Vertex Lie algebras and cyclotomic coinvariants

Abstract

Given a vertex Lie algebra $\mathcal{L}$ 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 $\mathsf{L}(\mathcal{L}{)}_{z_i}$ assigned to marked points $z_i$, by the action of a `global' Lie algebra ${\mathsf{L}}_{\{z_i\}}^{\Gamma }(\mathcal{L})$ of $\Gamma$-equivariant functions. On the other hand, the universal enveloping vertex algebra $\mathbb{V}(\mathcal{L})$ of $\mathcal{L}$ 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 ${\mathsf{L}}_{\{z_i\}}^{\Gamma }(\mathbb{V}(\mathcal{L}))$. 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 $\mathsf{L}(\mathcal{L}{)}^{\Gamma }$ of $\mathsf{L}(\mathcal{L})$. This module becomes a $\mathbb{V}(\mathcal{L})$-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

54 pages, latex

Cited by in corpus (6)

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• $({\mathfrak{gl}}_M,{\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities
• Affine opers and conformal affine Toda
• Populations of Solutions to Cyclotomic Bethe Equations