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$\newcommand{\SH}{\mathit{SH}}\newcommand{\Z}{\mathbb Z}$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 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. 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.

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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 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 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 two talks, and slides are here.

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  • $\begingroup$ This is a great answer; thank you! Is it known whether the 4d spin TFTs associated to fermionic modular tensor categories are invertible, like in the bosonic case? Because if so, I would think that it's possible to check whether these TFTs extend to characteristic-pair TFTs in terms of a question about bordism groups $\endgroup$ Commented Jul 6, 2021 at 17:13
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    $\begingroup$ @ArunDebray Yes, they are invertible. Interesting idea re turning the extension question into a (presumably simpler) question about bordism groups. Feel free to contact me if you want to discuss further. $\endgroup$ Commented Jul 7, 2021 at 14:30
  • $\begingroup$ Thank you! Just sent you an email. $\endgroup$ Commented Jul 8, 2021 at 22:52
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The anomaly theory (at least for the non-super level) is described in this paper:

Freed, Daniel S.; Hopkins, Michael J.; Lurie, Jacob; Teleman, Constantin, Topological quantum field theories from compact Lie groups, Kotiuga, P. Robert (ed.), A celebration of the mathematical legacy of Raoul Bott. Based on the conference, CRM, Montreal, Canada, June 9–13, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4777-0/pbk). CRM Proceedings and Lecture Notes 50, 367-403 (2010). ZBL1232.57022.

According to this, it can be modelled as an extended TQFT with values in the 4-category of braided tensor categories, and it corresponds to a fully dualizable object which is a certain braided tensor category $Sky^\tau[T]$ of skyscraper sheaves. Its value on a closed oriented 4-manifold $X$ is the number $$ \mu^{sgn(X)\cdot sgn(\tau)} $$ where $sgn(X)$ is the signature of $X$, $sgn(\tau)$ is a certain sign related to the bilinear form $\mathfrak{t} \times \mathfrak{t} \to \mathbb{Z}$ defined by $\tau$, and $\mu$ is a primitive 8th root of unity. In particular, the anomaly theory vanishes on spin manifolds, whose signature is a multiple of 8.

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    $\begingroup$ This is only for the case where $G$ is a torus, right? Also, if I'm understanding correctly, your final remark is about applying ordinary (non-spin) TQFTs to spin manifolds, but the question was about spin TQFTs (which are only defined on spin manifolds). $\endgroup$ Commented Jun 28, 2021 at 12:22
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    $\begingroup$ Right, as it is written in the paper this holds only for a torus. But for a compact simple Lie group the restriction $H^4(BG,\mathbb{Z}) \to H^4(BT,\mathbb{Z})$ is injective, which makes it likely that there is not much difference in this case. But I haven't thought more carefully about this. $\endgroup$ Commented Jun 29, 2021 at 6:59
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    $\begingroup$ Yes, my final remark was about the non-spin TQFT. I haven't seen a description of the anomaly for the spin TQFT. $\endgroup$ Commented Jun 29, 2021 at 7:04
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    $\begingroup$ But in the general case we do not expect that $\mu$ is a primitive 8th root of unity, right? $\endgroup$ Commented Jun 29, 2021 at 15:05

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