I recently attended an interesting seminar, where the concept of motivic Donaldson-Thomas invariants was explained ([0909.5088](http://arxiv.org/abs/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.)



Is there a "motivic Gromov-Witten invariant"?
Let me give an answer from a slightly different point of view. 


Let $M\_k$ be a moduli space as in your question; say it's a (compact) moduli space of sheaves on some (compact) Calabi-Yau threefold. In general, $M\_k$ is going to be very singular. However, it carries a so-called perfect deformation-obstruction theory of dimension zero. This gives a virtual fundamental class on $M\_k$, and the technical definition of the numerical invariant $e(M\_k)$ is that it is the degree of this virtual fundamental class. 


In the case of sheaves on CY3s, the deformation-obstruction theory has a duality property: it is a symmetric obstruction theory. In this case, according to a [result of Kai Behrend](http://xxx.lanl.gov/abs/math/0507523), $e(M\_k)$ can also be expressed as an Euler characteristic, albeit a weighted one:  

$e(M\_k)=\chi(M\_k, \nu\_{M\_k})$, 
where $\nu\_{M\_k}$ is the Behrend function of the singular space $M\_k$. In other words, one computes an Euler characteristic, but weighted with a numerical measure of how bad the singularities are. 


On can hope that this Euler characteristic definition can now be turned into something motivic. What one needs is a way to attach a motivic weight to points of $M\_k$. In some specific moduli problems, such as for Hilbert schemes of points where at least locally the moduli space can be expressed as a critical locus of a function on a smooth variety, this can be done using the tool of the motivic vanishing cycle; indeed, this is what our work does in the paper you cite. The general theory of how one attaches motivic weights is discussed in a (partially conjectural) [paper of Kontsevich and Soibelman](http://xxx.lanl.gov/abs/0811.2435).


The issue with Gromov-Witten theory on a CY3 is that the deformation-obstruction theory in that case, while it is of dimension zero, is not fully symmetric. It is symmetric on the open part corresponding to stable maps which are immersions from a smooth curve, but (as an expert assures me) not on the whole moduli space.


