5
$\begingroup$

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.

$\endgroup$

1 Answer 1

3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.