My question concerns some of the results of Costello's "Topological conformal field theories and Calabi-Yau categories" and how they are related/ can be rederived via the description of (fully extended) TFTs by locally constant factorization algebras. More precisely, I am wondering about the following points:
- Costello shows that for any open TCFT $Z: \mathcal{O} \rightarrow \operatorname{Ch}(\mathbb{C})$, one can construct a universal open-closed TCFT by performing a (homotopy) left Kan extension along the inclusion $\mathcal{O} \rightarrow \mathcal{OC}$. However, he does not motivate this in any way as far as I can see - is the reason why this works rooted in the fact that a factorization algebra can be reconstructed from its values on a Weiß covering by a Left Kan Extension? One would have to regard $Z$ as a factorization algebra on some category of 2D manifolds with corners to make this idea precise, I would use this paper to define this generally for stratified spaces (alternatively, this talk gives an alternative approach via structured spaces à la Lurie). I am not yet sure though if this would work out, and haven't understood the above well enough to do it myself.
- Open TCFTs are classified by Calabi-Yau $A_\infty$-categories, while closed TCFTs are classified by Frobenius algebras, as Costello shows. It seems to me that in the second case, the algebra structure should arise from the fact that pushing forward the locally constant factorization algebra (given by this 2D TFT) on a cylinder $S^1 \times I$ to the initial state $S^1$, we again get a loc. const. FA that is classified by an associative algebra (together with a modronomy operation). Intuitively, the algebras should be the same - is this true? On the other hand, loc. const. FAs on the interval are classified by an associative algebra $A$ and both a right and a left module over $A$ for both ends. How would I relate this to the category of D-branes (for open TCFTs), namely above mentioned CY $A_\infty$-category? Or, as a first step, how do I incorporate the branes my string ends on as labels? Finally, I would assume that to get the trace maps for the closed and the open case, I need to calculate factorization homology over surfaces - this paper does this, but again, I am puzzled how to proceed further.
- Finally, the left Kan extension above, followed by a restriction along $\mathcal{C} \rightarrow \mathcal{OC}$, gives a way to construct a universal closed TCFT from an open TCFT. Costello shows that the Frobenius algebra corresponding to the latter is the Hochschild Homology of the CY $A_\infty$-category corresponding to the former. Is this again just factorization homology in disguise? Namely, given a TFT on a circle that associates to opens homeomorphic to $\mathbb{R}$ the associative algebra $A$, the value on the whole circle is given by the Hochschild Homology of $A$. Using the above two points, we might get an alternative derivation of Costello's result - but again, my argumentation is very hand-wavy.
I know that I am presenting my ideas pretty roughly here; even though I have been thinking about this a lot, I am not yet able to make them more rigorous (maybe because I haven't yet understood all of the mentioned papers). Still, I am fairly sure that (following abstract results about TCFTS, TFTs and factorization algebras by Lurie, Scheimbauer and others, in particular the relation between 2D extended TFTs à la Lurie and TCFTs) that it should be possible, so I am asking if anyone knows more or might even provide a refence for such considerations. (Also, I hope that my wild speculations above are somewhat understandable/ well motivated.)
