# Conformal Quantum Field Theories as Chain-complex valued TQFTs?

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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.