Timeline for 4D TQFT from a modular tensor category
Current License: CC BY-SA 3.0
8 events
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| Nov 11, 2014 at 12:57 | comment | added | Manuel Bärenz | Jerome Petit claims that Crane-Yetter for premodular categories is still just signature and Euler characteristic (math.titech.ac.jp/~Jerome/article/4variete_unoriented.pdf), therefore $Z_{CYK}(S^1 \times M^3) = 1$, doesn't that mean that it's still an invertible theory? Or what do you mean by "not almost trivial"? | |
| May 17, 2013 at 2:47 | history | edited | Kevin Walker | CC BY-SA 3.0 |
added 124 characters in body
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| May 16, 2013 at 16:20 | comment | added | David Ben-Zvi | For a parallel example: given a (2-dualizable) associative algebra A, considered as an object of the Morita 2-category, we can construct a 2d TFT Z. But you can only recover the category A-mod functorially from Z, or A up to Morita equivalence, even though one could say A is Z(pt).. to see A itself you need extra structure: a specific boundary condition in the field theory Z, or equivalently a (compact) object of A-mod. In the Freed-Teleman language, that defines a theory relative to the 2d TFT Z, analogous to WRT being relative to CYK, and that relative theory is what knows A itself. | |
| May 16, 2013 at 16:16 | comment | added | David Ben-Zvi | I think we're simply working with different amounts of structure. In the Freed-Teleman setting (via cobordism hypothesis) Z_{CYK}(pt) is indeed C, but as an object of the Morita category, so is equivalent to any other modular tensor category with the same anomaly invariant. So in that setting you can't functorially recover C (hence WRT) itself from the CYK TFT, but only a single characteristic class. On the other hand if you give enough additional structure, you can rigidify from the Morita category to the category of braided tensor categories and then recover C. | |
| May 16, 2013 at 16:04 | comment | added | Kevin Walker | [continued] ... a preferred boundary condition corresponding to gluing together many copies of (iterated) identity morphisms of the input $n$-category, and evaluation at this preferred boundary condition gives a preferred map from Hilbert spaces to the ground ring. If W is a 4-manifold with boundary, then applying this preferred map to the element $Z_{CYK}(W)$ of the Hilbert space gives an element of the ground ring which is equal to $Z_{WRT}(\partial(W))$. | |
| May 16, 2013 at 15:58 | comment | added | Kevin Walker | @David: I'm not sure I fully understand your question but I'll answer as best I can. I would say that $C$, or rather $Rep(C)$, is isomorphic to $Z_{CYK}(pt)$, not $Z_{CYK}(D^2)$. Perhaps part of the confusion is due to the fact that my TQFT framework is not the Atiyah-Segal framework. Compared to Atiyah-Segal, I have some extra structure at my disposal: "fields" or boundary conditions on manifolds. (The graph $\Gamma$ above is an example of such.) My Hilbert spaces are realized concretely as functions on boundary conditions. There is often... | |
| May 16, 2013 at 15:28 | comment | added | David Ben-Zvi | @Kevin: very interesting! definitely had missed that point. I'm confused though since the 4d TFT only depends on your modular tensor category C up to Morita equivalence - are you saying it's Morita rigid? Or put another way, I think you're saying we recover C=Z_{CYK}(disc) - but I only see the latter as an object in D=Z_{CYK}(S^1), and unless I give the extra structure of forgetful functor from D to categories how do I recover the underlying category of C? That forgetful functor (domain wall) seems to me equivalent to the data of the anomalous theory Z_{WRT} itself, and not part of Z_{CYK}? | |
| May 16, 2013 at 14:55 | history | answered | Kevin Walker | CC BY-SA 3.0 |