Introducing inverse Hamiltonian reduction

Tuesday, 28 February 2023

Abstract

In this talk, I want to give an introduction to and overview of a new idea in mathematical physics and representation theory motivated by conformal field theory (CFT).

The problem at hand is classifying certain families of weight modules for affine Lie algebras, namely, those of interest in the mathematical study of CFT. For example, the well-studied Wess-Zumino-Witten (WZW) models are built from the nicest irreducible highest-weight modules over the corresponding affine algebra. However, in many models of interest in CFT, such as the so-called fractional-level WZW modules, these modules are insufficient to explain the physics completely; something more is required.

Related to this picture are the so-called W-algebras, which are constructed from affine Lie algebras by a procedure known as hamiltonian reduction. The reduction procedure also gives a way to construct W-algebra modules from affine Lie algebra modules. This is useful because the W-algebra modules of interest in CFT are easier to understand than the corresponding affine Lie algebra modules.

Recent work has shown that in certain cases, it is possible to invert the reduction procedure. This means one can reconstruct the messy representation theory of the affine algebras from their reduced W-algebra counterparts. Generalising this construction (which should be possible) would rise to a constructive and holistic approach to studying the weight modules over W-algebras and affine Lie algebras of interest in CFT. Inverse reduction involves many parts and is still very much a work in progress by myself and many others. Therefore, my goal for the talk is to try and impart some of the flavour of the problem along with some of the key ideas using simple examples.