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](http://lanl.arxiv.org/pdf/math/9810173.pdf), 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!



On the generating series of degree $d>1$ Gromov-Witten invariants of the local $\mathbb P^1$
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).


