Gromov-Witten classes (as opposed to invariants)?

Question

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 $$F_\ast^\text{vir}\left(\bigwedge_i \operatorname{ev}_i^\ast(\alpha_i)\right) \in H^\ast(\overline{M}_{g,n}).$$

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 $\overline{M}_{g,n}$?

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.

Answer 1

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.

Answer 2

I am not sure whether this answers your question, but you should perahps have a look at
Faber, Pandharipande: Relative Maps and Tautological Classes

"The push-forwards of all Gromov-Witten classes of compact homogeneous varieties X lie in the tautological ring by the localization formula for the virtual class (see [GrP1]). We do not know any example defined over $\bar{\mathbb{Q}}$ of a Gromov-Witten class for which the push-forward is not tautological."