Existence of orbifold vertex algebras – current status?

Question

Let the finite group $G$ act on a vertex algebra $V$. It is expected that there are certain vector spaces $V_g$ (with the structure of $g$ twisted $V$ modules), with $V_1=V^G$, and $$V/G\ :=\ \bigoplus_{g\in G}V_g$$ has a vertex algebra structure.

I think the idea is that, if $V$ is a chiral quantisation of the jet space $J_\infty X$ of a scheme $X$, then $V/G$ is the chiral quantisation of the quotient stack $X/G$, $J_\infty(X/G)$.

Question: What is the current status of the construction of $V/G$? When is it expected to exist?

E.g. in Frenkel-Ben Zvi 5.7.1, it is claimed that there is a construction when $V$ is holomorphic and $G$ is cyclic. But I am wondering if there have been advances since then.

Answer 1

The question of constructing $G$-orbifold vertex algebras amounts to the problem of producing a suitable multiplication operation on a sum of modules for the fixed-point vertex algebra $V^G$. The parts of the orbifold vertex algebra are not twisted modules for $V$, but instead are pieces of twisted modules (like fixed-point submodules for some group action).

When $V$ is holomorphic, we have a classification for all finite groups $G$, given in Evans-Gannon, under the assumption that $V^G$ is regular (which is unconditional when $G$ is solvable, by C-Miyamoto). In the cyclic case, this was a conjecture when Frenkel-Ben-Zvi was written, but it was solved by van Ekeren-Möller-Scheithauer.

When $V$ is not holomorphic, it is not clear what sort of object you expect to get. You are basically trying to make a commutative ring in a braided category of $V^G$-modules, but I don't know what constraints are natural.