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I know this isn't exactly a math question, but I am asking it here anyway. We define an extended TQFT to be a functor (preserving tensor products) from the $\left(\infty,n\right)$-category of cobordisms to a suitable $\left(\infty,n\right)$-category of vector spaces.

The original Atiyah-Witten definition was a functor for the category of $n$ dimensional cobordisms to $\mathrm{Vect}_{\mathbb C}$. This definition was justified from the path integral in physics.

Can we similarly get an physicsy intuition of extended TQFT from a path integral-like formulation? Any references to a general construction from physics that gives rise to such a functor, starting from the path integral?

Note: I don't want specific example relating to Chern-Simons theory or any other TQFT, but a general construction deriving the extended TQFT axioms from the path integral, or something similar.

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    $\begingroup$ It's been a while since I've read Freed 1992, but I thought he did argue for (once) extended TQFT from a path integral perspective. $\endgroup$
    – Theo Johnson-Freyd
    Commented May 25, 2020 at 18:42
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    $\begingroup$ Wild. I thought I knew where all of my old notes are, but I cannot find my own notes on Kevin's 2011 minicourse. $\endgroup$
    – Theo Johnson-Freyd
    Commented May 27, 2020 at 13:40
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    $\begingroup$ To answer your question, I think "equivalent" might be a heavy word. $\endgroup$
    – Theo Johnson-Freyd
    Commented May 27, 2020 at 13:41
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    $\begingroup$ Kevin has developed a very sophisticated theory of the extended operators in certain types of topological field theories. But he tends not to care about some "finiteness" conditions at the very top dimensions. $\endgroup$
    – Theo Johnson-Freyd
    Commented May 27, 2020 at 13:42
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    $\begingroup$ By now it is pretty well understood that Kevin's TQFTs extend "down" to points. There are some still missing theorems matching Kevin's axiomatization with Lurie's. In low spacetime dimensions, most of this has been proved by Cooke in her thesis. In full generality, the matching will eventually follow from the work by Ayala--Francis on what they have been calling "beta factorization algebras". $\endgroup$
    – Theo Johnson-Freyd
    Commented May 27, 2020 at 13:46

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The physics motivation for extended QFTs (and not just TQFTs) comes from the locality principle (no spooky action at a distance).

The mathematical expression of locality is the descent property for extended QFTs. For an early source, see, for instance, Higher Algebraic Structures and Quantization by Daniel S. Freed and Triangulations, Categories and Extended Topological Field Theories by Ruth J. Lawrence.

The descent property is proved in full generality in arXiv:2011.01208.

Specifically, the assignment to X of the symmetric monoidal (∞,n)-category of extended QFTs with bordisms equipped with a map to X is a stack of symmetric monoidal (∞,n)-categories with respect to X.

This claim fails unless we work with QFTs extended all the way down to points, since proving the descent property requires cutting bordisms all the way down to points.

An informal path integral construction produces an extended QFT starting from another functorial field theory (quantum or classical) on Y and performing pushforward along a map Y→X. The pushforward simply integrates over spaces of n-dimensional manifolds mapping to Y that have the same image in X.

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  • $\begingroup$ @Dmiti, what exactly is this 'descent property'? You're talking about evaluating the TQFT by gluing local pieces? I don't know about extended TQFTs with maps to spaces. Can you give me a reference to this stuff? And also the stack you're talking about? $\endgroup$
    – Chetan Vuppulury
    Commented May 25, 2020 at 17:41
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    $\begingroup$ @ChetanVuppulury: The descent property is explained in the third paragraph. I added a reference to a classical paper by Freed that explains how the locality property is connected to extended field theories. $\endgroup$
    – Dmitri Pavlov
    Commented May 25, 2020 at 17:58
  • $\begingroup$ @ChetanVuppulury: And now I added a reference to a new paper that explains and proves the descent property, giving complete details. $\endgroup$
    – Dmitri Pavlov
    Commented May 17, 2021 at 20:30
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The original motivation for extended TQFTs (as introduced by Freed, Lawrence, Baez-Dolan) is indeed giving a finer form of locality, as explained by Dmitri Pavlov. However I think there are two quicker, and arguably more physical, ways to see n-categorical structure in n-dimensional QFTs.

The first is not really about the states of a QFT (as axiomatized in the Atiyah-Segal formalism) but their algebras of observables (extending the distinction between geometric and deformation quantization in the context of quantum mechanics). Namely the theory of factorization algebras as developed in the book of Costello-Gwilliam extracts from the same data as the path integral an n-dimensional factorization algebra of observables. In the topological context such a factorization algebra is the same as an $E_n$ algebra, which is the same as a very connected $(\infty,n)$-category (one with one object, one 1-morphism, ...all the way down).

The second comes from thinking of what ELSE there is in a QFT beyond the path integral -- the most important being the structure of defects of various dimensions. Of these the richest is the notion of a boundary theory (or "boundary condition") for a QFT, which is very loosely "things we can put on the boundary and couple to our theory" -- something like a QFT of one dimension lower that lives on the boundary of manifolds where the bulk carries our given QFT.

In any case, boundary theories in an $n$-dimensional QFT naturally form something like an $(n-1)$-category, which in the cobordism hypothesis formalism for extended TQFT is closely related to what you'd attach to a point. Namely, as morphisms between any two boundary theories you can consider codimension 2 defects that are interfaces between the two theories (think of dividing the boundary of a half-space in $R^3$ into upper and lower halves with a 1-dim interface on the intersection). As 2-morphisms you can consider interfaces between interfaces, and so on and so forth.

To me this is the most compelling way to see that higher categorical structure is physically natural/meaningful. A (somewhat criminal) paraphrase of the cobordism hypothesis says that a fully extended TQFT is determined by its collection of boundary conditions. [Really boundary theories are morphisms between the unit and the object attached by the TQFT to a point, which in general needn't determine this object, but it's a decent ansatz.]

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  • $\begingroup$ Could you elaborate on the relation between defects and boundary theories (by which I understand holographically related theories, if not please do clarify)? $\endgroup$
    – Chetan Vuppulury
    Commented May 25, 2020 at 18:19
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    $\begingroup$ I don't know the physics but my impression was that holography is not the correct terminology since there's no gravity involved, just a bulk QFT and a theory coupled to it living on the boundary. This is a special case of a domain wall, a (codimension one) interface between two QFTs, where we let the theory on the other side of the wall be trivial. Domain walls can be considered codimension one defects (except for higher codimension defects there is only one theory involved, since space doesn't get disconnected by the defect). $\endgroup$
    – David Ben-Zvi
    Commented May 25, 2020 at 18:30

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