Twisted modules and co-invariants for commutative vertex algebras of jet schemes

TL;DR

This paper develops the orbifold theory for the vertex algebra structure of jet schemes of affine schemes, showing how twisted modules relate to fixed point schemes and coinvariants in the context of automorphisms.

Contribution

It introduces the construction of twisted modules for the vertex algebra of jet schemes under finite automorphisms and relates orbifold coinvariants to fixed point schemes.

Findings

Twisted modules for jet scheme vertex algebras are constructed.

Orbifold coinvariants are isomorphic to coordinate rings of fixed point schemes.

The theory extends vertex algebra orbifold concepts to jet schemes of affine varieties.

Abstract

Let $Z \subset \mathbb{A}^k$ be an affine scheme over $\C$ and $\J Z$ its jet scheme. It is well-known that $\mathbb{C}[\J Z]$, the coordinate ring of $\J Z$, has the structure of a commutative vertex algebra. This paper develops the orbifold theory for $\mathbb{C}[\J Z]$. A finite-order linear automorphism $g$ of $Z$ acts by vertex algebra automorphisms on $\mathbb{C}[\J Z]$. We show that $\mathbb{C}[\J^g Z]$, where $\J^g Z$ is the scheme of $g$--twisted jets has the structure of a $g$-twisted $\mathbb{C}[\J Z]$ module. We consider spaces of orbifold coinvariants valued in the modules $\mathbb{C}[\J^g Z]$ on orbicurves $[Y/G]$, with $Y$ a smooth projective curve and $G$ a finite group, and show that these are isomorphic to $\mathbb{C}[Z^G]$.