Generalised Smooth Cohomology Theories
Final Report Abstract
This project was concerned with obtaining a deeper understanding of differential cohomology theories in general, and differential K-theory in particular. These are tools that allow a fine scale probing of the geometry of manifolds, refining the global sensibility of cohomology theories to allow local geometric information also to be captured. This local sensitivity has proved very important to physical applications - the language of differential cohomology has proved itself ideally adapted to describe fields in abelian gauge theories like electromagnetism, and also more exotic theories like the theory of Ramond-Ramond fields and D-branes. Prior to the beginning of the project, much of the basic work had been done in the area, but there remained many interesting questions. In particular, while it was conjectured that various differential refinements of K-theory existed (for example, differential twisted K-theory), and in fact the existence of these theories was essential to describing certain aspects of string theory and supergravity, no adequate models of these refinements existed. Indeed, it was not very well understood what properties these refinements should have. One of the main goals of this project has been to fill this hole in the literature. We have used an approach to K-theory developed by Stolz and Teichner, where the basic “co-cycles" are in fact field theories (in the sense a physicist might think of these). This language has proven to be ideally suited to the task, and work by the principle author with Arturo Prat-Waldron has, at least at the conjectural level, a good model of the various refinements of K-theory above. The work, while mostly done, is however not complete. Even before attempting to provide such a refinement, it is important to know exactly what properties the refinement should enjoy. Elucidating these was a pleasing side effect of another part of this project. Along with Alessandro Valentino, the principle investigator set about investigating “T-duality" in the context of differential K-theory. T-duality is an important and deep symmetry in superstring theory, relating various seemingly disparate string theories. The basic requirement is that the spacetime have an action by isometries of an abelian group (the “T” in T-duality) - when this occurs one is supposed to have an equivalent “dual” description of the theory that looks quite different. Understanding T-duality in a mathematical context has proven to be challenging, and thus a source of many interesting results. One context where it is fairly well understood is in K-theory: here it predicts the isomorphism of twisted K-theory groups on “dual” principle torus bundles. In work with Alessandro Valentino, we refined this story to a story about differential K-theory. Physically this would mean shifting attention the charged objects (the D-branes) to the fields that they generate and interact with (the Ramond-Ramond fields). As expected, there is a mathematical refinement of T-duality to differential K- theory. However, in investigating this refinement we came across several surprising twists. For example, we needed an understanding of differential K-theory and twisted differential K-theory at a rather more refined level than usually considered in the literature (at the co-cycle level). In addition, we discovered that understanding various notions of invariance of differential cohomology groups under group actions proved key to a good formulation of the problem. This understanding of differential cohomology at the co-cycle level has been key to guiding the construction of twisted differential K-theory with Arturo Prat-Waldron. Finally, the work on T-duality led rather surprisingly to a direct application to superstring theory, and hopefully towards a better understanding of the coupling between Ramond-Ramond fields and D-branes. This is the subject of on-going work with Ruben Minasian, where we use the language of “generalised geometry” in the sense of Hitchin and twisted differential K-theory to write down the coupling in a way that reproduces new terms in the action predicted by perturbative analysis, as well as predict new terms that should lead to a fully T-duality and gauge invariant coupling.
Publications
- T-duality and Di erential K-theory.
Communications in Contemporary Mathematics 1350014
arXiv:0912.2516 (preprint)
Alexander Kahle and Alessandro Valentino
(See online at https://doi.org/10.1142/S0219199713500144)
