Some Comments on Branes, G-flux, and K-theory

TL;DR

This work surveys three interlinked avenues by which string theory and M-theory relate to $K$-theory. First, the RR flux and M-theory $G$-flux data organize into $K^0(X)$-labeled sectors, yielding a theta-function framework that aligns IIA and M-theory partition functions in the long-distance limit. Second, a 2D topological field theory perspective shows that open/closed string consistency naturally leads to a $K$-theory classification of D-branes via Frobenius algebras and their modules. Third, the $K$-theory of $C^*$-algebras provides a natural home for tachyon condensation in the presence of B-fields, with noncommutative geometry giving concrete realizations of brane charges and their orbifold generalizations. Together these results illuminate how K-theory encodes brane charges, flux quantization, and boundary conditions, and suggest rich generalizations to NS-NS charges, S-duality, and nonperturbative dynamics.

Abstract

This is a summary of a talk at Strings2000 explaining three ways in which string theory and M-theory are related to the mathematics of K-theory.