# (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