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?