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

2011-06-30T11:09:58Z

This is a follow up to this MO question: http://mathoverflow.net/questions/50497/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?

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

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