Today, Yuji Tachikawa gave a very nice talk at Strings in Tel Aviv. The slides are available on the website of the conference, and the talk is on Youtube along with all the other talks.

Usually, when we first learn about quantum anomalies, it is about a continuous symmetry in even number of simensions, with massless excitations which are fermions. Here Tachikawa looks at a very different kind of anomaly: discrete symmetry in odd number of dimensions without any massless excitations, and with anyons! The example he discusses is $3d$ Chern-Simons $U(n)_{2n}$, which has a secret parity symmetry but with an anomaly $\nu = \pm 2 \in \mathbb{Z}_{16}$.

The first surprising thing is that $U(n)_{2n}$ is parity symmetric! To show it, consider $4d$ $\mathcal{N}=1$ pure YM $SU(X)$ theory. It has $X$ vacua labeled by $\langle \lambda \lambda \rangle \sim e^{2 \pi i k/X}$, that can be separated by domain walls. Then you can ask: what is the $3d$ theory that lives on the domain wall between two vacua separated by $n$ steps? The answer is morally the $3d$ $\mathcal{N}=1$ $U(n)_X$ theory (as shown by Acharya and Vafa in 2001)! Therefore, it is natural to conclude that spacetime parity exchanges $$U(n)_X \leftrightarrow U(X-n)_X$$ and therefore $U(n)_{2n}$ is parity symmetric.

Now that the theory is defined, we can place it on a non-orientable space-time, for instance a Möbius strip times a circle. The boundary of this space is a torus $T^2$, so we have created a state in the Hilbert space $\mathcal{H}_{T^2}$, that we call $$| \mathrm{crosscap} \rangle \, .$$ Note that another way to visualize the three-dimensional space-time is as a solid torus with a crosscap running inside (indeed, the Möbius strip can be seen as a disk with a crosscap inside, and we take the product with a circle). We recall that this construction is general (see for instance the notes by Marcos Marino): if $M$ is a three-manifold with boundary $\Sigma$, a Hilbert space $\mathcal{H}(\Sigma)$ is obtained by canonical quantization of the theory on $\Sigma \times \mathbb{R}$. This space has been described precisely in a groundbreaking article by Witten (Quantum field theory and the jones polynomial): it is the finite-dimensional space of conformal blocks of a WZW model on $\Sigma$ with gauge group and level corresponding to the Chern-Simons theory! But let's go back to our current theory. The state $| \mathrm{crosscap} \rangle$ should in principle be invariant under transformations of the geometry that leave it unchanged, for instance the $S$ and $T$ transformations of the solid torus. In the case of $U(3)_6$, a precise calculation gives $$S^{-1} T S| \mathrm{crosscap} \rangle = e^{2 \pi i \times 2/16} | \mathrm{crosscap} \rangle \, .$$ This is a manifestation of the parity anomaly.

This may be important when one wants to use anomaly matching to constrain the dynamics. In that case, if the anomaly under consideration can be realized by a topological QFT, you can add it to match the missing anomaly. This means that it is very important to know when an anomaly can be realized by a TQFT.