Matsushima's theory making FMN-duality manifest again.