# (3,2,1)-TQFTs and Verlinde algebras

Given a modular category $\mathcal{C}$ there are two natural ways to get a Frobenius algebra out of $\mathcal{C}$. One is to take the Verlinde algebra (or `fusion algebra') of $\mathcal{C}$. The other consist in considering the $(3,2,1)$-dimensional TQFT associated with $\mathcal{C}$, and to get out of it a $(2,1)$-dimensional TQFT by multiplication by $S^1$ (and a $(2,1)$-dimensional TQFT is the same thing as the datum of a Frobenius algebra). It is well known in fully extended TQFT folklore that these two constructions coincide. Is anyone aware of a reference I could cite as a source for this statement? (I know Dan Freed's The Verlinde algebra is twisted equivariant K-theory, where this can be read between the lines)

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If: 1) you can't find a reference, and 2) you know how to prove this "folklore fact", then you should include a proof of it in your writing, maybe as an appendix. –  André Henriques Sep 5 '11 at 16:56
@Andre': that's precisely the strategy I had in mind. And MO is essential for step1! :) completely off-topic: have you recived my e-mail message from a few days ago? I'm not sure I was able to correcltly decipher your email address from your home page ;) –  domenico fiorenza Sep 5 '11 at 18:56
While I have the rough idea in mind and probably could work it out by myself, can someone give me a reference to the construction of the Frobenius algebra starting with the Verlinde algebra? –  Marcel Bischoff Dec 16 '13 at 9:26

## 1 Answer

I don't know a reference but it's not clear to me you need one for this statement, since it's close to definitional. More precisely you just need the fact that compactification on a circle corresponds to taking Hochschild homology, or in this case just complexified K-theory, of a category, for which there are lots of references (my kneejerk reaction is to quote Lurie's TFT manuscript though I'm sure for this you can find many older references). Then you're simply asserting that the K-groups tensor C inherit a commutative multiplication, a unit and a trace from the braided tensor category you started from (that's the definition of the Verlinde algebra), and that from the field theory these are given on the category as the pair of pants and (in or outgoing) disc, hence by the same pictures times S^1 on the Verlinde algebra, hence by the same pictures again in the dimensionally reduced theory.

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Here you're assuming that you already know that there's a orpo 321 theory with values in the 2-category of categories attached to any MTC which assigns the circle to the MTC and where the structural morphisms are assigned to the obvious bordisms (tensor to pants, etc.). That's the folklore result. –  Noah Snyder Sep 5 '11 at 17:53
I would think everything I need here is already in say the book of Bakalov-Kirillov - eg I don't need 1-2-3 or even all of 1-2, just that a braided tensor category is the same as an algebra over the E_2 operad, plus some finiteness of dimensions to allow me to close up discs, and I automatically have enough finiteness to be able to take product with a circle.. –  David Ben-Zvi Sep 5 '11 at 18:05
Hi David, thanks. Yes, I know the statement is almost tautological, but being able to cite a reference for this precise statement would be convenient to me. Otherwise I can follow Andre''s avdice and try to write a proof of this myself. –  domenico fiorenza Sep 5 '11 at 18:33
Oh, I see. If I understand correctly, you're saying that this really has very little to do with 3-dimensions at all. Dimensional reduction takes you from a 321 theory to a 210 theory, but the real calculation (going from the category that the 210 theory assigns to a point to the Frobenius algebra that it assigns to a circle) takes place entirely inside the 210 theory. –  Noah Snyder Sep 5 '11 at 19:05