Theta functions on Noncommutative T^4

TL;DR

This paper constructs theta vectors on noncommutative T^4, extending the concept of theta functions to noncommutative geometry with complex structures, using holomorphic connections and tensor products.

Contribution

It introduces a method to define theta vectors on noncommutative T^4, including the construction of holomorphic structures and tensor products, advancing noncommutative complex geometry.

Findings

Defined holomorphic connections on noncommutative T^4

Constructed theta vectors satisfying holomorphic conditions

Established tensor product of theta vectors

Abstract

We construct the so-called theta vectors on noncommutative T^4, which correspond to the theta functions on commutative tori with complex structures. Following the method of Dieng and Schwarz, we first construct holomorphic connections and then find the functions satisfying the holomorphic conditions, the theta vectors. The holomorphic structure in the noncommutative T^4 case is given by a 2x2 complex matrix, and the consistency requires its off-diagonal elements to be the same. We also construct the tensor product of these functions satisfying the consistency requirement.