Vertex Lie algebras and cyclotomic coinvariants

TL;DR

This paper develops a framework for cyclotomic coinvariants associated with vertex Lie algebras under cyclic group actions, proving their equivalence with big coinvariants and exploring their functoriality and module structures.

Contribution

It introduces the concept of cyclotomic coinvariants for vertex Lie algebras with cyclic group actions and proves their equivalence with big coinvariants, advancing the understanding of their structure and functorial properties.

Findings

Proved the equivalence of two definitions of cyclotomic coinvariants.

Established functoriality properties of cyclotomic coinvariants.

Derived an iterate formula for modules over the stable subalgebra at the origin.

Abstract

Given a vertex Lie algebra $\mathscr L$ equipped with an action by automorphisms of a cyclic group $\Gamma$, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over `local' Lie algebras $\mathsf L(\mathscr L)_{z_i}$ assigned to marked points $z_i$, by the action of a `global' Lie algebra ${\mathsf L}^{\Gamma}_{\{z_i \}}(\mathscr L)$ of $\Gamma$-equivariant functions. On the other hand, the universal enveloping vertex algebra $\mathbb V (\mathscr L)$ of $\mathscr L$ is itself a vertex Lie algebra with an induced action of $\Gamma$. This gives `big' analogs of the Lie algebras above. From these we construct the space of `big' cyclotomic coinvariants, i.e. coinvariants with respect to ${\mathsf L}^{\Gamma}_{\{z_i \}}(\mathbb V(\mathscr L))$. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in arXiv:1409.6937. At the origin, which is fixed by $\Gamma$, one must assign a module over the stable subalgebra $\mathsf L(\mathscr L)^{\Gamma}$ of $\mathsf L(\mathscr L)$. This module becomes a $\mathbb V(\mathscr L)$-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.