Chiral homology for the Virasoro algebra and/or affine Lie algebra

Question

I want to understand what concrete analytical objects are found in chiral homology of higher degree of a vertex algera (-module) $M$. More precisely: I can obtain conformal blocks on a surface $\Sigma$ for the Virasoro algebra as invariants of $M$ under the action of vector fields - for the torus one can then indeed work out a modular function depending on the structure of the torus and possibly the location of singularities. One can also compute elements in it by gluing the vertex operator on a 3-punctured sphere to a torus (a.k.a graded character of $M$)

But what about higher homologies: In the Lie algebra standard complex the chains are functions depending on an $n$-tuple of (here) vector fields up to...how does this relate for the torus to modular functions or something similar?

Same question for affine Lie algebra at negative level - I have found the great work of Gaitsgory, but I would like to know if there is any concrete analytic realization of the elements by...?

[But maybe it is simply the wrong question !?]

Also, I would already be very happy for reference like to as for references you might have on the chiral cohomology of the Virasoro algebra on surfaces (I know the computation for $M$ trivial, but for say $M$ a irrep of a minimal model?).

Thanks alot in advance, Simon Lentner

Answer 1

There isn't much written up on how to compute the higher chiral homology of vertex algebras. Besides the original work of Beilinson and Drinfeld that covers in particular the universal cases of Virasoro and Affine algebras you won't find much more. There's an unwritten theorem by Gaitsgory that proves the vanishing for simple Affine algebras at integral level.

In the case of elliptic curves and in the limit of the nodal elliptic curve, together with Van Ekeren we related the first chiral homology group to the first Hochschild homology of the Zhu algebra (just as comformal blocks are related to the zeroth Hochschild homology) in https://arxiv.org/abs/1804.00017

Unfortunately that relies on a technical condition of vertex algebras being "classically free", that is that their classical limit is isomorphic to the arc algebra of the C2 quotient. This was shown to be the case for minimal models $Vir_{p,p'}$ if and only if $p=2$. And some simple affine algebras at integral level.

We can now relax this condition to vertex algebras to the case then the classical limit is a quotient of the arc algebra by a finitely generated differential ideal. This is still very hard to prove. The only known example of this to my knowledge is the Ising model $Vir_{3,4}$ in

https://arxiv.org/abs/2005.10769

Using this we can now prove the vanishing of the first chiral homology group of an arbitrary elliptic curve (not necessarily the nodal limit) with coefficients in either the 2,p' minimal models or the Ising model. As well as recover some of Gaitsgory's results. That article should come out soon, it's being finalized for a while now.