What's the right way to think about "anomalies" in 3d TQFTs?

Question

3d TQFTs constructed from modular tensor categories don't in general give an honest 3d TQFT, instead they have an "anomaly." My vague understanding from Kevin Walker's talks and from skimming Freed-Hopkins-Lurie-Teleman is that what's really going on is that its a 4d TQFT that's almost boring on the 4d part and that's what the "anomaly" means. Does anyone know how to make this more precise?

Answer 1

You can find some of the details in a draft version of my TQFT notes on my web page. Here's a summary.

Given an n-category with strong duality (by which I mean, more or less, pivotal if $n=2$ and a higher dimensional version of pivotal for $n > 2$; this is stronger than what Lurie means by an n-category "with duals"), there's a standard procedure to construct an n+1-dimensional TQFT. This procedure works for free for the 0-through n-dimensional parts of the (extended) n+1-dimensional TQFT. In these dimensions, we need no additional assumptions on the n-cat, and there is no need to choose a decomposition of the manifolds, so there are no combinatorial topology moves to check. The construction is manifestly invariant.

To construct the top, n+1-dimensional part of the TQFT, we need to make some finiteness assumptions on the n-category. (This corresponds to the top-dimensional part of what Lurie et al mean by "fully dualizable".) If the n-category satisfies these assumption, then we get, for each handle decomposition of the n+1-manifold, a state sum type expression for the path integral of the n+1-manifold. It is not hard to show that this state sum is invariant under handle slides and handle cancelation, so we get a well-defined invariant of n+1-manifolds that interacts with the rest of the TQFT (via gluing formulas) in the correct way.

(Small technicality: the path integral construction depends on a choice of element in the Hilbert space of the n-sphere, corresponding to the path integral of the n+1-ball. Multiplying this choice by $\lambda$ changes the path integral by $\lambda^\chi$, where $\chi$ is the Euler characteristic of the n+1-manifold.)

A modular tensor category is a 3-category with strong duality and the right sort of finiteness properties, so we can apply the above construction to get a 3+1-dimensional TQFT.

In dimension 3 the vector space we construct is an old-fashioned skein module: finite linear combinations of ribbon graphs in M^3 modulo local relations. (Actually, the dual of this vector space.) If M is closed this in 1-dimensional. More generally, if M has boundary then it has the same dimension as the Witten-Reshetikhin-Turaev vector space associated to the boundary of M.

In dimension 4, the type of state sum we get depends on the type of handle decomposition. For a general handle decomposition we get the Crane-Yetter state sum. For 2-handles attached to a 4-ball along a framed link L we get the Reshetikhin-Turaev surgery formula for L. For a 4-dimensional neighborhood of a 2-complex we get the Turaev "shadow" state sum. For a closed 4-manifold we find that the path integral is equal $a^\chi b^\sigma$, where $\chi$ is the Euler characteristic and $\sigma$ is the signature of the 4-manifold. By choosing $\lambda$ above appropriately we can make a=1. b is related to the central charge of the MTC (or to the value of the RT surgery formula on framing +-1 unknot). For a 4-manifold with boundary we find that the state sum computes the Witten-Reshetikhin-Turaev invariant of the boundary of the 4-manifold.

In dimension 2, the 1-category associated to a closed surface is a full matrix category; i.e. it is Morita trivial. For a surface with boundary k circles the category is Morita equivalent to k copies of the MTC thought of as a 1-category.

In all of the above cases, we find that the TQFT invariant of $Z(X)$, where dim(X) = 2 or 3 or 4, depends strongly on the boundary of $X$ but only weakly (i.e. only up to bordism) in the interior of $X$. So we can define a new 2+1 dimensional TQFT $Z'$ via the formula

$$Z'(Y) := Z(\mathrm{boundary}^{-1}(Y))$$

This TQFT has an anomaly, since we need to enhance Y with enough extra structure to pick out an inverse-boundary, up to bordism.

Answer 2

There are several different kinds of TQFTs that can be defined. For example there are oriented theories, unoriented theories, framed theories, and many more. The 3D TQFTs coming from Modular Tensor Categories don't define oriented TQFTs, hence they have an "anomaly". One way to make this precise is to realize that they do give a TQFT, just not one based on oriented manifolds, but manifolds equipped with an additional structure.

This additional structure goes under various names, e.g. "p1-structure", "rigging" (there is a weaker version known as a "2-framing"). There are also several ways to define this structure. Some of these approaches do not yield equivalent structures, and it depends on your MTC as to which one will work (although there is a universal choice).

One common version of this structure on a 3-manifold is an equivalence class of choices 4-manifolds which bound it. The equivalence relation is that these 4-manifolds are taken up to bordism (hence you really only have the signature). This gives you a central extension of the bordism category, and (some of) the MTC TQFTs are defined on this central extension. But it also means that there is a way to interpret such TQFTs as assigning data to 4-manifolds, but just in a (mostly) trivial way.

Now the 4-manifold structure is fine for Chern-Simmons theory, but my understanding is that it is not the universal structure (which has to do with the determinant bundle on the moduli spaces of surface, see Segal's "Definition of Conformal Field Theory" manuscript).

Answer 3

My own understanding of anomalies in TQFTs:

In cases that I have seen, "anomaly" in general refers to central extensions and line bundles. Physicists have long thought of these issues in a very explicit way, in terms of integrals to compute Feynman diagrams. In many quantum field theories, the Feynman diagram integrals do not converge even though the quantum field theories are healthy just from counting degrees of terms in the Lagrangian (renormalizable). A modern explanation of this (that I do not know first-hand, but have been told) is that non-convergence comes from trying to compute something that is not a number, but rather a value in a non-trivial line bundle. "Anomaly cancellation" means that you find two things to compute that both have anomalies, but whose product does not because the line bundle is trivial.

A 3D TQFT is, among other things, a functor from the category of cobordisms between surfaces to the category of vector spaces. This category has central extensions, in the same sense that groups have central extensions. Its universal central extension over ℚ is given by tensoring the torsor of framings with ℚ and it is one-dimensional. (Here "torsor" just means an affine space over a group. Framings up to homotopy are a torsor over the abelian group of the homotopy space $[M,SO(3)]$.) So, if you want a central extension of cobordisms before making your functor, you want framings of 3-manifolds. The only point of 2-framings is to cancel the 2-torsion part of the torsor of framings by hand; it has always seemed better to me to tensor with ℚ instead.

This central extension appears naturally in the Chern-Simons QFT, a.k.a., Reshetikhin-Turaev invariants of 3-manifolds. And the Turaev-Viro invariant is a famous example of anomaly cancellation. If the input category to that invariant is modular (it need only be a spherical fusion category in general), then it is two factors of Reshetikhin-Turaev with opposite phase.

Answer 4

Here is my understanding from physics point of view.

A quantum field theory is anomalous if it lacks of a UV completion. In other words there is no lattice theory in the same dimension, whose continuum limit produces the quantum field theory.

However, even an anomalous quantum field theory can be produced at the boundary of a topological ordered state on lattice in one higher dimension. In other words, for a given quantum field theory, there is always a gapped lattice theory in one higher dimension, whose continuum limit at the boundary produces the quantum field theory. Thus

the types of anomalies in quantum field theories = types of topological orders in one higher dimension

This leads to a classification of gravitional anomalies in terms of topological orders in one higher dimension.

Also

the types of anomalies in topological quantum field theories （ie gapped quantum field thoeries）= types of topological orders with gappable boundary in one higher dimension （ie topological quantum field theories in one higher dimension that have state sum realization ）

I have two papers discussing those points in details:

arXiv:1303.1803 Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders (for gauge anomaly and group cohomology) Xiao-Gang Wen

arXiv:1405.5858 Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions (for gravitational anomaly and n-category, as well as the notion of H-type and L-type anomalies) Liang Kong, Xiao-Gang Wen

Kapustin also has a paper

arXiv:1404.3230 Anomalies of discrete symmetries in various dimensions and group cohomology (for gauge anomalies) Anton Kapustin, Ryan Thorngren