Vertex Algebras and Algebraic Curves
Lectures: MWF 11-12, DRLB 4C4
Course Description
Prerequisites: Some very basic knowledge of Lie algebras (just the
definition and some examples really) and Riemann surfaces.
Vertex algebras arose as an attempt to obtain a mathematical
description of certain algebraic structures in conformal field theory
in physics. They have since found many applications in representation
theory and in the study of moduli spaces in algebraic geometry. For
example, they play a central role in the proof of the Moonshine
conjecture, for which Borcherds received the 1998 Fields medal, and
more recently have made an appearance in the geometric Langlands
correspondence of Beilinson and Drinfeld. The latter is a geometric
analogue of the classical Langlands programme in number theory,
relating automorphic forms and Galois representations.
We will begin by introducing the notion of a vertex algebra
from a purely algebraic point of view, and discuss simple examples,
such as the Heisenberg, Kac-Moody, and lattice vertex algebras. We
will then construct sheaves of vertex algebras over algebraic curves
and obtain a coordinate-free description of vertex operators. This
will allow us to formulate the notion of conformal blocks, which are
the building blocks of correlation functions in the underlying field
theory. We will discuss the Knizhnik-Zamolodchikov equations and their
solutions. These are differential equations satisfied by conformal
blocks on the projective line. Further possible topics include the
relation between vertex algebras and moduli spaces of curves/bundles,
and applications to the geometric Langlands correspondence .
Text:
Frenkel, Ben-Zvi Vertex algebras and
algebraic curves
Evaluation:
Students enrolled in the course will be asked to
give a talk on a related subject of their choice.