Vertex Lie algebras and cyclotomic coinvariants

Given a vertex Lie algebra $\mathcal{L}$ equipped with an action by automorphisms of a cyclic group $\mathrm{\Gamma }$, 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 $\mathrm{\Gamma }$. 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 $\mathrm{\Gamma }$, one must assign a module over the stable subalgebra $\mathsf{L}(\mathcal{L}{)}^{\mathrm{\Gamma }}$ 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.