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

$\displaystyle \int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^\mathrm{vir}}
ev_1^*(\alpha_1) \cdots ev_n^*(\alpha_n)

where $ ev_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^*(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.

Jeffrey Herschel Giansiracusa 2005-06-27