Topological structures in string theory

Abstract

In string theory space–time comes equipped with an additional geometric structure called a B–field or ‘gerbe’. I describe this structure, mention its relationship with noncommutative geometry, and explain how to use the B–field to define a twisted version of the K–theory of space–time. String–theoretical space–time can contain topologically non–trivial dynamical structures called D–branes. These are simply accounted for in the framework of conformal field theory. In a highly simplified limiting casetopological field theory with a finite gauge group—the D–branes naturally represent elements of the twisted K–theory of space–time: the K–theory class is the ‘charge’ of the D–brane.