Gromov-Witten invariants

Gromov-Witten invariants count (in a loose sense only) holomorphic maps from genus $g$ Riemann surfaces to a variety $X$ which pass through a given collection of cycles on $X$. In order to define these, we compactify the space of maps from a variable pointed curve $C$ to $X$ by allowing the domain curve to degenerate to a nodal curve so that the corresponding map always has finite automorphism group. For a fixed genus $g$, image homology class $\beta$, and number of marked points $n$, this gives the moduli space of stable maps ${\overline{\mathcal{M}}}_{g,n}(X,\beta )$ which is typically a highly singular Deligne-Mumford stack. The Gromov-Witten invariants of $X$ are given by integrals

$${\int }_{[{\overline{\mathcal{M}}}_{g,n}(X,\beta ){]}^{\mathrm{vir}}}e{v}_1^{\ast }({\alpha }_1)\cdots e{v}_n^{\ast }({\alpha }_n)$$

where $e{v}_i:{\overline{\mathcal{M}}}_{g,n}(X,\beta )\to X$ is evaluation at the $i^{th}$ marked point and the ${\alpha }_i$ are elements of $H^{\ast }(X;\mathbb{Q})$. An important point of the theory is that this integral is defined via cap product with a distinguished homology class known as the virtual fundamental class of ${\overline{\mathcal{M}}}_{g,n}(X,\beta )$. The descendent Gromov-Witten Invariants are obtained by inserting monomials in the Witten classes ${\psi }_i$ into the integral.