Nonassociative tori and applications to T-duality

TL;DR

This paper develops a framework for nonassociative tori via twisted crossed product C*-algebras and applies it to T-duality in string theory, revealing noncommutative, nonassociative structures in T-duals of torus bundles.

Contribution

It introduces a novel construction of nonassociative tori using twisted crossed products and applies this to analyze T-duality in string theory with H-flux.

Findings

T-duals of principal torus bundles with H-flux are noncommutative, nonassociative tori.

The construction generalizes previous special cases.

Provides a mathematical framework for nonassociative geometries in string theory.

Abstract

In this paper, we initiate the study of C*-algebras endowed with a twisted action of a locally compact Abelian Lie group, and we construct a twisted crossed product, which is in general a nonassociative, noncommutative, algebra. The properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. We also show that this construction of the T-dual includes all of the special cases that were previously analysed.