Reduction and inverse-reduction functors I: standard $\mathsf{V^k}(\mathfrak{sl}_2)$-modules

TL;DR

This paper explores the application of quantum Hamiltonian reduction and its inverse to standard modules of affine vertex-operator algebras, providing a formalism and explicit computations for $ ext{sl}_2$.

Contribution

It introduces a formal framework for combining reduction and inverse-reduction functors and applies it to compute their action on standard modules of $ ext{sl}_2$.

Findings

Computed the action of reduction on standard modules of $ ext{sl}_2$

Presented a general formalism for reduction and inverse-reduction functors

Identified the role of unbounded spectral sequences in the formalism

Abstract

Quantum hamiltonian reduction is a fundamental tool of conformal field theory and vertex algebra representation theory. It has traditionally been applied to study highest-weight modules. On the other hand, inverse quantum hamiltonian reduction lends itself to the study of fully relaxed highest-weight modules and their spectral flows, sometimes called the standard modules. This is the first of several papers that study the composition of reduction and inverse-reduction functors. A general formalism is presented and exemplified with the simplest example, thereby computing the action of reduction on the standard modules of the affine vertex-operator algebra associated with $\mathfrak{sl}_2$. The appearence of unbounded spectral sequences in this formalism may be of independent interest.