String topology studies the algebraic structure of the homology of the free loop space $LM = Map(S^1,M)$ of a oriented closed manifold. One aspect of this structure is that the pair $(H\_\ast(LM;\mathbb{Q}),H\_\ast(M;\mathbb{Q}))$ forms a open-closed HCFT with positive boundary (work of Godin). This means that there are operations coming from the homology of moduli space of Riemann surfaces with each connected component having a non-empty outgoing or free boundary. The conjecture (although Blumberg-Cohen-Teleman claim it is a theorem) is that in fact $H\_\ast(M;\mathbb{Q})$ should be seen has $H\_\ast(P(M,M),\mathbb{Q})$, where $P(M,M)$ is the space of paths starting and ending in $M$, and the HCFT-structure can be extended to include open-closed cobordisms which have open boundaries labelled with a closed oriented submanifold $N$ of $M$. This then gives operations not only for $H\_\ast(LM;\mathbb{Q})$ and $H\_\ast(M;\mathbb{Q})$, but also on $H\_\ast(P(N\_1,N\_2);\mathbb{Q})$ for any two closed compact oriented $N\_1$, $N\_2$ in $M$. This is called the full set of branes. For $N\_1 = N\_2$ a single point and $M$ connected, $P(N\_1,N\_2) = \Omega M$, the based loop space. *In general $H\_\ast(P(N\_1,N\_2);\mathbb{Q})$ will therefore not be finite-dimensional.*


On the other hand, Costello has proven a classification theorem of open-closed TCFT. In this we don't work with homology, but chains in the moduli space of Riemann surfaces, and chain complexes instead of (graded) vector spaces. Costello has proven that a open-closed TCFT can be constructed from an open TCFT (cobordism without incoming or outgoing boundary components equal to the circle) and that an open TCFT is equivalent to a Calabi-Yau $A\_\infty$ category. *One of the properties of a Calabi-Yau $A\_\infty$ category is that all hom-spaces are finite-dimensional, forced by a certain non-degenerate pairing.*


One can construct a HCFT from a TCFT by applying homology everywhere. I think this HCFT will in fact be positive (or negative?) boundary, because the TCFT is defined from open-closed cobordisms where each connected component has at least one incoming boundary component. **Is this correct?** 


Furthermore, Costello conjectures in his paper that string topology (with the full set of branes) can be constructed as a TCFT, and applying homology then reduces to the HCFT given by Godin. But I can think of two reasons which make this conjecture seems false:
1) The naive choice of $C\_ast(P(N\_1,N\_2);\mathbb{Q})$ as Calabi-Yau $A\_\infty$ category is impossible, because these spaces will certainly be infinite-dimensional.
2) But no choice will work, because the homology of a finite-dimensional cell complex is finite-dimensional and we know some branes must be assigned infinite-dimensional spaces in the HCFT structure.


So my question is: **Is this reasoning enough to make my naive interpretation of Costello's conjecture false? If not, what is the mistake?**



Can string topology be a open-closed TCFT with the full set of branes?
No, the conjecture is not false. The mistake is the following: I said "On the other hand, Costello has proven a classification theorem of open-closed TCFT." He proves a classification of open TCFT's and gives a construction of an open-closed TCFT from an open TCFT. String topology can't be obtained by this specific construction.


However, Lurie(-Hopkins) gives a more general construction in section 4.2 of the article on the classification of TQFT's. This can deal with string topology. However, the Calabi-Yau objects in this case only require a non-degenerate cotrace, and string topology fits in now.


