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

Question

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?

Answer 1

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...

In the case of an affine (Z- graded) dg-category--- e.g. C= the category of perfect modules over a dg-algebra A, these conditions are guaranteed by A being homologically smooth. The reference for the first being implied is a famous paper of D. Kaledin and the second is a paper of D. Shklyarov http://arxiv.org/abs/math/0702590 (I think anyways).

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...)

What you do have from degenerate theories like this is an ideal in a certain Weyl algebra (see page 10 of the paper) I am pretty confident you could compute it at least for the case of spheres without too many problems, but I tend to be optimistic about these sort of things so... Hopefully some of this is right.