Conformal Quantum Field Theories as Chain-complex valued TQFTs?

Question

We will use the reference http://arxiv.org/abs/math/0412149v7.

Let \(\mathcal{M}\) be Segal's category of Riemann surfaces. Let \(C_{*}\) be a symmetric monoidal functor from \(\mathbf{Top}\) to the category of complexes of \(\mathbb{K} \)-vector spaces, with \(\mathbb{K}\) a base field of characteristic \(0\). Let \(\mathbf{Ch}_{n}\) denote the category of \(n \)-dimensional chain complexes. A CQFT is defined as a symmetric monoidal functor \(C_{*}(\mathcal{M})\to \mathbf{Ch}\). Note that \(C_{*}(\mathcal{M})=\mathbf{Ch}_{3}\), so the CQFT is a functor \(\mathbf{Ch}_{3}\to\mathbf{Ch}\). The CQFT maps \(\mathbf{Ch}_{3}\) to some subset \(\mathrm{Ch}_3\subset\mathbf{Ch}_{3}\). Using Lurie's classification allows us to define a CTQFT (a TQFT with the structure of a CQFT) as a functor \(\mathcal{C}:3\mathrm{Cob}\to\mathbf{Ch}_3\). This is a chain-complex valued TQFT, hence allowing us to conclude that

Idea: A CTQFT is a chain-complex valued TQFT. I.e., let \(\widetilde{3\mathrm{Cob}}\) denote the \((\infty,1)\) category with objects as two-dimensional manifolds, morphisms as bordisms, \(2\)-morphisms as diffeomorphisms, and \(3\)-morphisms as isotopies. Then, the functor \(\mathcal{C}:\widetilde{3\mathrm{Cob}}\to\mathbf{Ch}_3\bigcap n\mathrm{Hilb}\) is a CTQFT.

My question is, can this idea be extended to the more general class of CQFTs? I.e., can we get Idea 2?

Proposed Idea 2: A CQFT is a chain-complex valued (T)QFT, in the sense mentioned above.

Comments

Hi Sanath Devalapurkar; nice to see you participating here! 

Interesting idea... hope you get an answer soon!