D-Branes, RR-Fields and Duality on Noncommutative Manifolds

TL;DR

This paper develops a framework for string theory on noncommutative spacetimes, introducing a T-duality formulation, a D-brane charge formula, and a noncommutative Grothendieck-Riemann-Roch theorem, utilizing advanced K-theory and cyclic theory techniques.

Contribution

It presents a new axiomatic approach to T-duality and a general D-brane charge formula within noncommutative geometry, supported by a noncommutative Grothendieck-Riemann-Roch theorem.

Findings

Axiomatic formulation of T-duality for noncommutative spacetimes

A general formula for D-brane charges in noncommutative geometry

Development of a noncommutative Grothendieck-Riemann-Roch theorem

Abstract

We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincare duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant K-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.