Let $X$ be a smooth, projective, Calabi-Yau 3-fold (CY makes the exposition more elegant, I don't think it is necessary).

To define Gromov-Witten invariants, we consider moduli spaces of stable maps, $\bar{M}_g(X, \beta),$ indexed by the genus $g$ of the source curve and curve class $\beta$ of its image in $X$ . These moduli spaces admit virtual fundamental cycles $[\bar{M}_g(X, \beta)]^{vir},$ and the CY condition ensures that these are 0-cycles. The GW invariants are then defined to be

$$GW_{g, \beta} = \int_{[\bar{M}_g(X, \beta)]^{vir}} 1. $$ This number (as far as I understand) is a virtual count of the ($g$, $\beta$)-curves on $X$.

Similarly (here is where we really confined to 3-folds), to define Donaldson-Thomas invariants, we consider the moduli space of closed subschemes ---the Hilbert scheme --- of $X$, $I_n(X, \beta)$, indexed by the holomorphic Euler characteristic $n$ and the homology class $\beta$ of the subscheme. When $\beta$ is a curve class, these moduli spaces also admit zero dimensional virtual fundamental cycles, and the DT invariants are defined to be

$$DT_{n, \beta} = \int_{[I_n(X, \beta)]^{vir}} 1. $$ This number is a virtual count of subschemes of dimension $\leq 1$ of $X$.

In the paper Gromov-Witten theory and Donaldson-Thomas theory, I by Maulik, Nekrasov, Okounkov, and Pandharipande, these invariants were assembled into generating series:

$$Z_{DT}(q,v) = \Sigma_{\beta}\Sigma_n DT_{n, \beta}q^nv^{\beta} = \Sigma_{\beta}Z_{DT, \beta}(q)v^{\beta}$$ $$Z_{GW}(u,v)= 1+ exp( \Sigma_{\beta \neq 0} \Sigma_{g \geq 0} GW_{g, \beta}u^{2g-2}v^{\beta}) = 1+ \Sigma_{\beta \neq 0} Z_{GW, \beta}(u)v^{\beta}.$$

Here, $v^{\beta}$ is short-hand for $v_1^{\beta_1}\cdot \ldots \cdot v_k^{\beta_k}$, where $\beta_1, \ldots, \beta_k$ is a positive basis of $H_2(X, \mathbb{Z})$ mod torsion. Then, they make the following conjecture:

$$\frac{Z_{DT, \beta}(q)}{Z_{DT, 0}(q)} = Z_{GW, \beta}(u),$$ after the change of variables , $e^{iu} = -q$.

My question:

Can anyone explain why this change of variables is expected to work?

(You know, besides the fact that it has been proven in the toric case.)

  • 3
    $\begingroup$ My poor understanding is that in this game the "change of variables" is where the art resides. $\endgroup$ – José Figueroa-O'Farrill Feb 25 '10 at 0:24

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