250091 VO Introduction to conformal field theory (2014W)

This course is on two-dimensional conformal field theory (CFT), mostly from an axiomatic and algebraic point of view. After a short motivational discussion of axiomatic quantum field theory, we first study the group of conformal transformations and the associated Lie algebra in d-dimensional Euclidean space. The case d=2 is special and we will see how the infinite-dimensional Virasoro algebra arises. We then define a CFT axiomatically by imposing certain constraints on correlation functions. To construct CFTs we are lead to introduce the notion of conformal vertex operator algebras and their representations. In this setting we prove existence and uniqueness theorems, and study conformal Ward identities and conformal blocks. The latter will allow us to make the connection with three-dimensional topological quantum field theory. Ultimately we aim to describe full CFTs on arbitrary Riemann surfaces in terms of tensor categories and Frobenius algebras, as developed in the work of Fuchs-Runkel-Schweigert; this may only be achieved during the summer term 2015.

The main prerequisite for the course is an interest in algebraic structures. An interest in quantum field theory and a rudimentary familiarity with differential geometry and basic category theory is useful, but not very essential. Both pure mathematicians and theoretical physicists are welcome.