Intuitively, Gromov-Witten theory makes perfect sense. Via Poincare duality, we look at the cohomology classes $\gamma_1, \ldots, \gamma_n$ corresponding to geometric cycles $Z_i$ on a target space $X$, pull them back and then the integral
$$\langle \gamma_1 \cdots \gamma_n\rangle=\int_{\overline{\mathcal{M}}_{g,n}(X,\beta)^{vir}}ev_1^*\gamma_1 \smile \cdots \smile ev_n^*\gamma_n$$
should count the number of curves whose intersection with the given cycles is non-empty.
However, we also have the ψ-classes (or "gravitational descendants") arising from the moduli space $\overline{\mathcal{M}}_{g,n}$ which are the chern classes of the $i$-th cotangent line bundle to a given $(C, x_1, \ldots, x_n) \in \overline{\mathcal{M}}_{g,n}$.
So what, geometrically, do these represent? The fact that they arise from $\overline{\mathcal{M}}_{g,n}$ means that the inclusion of a ψ-class places restriction on the geometry of the curves which we count; that much is clear. What is this restriction?
The reason that I am curious is that I am trying to evaluate the GW-invariants corresponding to maps which have components collapsing to an A1 singularity (i.e. a $B\mathbb{Z}/2$), but such that not all of the curve collapses. It has been mentioned in passing that including a ψ-class could help with this, and while the little I understand makes this sound plausible, I don't exactly see why.
So what are ψ-classes? Can I use them to split my curve up into parts so that a fixed component lands on my stacky point, while the rest of it does whatever else curves do?
Best Answer
The following answer is unfortunately not quite correct, but it may be useful anyway. I will of course be ignoring any virtual fundamental class issues.
Imagine that you are computing a Gromov-Witten invariant where you require the i-th marked point to land at a specific point (i.e. your i-th insertion γi is the class of a point), and now lets add aditionally the i-th psi-class as an insertion. You can restrict to the subspace of maps with $f(x_i) = x$ for some generic choice of $x \in X$. Fixing an arbitrary non-trivial map $\Phi \colon T_x \to k$ gives you by composition a map from the relative tangent bundle of the universal curve over $M_{g, n}(X)$ at the section xi to the trivial line bundle, in other words a section $\phi$ of the relative cotangent bundle of the universal curve. It will vanish on curves which are tangent to a hypersurface through x with tangent direction matching the zero-locus of the map $\Phi$.
So you can think of Gromov-Witten invariants with psi-classes as counting maps which additionally satisfy tangency conditions at the marked points.
Why is this not correct? The zero locus of $\phi$ computes the Chern class of the relative cotangent bundle at $x_i$ over Mg, n(X), which is not the same as the pull-back of the $\psi$-class from Mg, n. Insertions of the former are sometimes called "gravitational ancestors", and the difference to gravitational descendants is described explicitly in alg-geom/9708024.