# What is the mathematical name for the anomaly for an action of a group on a lattice conformal field theory?

Suppose $V$ is a (bosonic) chiral conformal field theory which is "holomorphic" in the sense that its category of vertex modules is trivial. (The definition of "chiral conformal field theory" might be "vertex operator algebra", perhaps with "$C_2$-cofinite" added, but I am open-minded about the definition.) Suppose $G \subset \mathrm{Aut}(V)$ is a finite groups of automorphisms of $G$. Then there is (or there is expected to be, depending on the definition) a distinguished class in $\mathrm H^3(G,U(1))$ coming from the action, called (if I understand the physics correctly, but feel free to correct me) the 't Hooft anomaly. A sketch of the definition of this class can be found in the old MathOverflow question H^4 of the monster.

Suppose now that $V = V_L$ is the lattice CFT coming from some even unimodular lattice $L$ of rank $r$. Then $\mathrm{Aut}(V)$ contains a group of shape $2^r . \mathrm{Aut}(L)$. (Here $\mathrm{Aut}(L) \subset O(r)$ is the finite group of automorphisms of $L$. One must arbitrarily choose some signs in the construction of the lattice CFT, hence the extension.) Suppose $G \subset 2^r .\mathrm{Aut}(L) \subset \mathrm{Aut}(V)$.

Does the 't Hooft anomaly for the action of $G$ on $V_L$ have a name from finite group theory or homotopy theory or...? E.g. Is it a named characteristic class of some action? Does it have a straightforward description directly in terms of the lattice, without reference to conformal field theory?

I would be happy with a restricted version of this question:

Suppose furthermore that under the projection $2^r.\mathrm{Aut}(L) \to \mathrm{Aut}(L)$, we have $G \subset \mathrm{Aut}(L)$, i.e. $G \cap 2^r = \{e\}$. Does the 't Hooft anomaly have a name/description in terms of the action of $G$ on $L$? What about the action of $G$ on $L \otimes_{\mathbb Z} \mathbb R = \mathbb R^r$?

Given that there are different mathematical models of chiral conformal field theory, some better for computations (e.g. VOAs) and some better for abstract nonsense (e.g. CNs), I should also ask:

For which models is it mathematically known (at the level of theorem) that the name/description of the 't Hooft anomaly for $V_L$ is correct, and for which models its it expected (physically known) but not a theorem?

Addendum: Here is an example of the type of answer that one could hope for (although in this case I think the hope is too optimistic). Suppose $G \cap 2^r = \{e\}$ so that $G$ acts faithfully on the lattice $L$. By definition, then, we have a map $G \to \mathrm{Aut}(L) \subset {GL}(L) = {GL}(r,\mathbb Z)$. When $r \gg 0$, $\mathrm H^4(GL(r,\mathbb Z),\mathbb Z) = \mathbb Z/2 \oplus \mathbb Z/24$, if my memory is correct. Under the map ${GL}(r,\mathbb Z) \to GL(r,\mathbb R) \simeq {O}(r)$, the generator $p_1 \in \mathrm H^4({BO}(r)) = \mathbb Z$ pulls back to the generator of the $\mathbb Z/24$. It therefore pulls back to a class in $\mathrm H^4(G,\mathbb Z) = \mathrm H^3(G,U(1))$. One could hope that the 't Hooft anomaly might be this class, or some multiple of it, or perhaps I also need to use the $\mathbb Z/2$ (which comes from the reflections).