What Does(n't) K-theory Classify?

TL;DR

This paper reviews various conjectures and proposals for classifying D-branes and RR fluxes in string theory using K-theory, highlighting successes, limitations, and applications in dualities and specific geometries.

Contribution

It provides a comprehensive review of K-theory classification proposals in string theory, analyzing their consistency, limitations, and implications for dualities and specific backgrounds.

Findings

Some proposals classify all twisted K-theory classes via boundary states.

Certain proposals fail to respect RR field self-duality and S-duality.

Twisted K-theory predicts topology of T-duals and classifies vacua in specific geometries.

Abstract

We review various K-theory classification conjectures in string theory. Sen conjecture based proposals classify D-brane trajectories in backgrounds with no H flux, while Freed-Witten anomaly based proposals classify conserved RR charges and magnetic RR fluxes in topologically time-independent backgrounds. In exactly solvable CFTs a classification of well-defined boundary states implies that there are branes representing every twisted K-theory class. Some of these proposals fail to respect the self-duality of the RR fields in the democratic formulation of type II supergravity and none respect S-duality in type IIB string theory. We discuss two applications. The twisted K-theory classification has led to a conjecture for the topology of the T-dual of any configuration. In the Klebanov-Strassler geometry twisted K-theory classifies universality classes of baryonic vacua.