Abstract

This is an expository paper which aims at explaining a physical point of view on the K-theoretic classification of D-branes. We combine ideas of renormalization group flows between boundary conformal field theories, together with spacetime notions such as anomaly cancellation and D-brane instanton effects. We illustrate this point of view by describing the twisted K-theory of the special unitary groups SU(N).

Introduction

This is an expository paper devoted to explaining some aspects of the K-theoretic classification of D-branes. Our aim is to address the topic in ways complementary to the discussions of,. Reviews of the latter approaches include. Our intended audience is the mathematician who is well-versed in conformal field theory and K-theory, and has some interest in the wider universe of (nonconformal) quantum field theories.

Our plan for the paper is to begin in Section 2 by reviewing the relation of D-branes and K-theory at the level of topological field theory. Then in Section 3 we will move on to discuss D-branes in conformal field theory. We will advocate a point of view emphasizing 2-dimensional conformal field theories as elements of a larger space of 2-dimensional quantum field theories. ‘D-branes’ are identified with conformal quantum field theories on 2-dimensional manifolds with boundary. From this vantage, the topological classification of D-branes is the classification of the connected components of the space of 2-dimensional theories on manifolds with boundary which only break conformal invariance through their boundary conditions.