Let $\overline{M}\_{g,n}(X,\beta)$ be the moduli of stable maps into $X$ of class $\beta \in H\_2(X)$. We have the evaluation maps $\operatorname{ev}\_i : \overline{M}\\_{g,n}(X,\beta) \to X$. Given $\alpha\_i \in H^\ast(X)$, the *Gromov-Witten invariant* corresponding to the tuple $(X,\beta,g,n,\alpha\_i)$ is the integral $$\int\_{[\overline{M}\_{g,n}(X,\beta)]^\text{vir}}\bigwedge\_i \operatorname{ev}\_i^\ast(\alpha\_i).$$


There is also the "forgetful" map (or "stabilization" map) $F : \overline{M}\_{g,n}(X,\beta) \to \overline{M}\\_{g,n}$. I don't know if this is the standard terminology (is it?), but one can define the *Gromov-Witten class* corresponding to the tuple $(X,\beta,g,n,\alpha\_i)$ to be the (virtual) pushforward $$F\_\ast^\text{vir}\left(\bigwedge\_i \operatorname{ev}\_i^\ast(\alpha\_i)\right) \in H^\ast(\overline{M}\_{g,n}).$$



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> Question: Are there any nontrivial cases in which these Gromov-Witten classes have been identified explicitly, e.g., in terms of say tautological classes on $\overline{M}\_{g,n}$?
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Just as a very naive motivation, note that in many situations Gromov-Witten *invariants* are zero simply because of "stupid reasons", like degree/dimension reasons (i.e. the degree of the integrand doesn't match the virtual dimension), and so provide no information. But the Gromov-Witten *classes* may still be nonzero and contain some information.



Gromov-Witten classes (as opposed to invariants)?
Consider $\overline{M}\_{g+1}(C,1)$ where $C$ is a (fixed) genus g curve. Geometrically, the points in this moduli space correspond to maps $E\cup\_p C \to C$ whose domain is the union of an elliptic curve $E$ attached to $C$ at a node $p$. The map is an isomorphism on $C$ and collapses the component $E $ to the point $p\in C$. Thus the moduli space is isomorphic to $\overline{M}\_{1,1} \times C$ 


which is of the expected dimension 2. Consequently, the virtual class is just equal to the usual class and the map 


$ \overline{M}\\_{1,1} \times C \to \overline{M}\_{g+1}$ 


is an inclusion and its image is clearly an explicit stratum in the boundary.


