Gromov-Witten invariants

Gromov-Witten invariants count (in a loose sense only) holomorphic maps from genus Riemann surfaces to a variety which pass through a given collection of cycles on . In order to define these, we compactify the space of maps from a variable pointed curve to 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 , image homology class , and number of marked points , this gives the moduli space of stable maps which is typically a highly singular Deligne-Mumford stack. The Gromov-Witten invariants of are given by integrals

where is evaluation at the marked point and the are elements of . 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 . The descendent Gromov-Witten Invariants are obtained by inserting monomials in the Witten classes into the integral.

Jeffrey Herschel Giansiracusa 2005-06-27