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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.)

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3 Answers

up vote 8 down vote accepted

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, $e(M_k)$ 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.

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.

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I don't know anything about motivic Donaldson-Thomas invariants, but it is possible to define the notion of GW invariants on the level of motives. You should check out Behrend and Manin, "Stacks of stable maps and Gromov-Witten invariants" sections 8 and 9, as well as Toën, "On motives for Deligne-Mumford stacks". Namely, instead of considering GW invariants as a collection of maps $$ I^X_{g,n,\beta} : H^\ast (X^n) \to H^\ast(\overline M_{g,n})$$ you can define them as morphisms between the Chow motives associated to $X^n$ and $\overline M_{g,n}$. The notion of a motive associated to a DM-stack is explained in Toën's article. In the case of $\overline M_{g,n}$ it is easy to say explicitly what this means: there is a finite cover $f \colon M \to \overline M_{g,n}$ by a smooth projective scheme M, and we take as the motive associated to $\overline M_{g,n}$ the motive associated to the scheme M and the projector $\frac{1}{\deg f} f^\ast f_\ast$. The correspondence inducing the morphism is given by the pushforward of the virtual fundamental class of $\overline M_{g,n}(X,\beta)$ to $A^\ast(X^n \times \overline M_{g,n})$ along the product of the evaluation maps and the forgetful map.

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See this great talk by Manin: dailymotion.com/video/… –  Peter Arndt Apr 4 '11 at 21:48
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Let me just add a bit to what Balazs said. The fact that the moduli spaces of sheaves on a CY3 have this symmetric obstruction theory is a reflection of properties of the category they live in, namely the derived category of coherent sheaves on the CY3. Indeed, Kontsevich and Soibelman's general construction of motivic DT invariants applies to a general class of CY3 categories and the motivic invariants live in a Hall algebra associated to this category.

My point is that the natural category associated to Gromov-Witten invariants is the Fukaya category and so I would expect any sort of "motivic" version of GW theory to live in something associated to the Fukaya category. In particular, I don't think these invariants will live in the Grothendieck group of varieties the way the motivic DT invariants do. GW theory (A-model) is inherently analytic (or symplectic) whereas DT theory (B-model) is inherently algebraic.

It would be really interesting to figure out what the analog of the Hall algebra is in the Fukaya category. Maybe the symplectic geometers already know it?

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Jim - I'm not sure I follow the reasoning formally. The Fukaya category of a compact symplectic 3-fold is a CY3 category, indistinguishable intrinsically from coherent sheaves on a (smooth proper) CY3. It has a moduli (derived higher) stack of objects, and one can speak of a Hall algebra of motivic functions on it. Perhaps you're suggesting that one uses something more (like knowledge of a particular class of objects) in defining the motivic DT invariants? or that the extraction of GW invariants from the Fukaya category is not fully analogous to the categorical extraction of DT invariants? –  David Ben-Zvi Apr 6 '11 at 0:44
The extraction of the invariants from the categories are definitely not analogous. The DT invariants come directly from the moduli of objects (i.e. sheaves) in the CY category. In contrast, the GW invariants come from moduli of stable maps, which are \em{not} the moduli of objects in the Fukaya category (those would be special Lagrangians). I guess this discussion does make my reasoning seem faulty --- one should not use the A-model / B-model equivalence to understand the DT/GW correspondence (it is not mirror symmetry -- the correspondence takes place on the same CY3 after all). –  Jim Bryan Apr 6 '11 at 5:15
I'd like to thank everybody for the answers. My most naive expectation was that the DT/GW correspondence would be extended to the motivic DT/motivic GW correspondence, but that idea doesn't sound promising, judging from the answers... –  Yuji Tachikawa Apr 10 '11 at 18:06
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