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

2011-09-05T16:08:30Z

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)

Answer by David Ben-Zvi for (3,2,1)-TQFTs and Verlinde algebras

2011-09-05T17:40:28Z

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.

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

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

Answer by David Ben-Zvi for (3,2,1)-TQFTs and Verlinde algebras