APS $\eta$-invariant and spin-Ising TQFT

Question

I am interested in the relation between the Atiyah-Patodi-Singer-$\eta$ invariant and spin topological quantum field theory. In the paper Gapped Boundary Phases of Topological Insulators via Weak Coupling by Seiberg and Witten, they presented such relation between the two.

Let the $3+1$ dimensional manifold $X$ be the world volume of a topological insulator. Its bounday $W$ is a $2+1$ dimensional manifold. Let $\chi$ be a massless Dirac fermion on the bounday manifold $W$. Then, integrating out the boundary fermion

$$\int_{W}d^{3}x\,i\bar{\chi}D \!\!\!\!/\,\chi\,, $$

where $D_{\mu}=\partial_{\mu}+iA_{\mu}$, one has the partition function

$$\mathcal{Z}=|\det(iD)|e^{-i\pi\eta(A)/2}.$$

So far, this is just the standard parity anomaly in odd dimensions.

On page 35, the authors claimed that the factor $\Phi=e^{-i\pi\eta/2}$ is actually the partition function of a topological quantum field theory, called spin-Ising TQFT. The name comes from the fact that it is related with the 2D Ising model of CFT. The authors explained that this is due to the Free-Dai theorem.

I don't really understand much from the paper of Freed-Dai theorem because of its heavy mathematics. But from my understanding, it is saying that the $\eta$ invariant is some kind of topological invariant and cobordism invariant, which satisfies the gluing and surgery axioms of TQFT. Thus, the factor $\Phi=e^{-i\pi\eta/2}$ can be treated as the partition function of some TQFT.

Now the question is why this TQFT is the so-called spin-Ising TQFT. The authors claim that the partition function of the spin-Ising TQFT should be of modulus $1$ because the $\mathbb{Z}_{2}$ chiral ring generated by the field $\psi$ (from the 2D Ising model $\left\{1,\sigma,\psi\right\}$) has only one representation.

Question 1: Why does the fact that the chiral algebra has one representation makes its partition function being of modulus $1$?

The authors then showed an example, taking the manifold $W$ to be $S^{2}\times S^{1}$, that the partition function of the corresponding spin-Ising TQFT is $\pm 1$, which is indeed a phase. Then, by Freed-Dai theorem, they claimed that in general the partition function of a spin-Ising TQFT should be $\Phi=e^{-i\pi\eta/2}$.

I don't really understand much from the paper of Freed-Dai theorem. Could anyone please enlighten me on how one should apply that theorem to this case?

The authors explained in the following that if $W$ has a boundary $\Sigma$, then the product of the path integral of the chiral fermion $\psi$ on $\Sigma$ and the factor $\Phi$ is smooth and well-defined because of the Freed-Dai theorem.

However, in our case, the manifold $W$ itself is the boundary of the $3+1$-manifold $X$, and so $W$ has no bounday at all. How should one understand the explanation provided by the authors?

Answer 1

1. For 3-manifolds which are circle fibrations of 2-manifolds, this follows from the fact that any such partition function may be computed as trace of an operator on the Hilbert space of the 2-manifold. More generally however, we need to understand how to cut up our 3-manifolds and compute the partition function of the pieces. Using a Heegaard splitting for instance, we can express the partition function on any 3-manifold as an inner product between two states in the Hilbert space of a surface. The states are associated to the path integral on the two handle-bodies on either side of the splitting, and one can show that they are normalized by considering the path integral on the double $M \cup \bar M$ of either handle-body, which equals 1 by reflection positivity. Since the Hilbert space of states is one-dimensional on the separating surface, it follows that the partition function is a phase. You can think of the Dai-Freed theorem as giving an explicit representation of the states associated to these handle-bodies. This is a version of the usual TQFT-RCFT correspondence where conformal blocks of the RCFT give states of the TQFT.

2. There are equivalent formulations of this $\eta$ invariant which are useful for different purposes. It is true you can use the APS index theorem to express $e^{i\eta/2}$ of a closed 3-manifold as an integral over a bounding 4-manifold. This is useful to show its quantization properties. On the other hand, $e^{i\eta/2}$ has an intrinsically 3-dimensional definition which is the spectral asymmetry of the associated Dirac operator. Dai-Freed tell you what happens when you try to consider this definition on a 3-manifold with boundary. You cannot use the 4-dimensional definition in this case (which is manifestly gauge-invariant), and so there is room for some gauge non-invariance.

Answer 2

I will recapitulate the argument of the authors. I can not, however prove their statements. I hope this helps anyhow.

The authors write

Let's recapitulate the argument. Take a three-manifold $W$ with boundary $\Sigma$. Then, they say, the Dai-Freed theorem declares that the following is well defined:

$$ \exp(-i \pi \eta(W) /2) Z_\psi(\Sigma) \ ,$$

where $Z_\psi(\Sigma)$ is the partition function of the Ising spin-CFT on the 2-manifold $\Sigma$. This means that, for the special case that $\Sigma = \{ \}$, the empty manifold, $\exp(-i \pi \eta(W) /2)$ is well-defined and is associated to the Ising spin-CFT. This is a statement purely about the $\eta$-invariant of a closed three-manifold $W$ and thus holds regardless whether $W$ is a boundary or not.

For the questions about the representations: It doesn't seem to me that this is what they are claiming. They start with some the $\eta$-invariant, and then they show how this is related to this spin-CFT.