Generalized theta functions, projectively flat vector bundles and noncommutative tori

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 (1976) 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 (Florentino, Mourão, Nunes (2002), Tyurin (2003)), we extend the latter to vector valued thetas and develop an additional algebraic reinterpretation of Matsushima’s theory making FMN-duality manifest again.