Conformal blocks attached to twisted groups

Abstract

The aim of this paper is to generalize the notion of conformal blocks to the situation in which the Lie algebra they are attached to is not defined over a field, but depends on covering data of curves. The result will be a sheaf of conformal blocks on the stack Hur(Γ, ξ) g,n parametrizing Γ-coverings of curves. Many features of the classical sheaves of conformal blocks are proved to hold in this more general setting, in particular the fusion rules, the propagation of vacua and the WZW connection.Parahoric Bruhat-Tits groups arising from coverings. As already mentioned, in our generalization we replace the group G with a parahoric Bruhat-Tits group H defined over a curve X.Since the group H depends on the geometry of the curve, our version of the sheaf of conformal blocks will be in general not defined over M g,n but on a moduli space which encodes also the information on H. Inspired by [BS15], we restrict ourselves to consider only those groups arising from coverings in the following sense. We fix the cyclic group Γ := Z/pZ of prime order p and a group homomorphism ρ : Γ → Aut(G). Let q : X → X be a (ramified) Galois covering of nodal curves with Galois group Γ and denote its moduli stack by Hur(Γ, ξ) g . We remark that in contrast to [BR11], we assume that the nodes of X are disjoint from the branch locus R of q. Then we say that a group H on X arises from q and ρ if it is isomorphic to the group of Γ-invariants of the Weil restriction of X × k G along q, i.e. H = q * ( X × k G) Γ .We observe that the groups H that we consider are parahoric Bruhat-Tits groups which in general are not generically split, while in [BS15] the authors only work in the split situation. This reflects the condition that in their paper they only allow Γ to act on G by inner automorphisms, i.e. ρ is a group homomorphism Γ → G. The following statement is a particular instance of Theorem A.0.7 which generalizes [BS15, Theorem 4.1.6].Theorem. Let q : X → X be a Γ covering of curves and ρ : Γ → Aut(G) be a homomorphism of groups. Set H = (q * (X × G)) Γ . Then the functor q * (−) Γ induces an equivalence between Bun H (X) and the stack Bun G (G,Γ) ( X) parametrizing G-bundles on X equipped with an action of Γ compatible with the one on G.The notion of compatibility stressed in the above Theorem will be clarified in Appendix A in terms of local type of (Γ, G)-bundles.Main results. In order to define the generalized sheaf of conformal blocks, we first of all need to introduce the pointed version of Hur(Γ, ξ) g and in second place replace P ℓ with an appropriate set of representations of H. We denote by Hur(Γ, ξ) g,1 the stack parametrizing Γ-coverings of nodal curves X → X, where X is marked by a point p which is disjoint from the branch locus R. In similar fashion we define Hur(Γ, ξ) g,n for n ≥ 1. Let H be the group on X univ arising from the universal covering ( X univ → X univ , p) on Hur(Γ, ξ) g,1 and the homomorphism ρ : Γ → Aut(G). Set h := Lie(H) and denote by IrRep ℓ (h| p ) the set of irreducible representations V of h| p of level at ...

Conformal blocks for Galois covers of algebraic curves

“…In the non-equivariant case, this smoothing construction as formal deformation is sketched by Looijenga in [Loo13, § 6], and detailed argument from formal deformation to algebraic deformation can be found in [Dam20, § 6.1]. These constructions/arguments can be easily generalized to the equivariant setting when acts on the node stably and does not exchange the branches.…”

Local types of (Γ,G)$(\Gamma ,G)$‐bundles and parahoric group schemes

“…Among others, they suggest the description of spaces of generalized theta functions on $$\mathrm{Bun}_\mathcal {G}$$ via an appropriate notion of conformal blocks, which would be the first step to obtain a Verlinde‐type formula to compute their dimension. Motivated by this paper and by the uniformization theorem for $$\mathrm{Bun}_\mathcal {G}$$ [16]—conjectured as well in [24]—many mathematicians have worked on twisted conformal blocks [12, 14, 18, 19, 38]. These are a natural generalization of the conformal blocks attached to a curve $$C$$ and to representations of simple Lie algebras (see, e.g., [35]), where the simple Lie algebra is replaced by a pair $$(\Gamma ,\mathfrak {g})$$ consisting of a simple Lie algebra $$\mathfrak {g}$$ and a finite group $$\Gamma$$ acting on $$\mathfrak {g}$$.…”

On equivariant bundles and their moduli spaces

“…In this setting, [BS,Theorem 5.2.7] states that all split parahoric Bruhat-Tits can be recovered from coverings, provided that the Galois group Γ acts on G via inner automorphisms only. However it is natural to include also automorphism which are not necessarily inner (see for instance [Dam,Example 2.3]). It follows that the category of groups H which can be constructed through coverings, and to which we can apply our main result, is much larger than the one studied in [BS].…”