# What is the Gromov-Witten potential associated to String Topology?

Kevin Costello's article on the Gromov-Witten potential associated to a TCFT constructs for each TCFT, i.e. a functor from chains on Riemann surfaces with boundary to chain complexes satisfying certain conditions, a canonical formal power series $D$ with coefficients in a certain Fock space $\mathcal{F}$, which is constructed from the chain complex $V$ that a TCFT associates to the circle.

When one can construct the Gromov-Witten invariants for a manifold, we get a TCFT from Gromov-Witten theory. In that case a certain choice of polarization allows us to identify this potential $D$ with the Gromov-Witten potential. This potential encodes the intersection numbers of $\Psi$-classes and the fundamental class.

According to Costello's earlier article on TCFT's and $A_\infty$ Calabi-Yau categories the constructions of string topology allow us to define a TCFT for each oriented closed manifold $M$ (basically the chain level operations of Godin's operations in homology). One also gets a Gromov-Witten potential in this case. Is there an easier expression known for this potential? What geometric information does it encode?

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I think that these are all open questions. In principle, following Costello, we can write down a GW potential for any given TCFT or Calabi-Yau A-infty category, but I don't think this has been done yet for any concrete examples whatsoever, or at least nothing has been written up. I would be very happy, though, if I were wrong. –  Kevin H. Lin Jun 10 '10 at 19:07
Does the Costello's construction of B model agree with Kontsevich-Baranikov's one in genus 0 (asking by chance)? –  HYYY Jun 11 '10 at 17:32

Here's how I understand the situation(I'm not very deep and might well be wrong) --- Costello's paper explains how to construct the Gromov Witten potential from a compact A(infinity) Calabi Yau category given two little additional conditions 1) the Hodge to de-Rham spectral sequence degenerates and 2) the induced pairing as defined in his first paper on $HH_*$ is non degenerate. This is explained for example on the bottom of page 9...
None of these conditions are satisfied for string topology of a manifold(probably never somehow). The smoothness for example breaks pretty clearly --- think about $S^n$, C*(X) is $Q[x]/x^2$ with no higher operations.) The calculation of the HH to cyclic spectral sequence is a tad easier for odd spheres--- it is the case of a free commutative algebra on an odd variable which you can find in say Loday's book. HH is differential forms on the superspace $R(0,1)$ and normally the Connes operator acts by de Rham d extended to superdifferential forms(I haven't checked this little part but it's the only thing that makes any sense...)