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Let $N$ be the total space of the vector bundle $\mathscr O_{\mathbb P^1}(-1)\oplus \mathscr O_{\mathbb P^1}(-1)$ over $\mathbb P^1$, and let $C_0\subset N$ be the zero section. Then $N$ is a quasi-projective Calabi-Yau threefold, and the moduli space of stable maps $\overline{\mathcal M}_g(N,[dC_0])$ is quasi-projective. It contains, as an open and closed substack, the space $$\overline{\mathcal M}_g(\mathbb P^1,d)\subset \overline{\mathcal M}_g(N,[dC_0]).$$ Faber and Pandharipande, in this paper on Hodge integrals, compute the contribution of the stable maps to $N$ factoring through the zero section $C_0\cong\mathbb P^1$ to the Gromov-Witten invariants of $N$. If the data $$\pi:U\to \overline{\mathcal M}_g(\mathbb P^1,d),\qquad f:U\to\mathbb P^1$$ describe the universal stable map to $\mathbb P^1$, this contribution is defined to be: $$C(g,d)=\int_{[\overline{\mathcal M}_g(\mathbb P^1,d)]^{\textrm{vir}}}e(R^1\pi_\ast f^\ast N).$$ In the paper, they write explicitely the generating series for these contributions in degree $1$. It is: $$\sum_{g\geq 0}C(g,1)t^{2g}=\Bigl(\frac{t/2}{\sin(t/2)} \Bigr)^2.$$

Question. Does anybody know (where to find) a similar formula for the generating series $\sum_{g\geq 0}C(g,d)t^{2g}$ when $d>1$? Maybe, it is hidden in that same paper, but I cannot see it. However, I am pretty sure this has been computed, as the Gromov-Witten theory of the local $\mathbb P^1$, as far as I know, is well understood.

Thank you very much!

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The formula is given by

$$ \sum_{g\geq 0} c(g,d) t^{2g-2} = \frac{1}{d}\left( 2\sin \left(\frac{dt}{2}\right)\right)^{-2}$$

It is in Faber-Pandharipande http://arxiv.org/pdf/math/9810173v1.pdf. See also equations (1) and (2) in http://arxiv.org/pdf/math/9811140.pdf.

By the way, the inclusion you write $\overline{M}_g(\mathbb{P}^1,d) \subset \overline{M}_g(N,d[C_0])$ is actually an equivalence --- every stable map to $N$ lies in the zero section. The two moduli spaces are the same, but their virtual fundamental classes are different. The obstructions to deforming a map $f:C\to N$ off of the zero section lie in $H^1(C,f^*(N))$. These spaces are the fibers of a bundle whose Euler class capped against the virtual fundamental class of $\overline{M}_g(\mathbb{P}^1,d)$ gives the virtual class of $\overline{M}_g(N,d[C_0])$. Thus the integral you write for $C(g,d)$ is not a definition but actually a theorem (expressing the above relation of the virtual classes).

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  • $\begingroup$ Thank you very much for your answer, and for adding the last clarifying paragraph! $\endgroup$ – Brenin Oct 4 '14 at 17:34
  • $\begingroup$ I just wonder: how is, then, the contribution $C(g,d)$ actually defined? For instance: what about the "non-local" case of a rigid curve $C\subset Y$ of genus $g$ in a CY3 $Y$? It seems that the substack $\overline M_{g+h}(C,d[C])\subset \overline M_{g+h}(Y,d[C])$ being a connected component is essential for the contribution $C_g(h,d)$ to be well-defined. $\endgroup$ – Brenin Oct 4 '14 at 17:42
  • $\begingroup$ Right, for the contribution of $C\subset Y$ to the GW invariants of degree $d[C]$, one needs the condition of $d$-rigidity. $O(-1)\oplus O(-1)$ curves are super-rigid, they satisfy the $d$-rigidity condition for all $d$. Precisely this issue is discussed in detail in one of my papers with Rahul: arxiv.org/pdf/math/0405204.pdf $\endgroup$ – Jim Bryan Oct 4 '14 at 23:44

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