T-Duality in Type II String Theory via Noncommutative Geometry and Beyond

Abstract

It is by now well known that there are five (super) string theories, which are related to each other by transformations called dualities. If two such string theories are related by a duality transformation, it means that they are mathematically equivalent descriptions, and the string theories are said to be dual to each other. Here we review global aspects of one such duality, called T-duality in type II string theory, involving compactifications of spacetimeX with H -flux. That is,X is a principal torus bundle over M , with H -flux, and applying T-duality transformation along the fibre in general gives a space which does not admit a global manifold description. We precisely identify the cases in which the T-dual is another principal torus bundle with H-flux. This is always the case when when the torus bundle is of rank one, i.e. in the case of circle bundles. In general, the T-dual spacetime will have different topology to the original spacetime. For the case of torus bundles of rank greater than or equal to 2, we precisely identify the cases when the Tdual is a continuous field of stabilized noncommutative tori fibred over M , which globally corresponds to a certain crossed product C∗-algebra. This is always the case when the torus bundle is of rank two or less. The general case, when the rank or the principal torus bundle X with H-flux is greater than or equal to 3, leads to continuous fields of nonassociative and noncommutative tori in the fibre directions. Even in the general case, the action of T-duality is again globally realized by taking a certain twisted crossed product C∗-algebra (but now in a tensor category), and motivates an axiomatic definition of string T-duality. In this way, we see that both noncommutative and nonassociative geometry appears naturally in the study of Tduality. This article comprises essentially of a transcript of my talks that were delivered at the 21st Nishinomiya-Yukawa Memorial Symposium on Theoretical Physics: “Noncommutative Geometry and Quantum Spacetime in Physics”, Japan, 11–15 Nov 2006. I am grateful to the organizers for the invitation to give a series of survey talks based on the joint work with my collaborators, P. Bouwknegt, J. Evslin, K. Hannabuss and J. Rosenberg. In the title, the phrase “and beyond” refers to nonas-