Topological quantum field theories (TQFTs) are studied for different reasons, as exemplified in the following places:
- Atiyah, Topological quantum field theory
- Lurie, Topological Quantum Field Theory and the Cobordism Hypothesis
- Math Overflow, Usefulness of using TQFTs
- Sonnenschein, Topological quantum field theories, moduli spaces, and flat gauge connections
- Baez, Topological Quantum Field Theory
One notable application is that a closed manifold is (can be thought of as) a cobordism from $\emptyset$ to $\emptyset$, hence a TQFT assigns to it an element of $\operatorname{Hom}(\mathbb{C},\mathbb{C}) = \mathbb{C}^\times$, that is, a nonzero complex number. This is a diffeomorphism invariant of the manifold, which is a highly useful thing to have, especially if it can be computed in practice.
The introductory accounts that I have looked at usually treat the case of orientable $2$-manifolds. These can be decomposed into disks and pairs-of-pants ("trinions"). The datum of a TQFT in this case turns out to be a finite-dimensional Frobenius algebra, and the invariants in question are expressible in terms of its structure constants.
As pleasing as this picture is, the manifolds that I encounter in the wild are not easily decomposed into standard pieces. This raises questions about the practical computability of TQFT invariants. I gather that extended topological quantum field theories allow for more general kinds of decompositions (than gluing along closed submanifolds of codimension $1$), but it remains unclear (to me) when I would find myself in a position to use these tools.
I realize, of course, that there are many interesting questions to ask about TQFTs that are not directly related to distinguishing particular manifolds. One can ask which invariants are obtained in this way, or which algebraic structures correspond to Frobenius algebras when the dimension is higher than two—to say nothing about questions motivated by physics!
Still, my question is: do people who encounter specific, unknown manifolds ever use TQFTs to extract information about them? If so, how?
An ideal answer would be something like the following: "I was doing X when I encountered two manifolds $M_1$ and $M_2$. Because of the way that the manifolds presented themselves to me, I had access to the additional structure Y. Using that structure, I computed this TQFT invariant, thereby showing that $M_1$ and $M_2$ are not isomorphic." That is, I am interested in the context, and the specifics about how the manifolds arise in nature.
Let me say a few words about what I have found already, and why it isn't quite what I'm looking for. (Ian Algol's nice answer to the Math Overflow question linked above was useful.)
- First, there is a lot a work where TQFTs are themselves the objects of study. Like I said, this is not what I'm currently looking for.
- There is an interesting article, Quantum invariants of random 3-manifolds by Dunfield and Wong, which studies precisely what the title says. I am interested in examples where specific manifolds are studied using TQFT invariants.
- There seems to be work done in knot theory where existing invariants are studied in relation to TQFT invariants, but this is again more of an application of examples to theory than the other way around.
- There are theories like Donaldson-Thomas and Seiberg-Witten which produce invariants but these are not themselves quite TQFT invariants (if I understand this answer and its comments correctly.)
