Is there a "motivic Gromov-Witten invariant"?
I recently attended an interesting seminar, where the concept of motivic Donaldson-Thomas invariants was explained (0909.5088).
Very roughly, the DT invariant is a generating function $\sum q^k e(M_k)$ of a numerical invariant $e(\cdot)$ of a sequence of moduli spaces $M_k$. The motivic DT invariant is obtained by considering $\sum q^k [M_k]$ where $[M_k]$ is the image in $K(Var)$. This contains more info than the ordinary DT invariant.
Can this idea be applied for, say, the GW invariant of Calabi-Yau 3-folds, to get a finer invariant?
(Sorry for my vague question.)