In his 2005 paper ”The Gromov-Witten potential associated to a TCFT” Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a Calabi-Yau category and a splitting of the Hodge filtration. The main difficulty to be overcome in that paper was dealing with the fact that TCFT’s constructed form Calabi-Yau categories are always required to have at least one input. This problem was originally solved in a non-constructive fashion using dg-Weyl algebras and associated Fock spaces.

In my talk I shall describe recent work on giving a new definition of Costello’s invariants. We bypass the dg-Weyl algebra approach completely. Instead we use a Koszul resolution of the space of S_n-invariant chains on M_{g,n}. This approach involves no choices, and makes the new invariants amenable to explicit computer calculations. I will list some of the higher genus invariants that we computed; they agree with predictions from mirror symmetry.

(This talk is based on joint works with Junwu Tu and with Kevin Costello).