Nonassociative tori and applications to T-duality
Peter Bouwknegt, Keith Hannabuss, Varghese Mathai
TL;DR
This work extends T-duality in string theory to a nonassociative C*-algebraic framework by introducing twisted Busby–Smith crossed products $\mathcal{B}\rtimes_{\beta,v}G$ governed by a 3-cocycle obstruction $\phi$. It develops twisted induced algebras, stabilisation, and a comprehensive duality theory, proving that the T-dual of a principal torus bundle with H-flux can be a bundle of nonassociative tori, with nonassociativity controlled by the flux components $H_0$, $H_1$, $H_2$, and $H_3$. The results unify prior special cases (noncommutative tori and classical commutative cases) via a twisted Takai duality and explicit kernel descriptions, and they pave the way for a monoidal-category reformulation to manage associativity. Overall, the paper provides a rigorous algebraic mechanism for nonassociative T-duality and highlights avenues for K-theory, invariants, and categorified dualities in noncommutative geometry.
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.