Relation between fully-extended TQFT and a “topless” TQFT

Consider 3-dimensional TQFTs for example. One version of them is the 3-2-1-0 fully extended TQFT. Do we have another version: 2-1-0 extended "TQFT"?

If yes, do we have an example of 2-1-0 extended TQFT that is not 3-2-1-0 fully extended TQFT?

By 2-1-0 extended 3-dim "TQFT", we mean that we can assign a Hilbert space to every closed orientable 2-manifold, but we do not require the path integral to be well defined for every closed orientable 3-manifold. Certainly, there is a higher dimensional analogue of this.

Unitary condition: The Hilbert space mentioned above has a well define inner product so that the norm are all positive. Also the Hilbert space has a finite dimension on all closed surfaces.

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By "2-1-0 extended", do you mean that we can assign a Hilbert space to a 2-manifold, but that there is no path integral for a 3-manifold? –  Kevin Walker May 11 at 12:59
Yes. We can assign a Hilbert space to every closed orientable 2-manifold, but we do not require the path integral to be well defined for every closed orientable 3-manifold. Certainly, there is a high dimensional analogue of this. The issue is that, for 3-dim TQFT, if 2-1-0 extended already implies 3-2-1-0 extended. –  Xiao-Gang Wen May 11 at 13:31
I think when people say "not-fully-extended" they usually mean something like 3-2-1 or 3-2; that is, the "not" usually refers to not being able to extend down, rather than up. Right? –  Qiaochu Yuan May 11 at 17:47
@QY: Yes, that's correct –  Kevin Walker May 11 at 18:53
Also, I think the objects the OP is looking for should be classified by $2$-dualizable objects which aren't $3$-dualizable in symmetric monoidal $3$-categories, or something like that. In particular there's a symmetric monoidal $3$-category of monoidal categories and bimodule bicategories over these which I think is where Noah's example comes from. –  Qiaochu Yuan May 12 at 3:27

If I understand you correctly, your "2-1-0" TQFTs are what are frequently called "2+$\epsilon$-dimensional TQFTs" in the mathematical literature. (The $\epsilon$ means that very thin 3-manifolds, e.g. the mapping cylinder of a homeomorphism of 2-manifolds, can have their path integral defined.)

If you try to construct a Turaev-Viro TQFT (Levin-Wen model) based on $Rep(U_q(g))$ for q not a root of unity (so there are infinitely many simple objects), then you can construct an (infinite-dimensional) Hilbert space for any 2-manifold, and also assign a 1-category to 1-manifolds and 2-category to 0-manifolds. But you cannot construct the path integral of, say, $Y\times S^1$, since this should be equal to the dimension of $Z(Y)$ which is infinite (unless $Y$ is very simple). So this is an example of a "2-1-0" theory which is not a "3-2-1-0" theory.

In response to your comment asking for a unitary example with finite-dimensional Hilbert spaces:

If your definition of "unitary" in this context is the same as mine, then the answer is that any finite unitary 2+$\epsilon$ ("x-2-1-0") theory extends to a full "3-2-1-0" theory. More specifically, if the input 2-category (e.g. tensor category) has a collection of positive-definite inner products which are compatible with the tensor category structure (I assume "unitary" implies this) and the Hilbert space for any surface is finite-dimensional, then it follows from Theorem 6.3.1 of this that the theory can be extended to a full "3-2-1-0" theory.

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Thank you very much. an infinite-dimensional Hilbert space for a 2-manifold implies that the system is gapless in physics. I am looking for a finite unitary example. –  Xiao-Gang Wen May 12 at 1:00
Can you clarify precisely what you mean when you say the theory should be unitary? –  Chris Schommer-Pries May 12 at 7:11
Thanks Kevin. That is very helpful. In condensed matter physics, we are dealing with x-2-1-0 TQFT, while in high energy and math, people mainly deal with 3-2-1-0 TQFT. Understanding their relation is very important. –  Xiao-Gang Wen May 13 at 23:53

With Douglas and Schommer-Pries construct such a 210 TFT for every finite tensor category (in the sense of Etingof-Ostrik). When the category is not semisimple there's no 3210 TFT.

My understanding from conversations with Kevin Walker is that one should have a similar story for (some) infinite tensor categories, but in our setup we are not yet able to prove such a result rigorously (Kevin's formalization of local TFT is somewhat different).

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Thanks! I wonder if the 210 TFT unitary when the category is not semisimple? –  Xiao-Gang Wen May 11 at 22:30
@Xiao-GangWen I think unitarity should force semisimplicity for the usual reasons, so I don't think these examples will be unitary. –  Noah Snyder May 12 at 4:15