Conformal blocks on smoothings via mode transition algebras

TL;DR

This paper introduces mode transition algebras associated with vertex operator algebras, linking algebraic structures to geometric moduli of curves, and applies this to describe higher level Zhu algebras, confirming a conjecture.

Contribution

It defines mode transition algebras that connect algebraic and geometric aspects of vertex operator algebras and applies this to describe higher level Zhu algebras, including a complete description for the Heisenberg algebra.

Findings

Mode transition algebras reflect properties of vertex operator algebras and moduli of curves.

Sheaves of coinvariants deform on families of curves with nodes when certain algebraic conditions are met.

Complete description of higher level Zhu algebras for the Heisenberg vertex algebra, confirming a conjecture.

Abstract

Here we define a series of associative algebras attached to a vertex operator algebra $V$, called mode transition algebras, showing they reflect both algebraic properties of $V$ and geometric constructions on moduli of curves. One can define sheaves of coinvariants on pointed coordinatized curves from $V$-modules. We show that if the mode transition algebras admit multiplicative identities with certain properties, these sheaves deform as wanted on families of curves with nodes (so $V$ satisfies smoothing). Consequently, coherent sheaves of coinvariants defined by vertex operator algebras that satisfy smoothing form vector bundles. We also show that mode transition algebras give information about higher level Zhu algebras and generalized Verma modules. As an application, we completely describe higher level Zhu algebras of the Heisenberg vertex algebra for all levels, proving a conjecture of Addabbo--Barron.