Principal $G$-bundles as fully extended TQFTs, and $n$-representations - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:16:39Z http://mathoverflow.net/feeds/question/69175 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69175/principal-g-bundles-as-fully-extended-tqfts-and-n-representations Principal $G$-bundles as fully extended TQFTs, and $n$-representations domenico fiorenza 2011-06-30T11:09:58Z 2012-03-13T18:58:42Z <p>This is a follow up to this MO question: <a href="http://mathoverflow.net/questions/50497/fully-dualizable-objects-in-classical-field-theories" rel="nofollow">http://mathoverflow.net/questions/50497/fully-dualizable-objects-in-classical-field-theories</a></p> <p>Assuming the notation there (which in turn come from <a href="http://arxiv.org/abs/0905.0731" rel="nofollow">Topological Quantum Field Theories from Compact Lie Groups</a>), 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 <code>$F(pt^+)=*//G$</code>, and there exists such a functor precisely when $*//G$ is a fully dualizable object in $Fam_n$. </p> <p>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 <code>$F(pt^+)=*//G$</code> this should mean that:</p> <p>i) for any finite group $G$, the groupoid <code>$*//G$</code> is a fully dualizable object in $Fam_n$;</p> <p>ii) $Bun_G$ is the unique <code>$Fam_n$</code>-valued fully extended TQFT determined by $G$ (i.e., with <code>$F(pt^+)=*//G$</code>).</p> <p>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. </p> <p>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?</p>