Vertex Lie algebras and cyclotomic coinvariants

Abstract

Abstract. Given a vertex Lie algebra L equipped with an action by automorphisms of a cyclic group Γ, we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over 'local' Lie algebras L(L )z i assigned to marked points zi, by the action of a 'global' Lie algebraOn the other hand, the universal enveloping vertex algebra V(L ) of L is itself a vertex Lie algebra with an induced action of Γ. 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 L Γ {z i } (V(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 [VY14].At the origin, which is fixed by Γ, one must assign a module over the stable subalgebra. This module becomes a V(L )-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

Cyclotomic Gaudin models with irregular singularities

“…Many of the statements and proofs of [VY16b] concerning cyclotomic coinvariants which did not include the point at infinity can be seen to carry over with minor modifications to the present case.…”

“…There is a Z-grading on W 0 defined by deg | = 0 and deg a α [n] = deg a * α [n] = deg b i [n] = n. Let us recall some facts about the free-field realization of g. The H(g)⊕h⊗C((t))-module W 0 is endowed with the structure of a vertex algebra. In the notation of [VY16b], we have the "big" Lie algebra L(W 0 ) := Lie C((t)) W 0 of all formal modes of states in W 0 . For each of the marked points x i , i = 1, .…”

“…In this appendix we use the notion of a vertex Lie algebra and related concepts as defined in [VY16b]. In particular, all our vertex Lie algebras are finitely generated as C[D]-modules, of the form C[D] ⊗ (L o ⊕ Cc) C[D] ⊗ Cc, where L o is a finite dimensional vector space and D is the translation operator of the vertex Lie algebra.…”

“…where u is a formal variable and A W (n) are endomorphisms of M ∞ for each A ∈ V(L ) and n ∈ Z, by direct analogy with the definition given in [VY16b] for left smooth modules over L(L ) Γ as follows. For any a ∈ L we set The following is a direct analogue of [VY16b, Proposition 3.6] for the point at infinity, providing an alternative definition the above Y W -map on co-smooth modules using co-invariants.…”

“…The cyclotomic Gaudin algebra was constructed in [VY16a] using the technology of coinvariants/conformal blocks of g-modules of critical level, following [FFR94]. The relevant coinvariants in this case are Γ-equivariant; see [VY16b]. Now, in fact, this approach using coinvariants naturally gives commutative subalgebras not just of U (g ⊕N ) but of the larger algebra U…”