# Existence of orbifold vertex algebras – current status?

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.

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.