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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?

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What does "I think this HCFT will in fact be positive (or negative?) boundary" mean? – Kevin H. Lin Jun 25 '10 at 22:11
A HCFT with positive (resp. negative) boundary is one that has operations for homology classes in the moduli space of open-closed cobordisms such that the outgoing or free (resp. incoming or free) boundary is not-empty. For example, in Godin's article this prevents operations from the cap, which would imply that looking at just $H_0$ and closed cobordisms, string topology would be an ordinary two-dimensional TQFT. But this is clearly not true, as the homology of the free loop space is in general not finite-dimensional. – skupers Jun 26 '10 at 20:00
up vote 2 down vote accepted

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.

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