$\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$Let $G$ be a compact Lie group and $\lambda\in
H^4(BG;\Z)$. The data $(G, \lambda)$ determine a 3d topological field theory called Chern-Simons theory, except not
quite: there is an obstruction to defining it on general closed, oriented $3$-manifolds called the *anomaly*.
[Freed-Teleman](https://arxiv.org/abs/1212.1692) characterize anomalies of $n$-dimensional field theories as
$(n+1)$-dimensional invertible field theories, which have been classified. I think in this case the anomaly
field theory should be unitary, so is classified up to isomorphism by a torsion element of
$\mathrm{Hom}(\Omega\_4^{\mathrm{SO}}, \mathbb C^\times)$, by a theorem of
[Freed-Hopkins](https://arxiv.org/abs/1604.06527). If I choose $G$ and $\lambda$, is the isomorphism type of the
anomaly field theory, as a bordism invariant, known?


I'm actually interested in a slightly more general story, where $\lambda\in\SH^4(BG)$. (Here $\SH$ is a generalized
cohomology theory called *supercohomology*: $\pi\_0\SH = \Z$, $\pi\_1\SH = \Z/2$, and the $k$-invariant is nonzero.
When $G$ is simple and simply connected, using $\SH$ instead of $H$ amounts to choosing a half-integer rather than
an integer.) Then there is a 3d spin TFT called spin Chern-Simons theory, which is again anomalous. Now the anomaly
is a torsion element of $\mathrm{Hom}(\Omega\_4^{\mathrm{Spin}}, \mathbb C^\times)$.


**For spin Chern-Simons theories, is the isomorphism type of the anomaly known? If not, is there an explicit
conjectured description?**


I'm primarily interested in the spin case when $G$ is a torus, but any information (e.g. $G$ simple and simply
connected, only for the oriented case, etc.) is helpful.



Formula for the anomalies of spin Chern-Simons theories?
This is not a direct answer to your question, but I think it's relevant.


One way of thinking about the anomaly for ordinary (oriented) Chern-Simons theories is that it's the evaluation of the associated 3+1-dimension Crane-Yetter theory (what Freed would call the anomaly theory) on a generator of 4d oriented bordism, i.e. $CP^2$. Choosing the standard handle decomposition of $CP^2$, one calculates that
$$
 Z\_{3+1}(CP^2) = D^{-1/2} \sum\_a \theta\_a d\_a^2 =: C
$$
(The sum is over simple objects $a$ of the modular category associated to the Chern-Simons theory, $D$ is the global dimension of that category, $\theta\_a$ is the ribbon twist of $a$, and $d\_a$ is the quantum dimension (loop value).) The bordism invariant in this case is $C^{\sigma(W)}$, where $W$ is a closed 4-manifold and $\sigma$ is its signature.


Imitating this approach for 2+1-dimensional Spin Chern-Simons TQFTs we can again evaluate the associated 3+1-dimensional Spin Crane-Yetter theory on a generator of 4-dimensional Spin bordism, such as the K3 surface. So
$$
 C := Z\_{3+1}(K3) = \;\; ???
$$
The reason I wrote ??? above rather than an explicit formula is that the simplest handle decomposition of the K3 surface (which is in some sense the simplest generator of 4d Spin bordism) has 22 2-handles. Translating that complicated handle structure into a formula involving quantum dimensions, ribbon twists, $6j$-symbols, etc. would not be very enlightening.


One way around this difficulty would be to enlarge the class of manifolds (on which the TQFT is defined) from Spin manifolds to "characteristic pairs" -- pairs $W^k \supset V^{k-2}$ such that $V$ is a Poincare dual to the second Stiefel-Whitney class of $W$ and $W\setminus V$ is equipped with a spin structure which does not extend over $V$. (See [this paper](https://www.maths.ed.ac.uk/%7Ev1ranick/papers/kirbytaylorpin.pdf) by Kirby and Taylor for details.) The generators of the 4-dimensional characteristic pair bordism group are much simpler: $(CP^2,CP^1)$ and $(S^4,RP^2)$. The central charge associated to $(S^4,RP^2)$ can be normalized away. The central charge associated to $(CP^2,CP^1)$ looks very similar to the oriented case:
$$
 C:= Z\_{3+1}(CP^2, CP^1) = D^{-1/2} \sum\_v \frac{\theta\_v d\_v^2}{|\mbox{End}(v)|}
$$
Here the sum is over only "vortex" simple objects (associated to the pair $(D^2,pt)$). $|\mbox{End}(v)|$ denotes the dimension of the endomorphism algebra of $v$ -- 2 for Majorana simple objects and 1 for ordinary simple objects.


Can we always enlarge the domain of definition of a Spin TQFT associated to a fermionic modular category to characteristic/vortex pairs? I think that a [recent paper](https://arxiv.org/abs/2105.15167) of Johnson-Freyd and Reutter implies the answer is "yes", and Corey Jones, David Aasen and I think we have a more constructive proof (which we currently are in the process of writing up).


More details on the above approach can be found in [these](https://mathinstitutes.org/videos/videos/16665) two [talks](http://scgp.stonybrook.edu/video_portal/video.php?id=3238), and [slides](https://canyon23.net/math/talks/SCGP%20talk%20201706%20compressed.pdf) are here.


