# Principal $G$-bundles as fully extended TQFTs, and $n$-representations

This is a follow up to this MO question: Fully dualizable objects in classical field theories

Assuming the notation there (which in turn come from Topological Quantum Field Theories from Compact Lie Groups), let $G$ be a finite group. The one-object groupoid $*//G$ is then an object of the symmetric monoidal category $Fam_n$ for any fixed $n$. Then by the cobordism hypotesis, there is at most one (up to isomorphism) symmetric monoidal functor $F: Bord_n^{SO}\to Fam_n$ with $F(pt^+)=*//G$, and there exists such a functor precisely when $*//G$ is a fully dualizable object in $Fam_n$.

So since the functor $Bun_G$ assigning to a manifold $X$ the groupid of principal $G$-bundles on $X$ is clearly a symmetric monoidal functor $F: Bord_n^{SO}\to Fam_n$ with $F(pt^+)=*//G$ this should mean that:

i) for any finite group $G$, the groupoid $*//G$ is a fully dualizable object in $Fam_n$;

ii) $Bun_G$ is the unique $Fam_n$-valued fully extended TQFT determined by $G$ (i.e., with $F(pt^+)=*//G$).

If so, a classical (fully extended) topological field theory from a finite group $G$ in the sense of Freed-Hopkins-Lurie-Teleman would reduce to the datum of a (fully dualizable) $n$-representation $G\to \mathcal{C}$, for $\mathcal{C}$ a symmetric monoidal $n$-category.

My question is: are there other classical examples of these (fully dualizable) $n$-representations of finite groups than those considered in Freed-Hopkins-Lurie-Teleman's paper? which are the TQFTs associated with these examples?

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For readers who have not recently read FHTL, I recall that $Fam_n$ is the $(\infty,n)$-category whose objects are finite groupoids, 1-morphisms are correspondences of finite groupoids, 2-morphisms are correspondences of correspondences, and so on up to n-morphisms; the (n+1)-morphisms and higher are equivalences. The "SO" on $Bord^{SO}_n$ reminds the reader that this is the category of n-framed bordisms. –  Theo Johnson-Freyd Jun 30 '11 at 14:34
By "other examples", I take you to mean the following: FHLT translate between some homological data and representations $G \to$ some delooping of Vect. You are asking if this is all the basic representations? Or maybe I misunderstand you and/or FHLT. –  Theo Johnson-Freyd Jun 30 '11 at 14:40
Hi Theo, thanks a lot for your comments. You perfectly read my question: it is a twofold question: i) does the one-line summary of FHLT says that a classical field theory is a representation $G\to$ some delooping of Vect? (seems so, but such a bold statement is not explicitly in FHLT) ii) are there in the algebra/representation theory literaure examples of representation $G\to$ some delooping of Vect not discussed in the examples of FHLT? if so, are the associated TQFT identified with known TQFTs? –  domenico fiorenza Jun 30 '11 at 18:10
I am having trouble extracting a precise statement from conclusion ii. In particular, "determined by G" does not seem to be a precisely formulated condition. –  S. Carnahan Jul 1 '11 at 7:25
"determined by $G$" there stands for $F(pt^+)=*//G$, which is the form I'm using a few lines above. I'm now editing the question to specify this. –  domenico fiorenza Jul 1 '11 at 16:16
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