Dualities in Field Theories and the Role of K-Theory

s of the Lectures Lecture 1 The basic idea of dualities in field theories and T-duality It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, and then give a (simplified) outline of the case of string theory and T-duality (“target space” duality). This duality is closely related to other dualities in string theory, such as mirror symmetry. Lecture 2 K-Theory and its relevance to physics K-theory is the extraordinary cohomology theory that arises out of classifying vector bundles. It thus has a very simple geometric interpretation. We will discuss some of the ways vector bundles arise in physics and how the study of “charge conservation” forces one to consider the role played by K-theory.