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Let $G$ be a (finite, say) group and $\alpha \in \mathrm{H}^3(\mathrm{B}G; \mathrm{U}(1))$ a 3-cohomology class. For each oriented 3-manifold $X^3$ equipped with a $G$-bundle $P : X \to \mathrm{B}G$, one can pull back $\alpha$ and integrate: $\int_X P^*\alpha \in \mathrm{U}(1)$.

I would like to restrict attention just to genus-one mapping tori. By this I mean: given $\gamma \in \mathrm{SL}(2,\mathbb Z)$, build the manifold $T(\gamma) = T^2 \times [0,1] /\sim$ where for $(x,y) \in T^2$, I set $((x,y),0) \sim (\gamma(x,y),1)$.

Question: Is the class $\alpha$ necessarily determined by the values of $\int_{T(\gamma)} P^*\alpha$ for all $\gamma \in \mathrm{SL}(2,\mathbb Z)$ and $P : T(\gamma) \to \mathrm{B}G$?

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    $\begingroup$ @johnmangual The context of my question is the following. Suppose you have a 2d (i.e. (1+1)d) QFT with a global $G$ symmetry. You can try to "gauge" that symmetry. If the QFT were trivial, then you'd be trying to write down a pure 2d $G$-gauge theory, i.e. a 2d Dijkgraaf--Witten theory. These are, of course, classified by classes in $H^2(BG; U(1))$. This class is the data of the "Lagrangian" for the gauge fields. If your QFT is nontrivial, then you already have some fields and Lagrangian for your QFT, and gauging the $G$ symmetry requires choosing an additional term in the Lagrangian for ... $\endgroup$ Oct 26, 2017 at 18:06
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    $\begingroup$ ... the gauge fields. This additional term is something like a 2-cocycle on $BG$ modulo coboundaries. Actually, it isn't quite. It is a 2-cochain $\beta$ on $BG$ modulo coboundaries with prescribed derivative $d\beta = \alpha$. The 3-cocycle $\alpha$ is determined explicitly from the data of how $G$ acts on your QFT. So if $\alpha$ vanishes in cohomology, then you can find a $\beta$ and gauge the symmetry. If $\alpha$ is cohomologically nontrivial, then you cannot gauge the symmetry. The cohomology class $\alpha \in H^3(BG;U(1))$ is called the anomaly or gauge anomaly of the symmetry. $\endgroup$ Oct 26, 2017 at 18:09
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    $\begingroup$ In examples, one only has partial information about a QFT. For example, one might know the behavior of the QFT on closed tori ("genus-one data") without knowing a fully-local description of the QFT. In particular, in the examples I care about, I know how to compute the numbers $\int_{T(\gamma)}P^*\alpha$, but I don't have a direct description of the anomaly $\alpha$. (I would have such a description if I had a fully local, $G$-symmetric description of the QFT.) Hence my question: is the anomaly determined by genus-one data? $\endgroup$ Oct 26, 2017 at 18:12
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    $\begingroup$ BTW, Witten has a nice ten-page note at arxiv.org/abs/1710.01791 describing some of this physics. The point is that a global action of $G$ on a 2d QFT is "the same" as a way of placing your QFT at the "boundary" of a 3d pure gauge theory with gauge group $G$. In this description, $\alpha$ is nothing but the action for that 3d pure gauge theory (= 3d Dijkgraaf--Witten theory). See also section 2 of my paper arxiv.org/abs/1707.08388 and I'm sure many other places. $\endgroup$ Oct 26, 2017 at 18:17
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    $\begingroup$ Hi @johnmangual, I don't quite understand your comment. The mathematical question I want to ask is whether the restrictions to genus-one, in the sense I gave, determines classes in $H^3(BG, U(1))$. My comments tried to outline why that question was summarized by my title. The relation between (anomalies for) symmetries of 2d QFTs are (actions for) 3d gauge theories is pretty well known, although I'm happy to try to explain it better. (E.g. my comments pretended that the 2d theory necessarily had a fields+Lagrangian description compatible with $G$, which isn't true in general.) $\endgroup$ Oct 27, 2017 at 13:38

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