Let $(L,\langle -,-\rangle)$ be an even integral lattice, and let $(A,q)$ be the associated discriminant form: $$
A=L^\*/L, \quad q(a)=e^{\pi i \langle a,a\rangle}.
$$
We let $\hat L$ to be the extension of $L$ by $\{\pm1\}$ such that $\hat a\hat b=(-1)^{\langle{a,b}\rangle} \hat b\hat a$.


Then we have three natural "isometry groups", $O(L)$, $O(A)$ and $O(\hat L)$; the first two are subgroups of automorphism groups preserving $\langle,\rangle$ or $q$; the last is a natural extension of $O(L)$ by $\mathrm{Hom}(L,\{\pm1\})$.


Let us now consider the modular tensor category $\mathcal{C}(A,q)$ associated to $(A,q)$. This is also the category of modules of the lattice VOA $V\_L$ constructed from $L$.


Let $\mathrm{Aut}(\mathcal{C})$ be the group of braided auto-equivalences of a modular category $\mathcal{C}$ up to natural transformations. $\mathrm{Aut}(\mathcal{C}(A,q))$ is known to be equal to $O(A)$.


So we have a natural sequence of group homomorphisms $$
O(\hat L)\to O(L)\to O(A)=\mathrm{Aut}(\mathcal{C}(A,q)) .
$$


In [[ENO, arXiv:0909.3140]](https://arxiv.org/abs/0909.3140), extendsion of a modular tensor category $\mathcal{C}$ by any group $G$ with a homomorphism to $\mathrm{Aut}(\mathcal{C})$ was studied, and two obstructions are identified. The first obstruction takes values in $H^3(G,\mathrm{Inv}(\mathcal{C}))$ where $\mathrm{Inv}(\mathcal{C})$ is the group of invertible objects of $\mathcal{C}$.


So it seems natural to study the obstructions to extending $\mathcal{C}(A,q)$ by any subgroup of $O(A)$, $O(L)$ or $O(\hat L)$. The first obstruction takes values in $H^3(G,A)$. (I believe the first obstruction vanishes for (subgroups of) $O(\hat L)$.)


Does anybody know of any references concerning this point? I would be happy with any partial results, or any starting point for me to explore the literature. 
(Also, does the extension theory of ENO appear somewhere in the theory of lattice VOA?)



Modular tensor category associated to an even integral lattice and the lattice automorphism
**Edit:** I've thought about this question again, and I think the answer is more positive than what I said in an earlier version.


I will assume $L$ is positive-definite, since we need that to make $V\_L$ into an honest VOA. I think what you say is still true using vertex algebras of indefinite lattices, but one has to be very careful with definitions to make fusion work well, and no one has done so in that setting yet.


Here is what we expect to be true: For subgroups of $\operatorname{Aut}(\mathcal{C}(A,q))$ in the image of the map from $O(\hat{L})$, we expect an extension to exist and to be given by the category of twisted $V\_L$-modules (where the twisting ranges over automorphisms of $V\_L$). The two obstructions $O\_3$ and $O\_4$ in the ENO paper concern associativity of tensor product, and are expected to vanish in all cases of interest to us. I have no idea what happens for elements not in the image of $O(\hat{L})$.


Here is what we know: If the twisting ranges over a solvable group $G$, then the existence of an extension essentially follows from the regularity (i.e., complete reducibility) properties of the fixed-point subVOA $(V\_L)^G$ established in [C-Miyamoto](https://arxiv.org/abs/1603.05645), together with [Huang's work on modular tensor structure](https://arxiv.org/abs/math/0502533). There is some technical work involving compatibility of twisted intertwining operators, but I don't think anything important is missing. In other words, if the determinant of $L$ is small, you get an extension unconditionally. If the determinant is large, then the existence of an extension given by twisted $V\_L$-modules is conditional on the conjecture that taking fixed points under any finite group action preserves complete reducibility.


The technical work with intertwining operators comes from the fact that a full definition was only given in [Huang's preprint from last year](https://arxiv.org/abs/1702.05845). On the other hand, you can simplify the definition significantly under the assumption that your twisted modules are just finite direct sums of irreducible $(V\_L)^G$-modules, and various conjectured properties do not need new proofs.


