Tangency quantum cohomology and characteristic numbers - Joachim Kock (Anais da Academia Brasileira de Ciências, 2001)
This work establishes a connection between gravitational quantum cohomology and enumerativegeometry of rational curves (in a projective homogeneous variety) subject to conditions of infinitesimalnature like, for example, tangency. The key concept is that of modified psi classes, whichare well suited for enumerative purposes and substitute the tautological psi classes of 2D gravity.The main results are two systems of differential equations for the generating function of certaintop products of such classes. One is topological recursion while the other is Witten-Dijkgraaf-Verlinde-Verlinde. In both cases, however, the background metric is not the usual Poincaré metricbut a certain deformation of it, which surprisingly encodes all the combinatorics of the peculiarway modified psi classes restrict to the boundary. This machinery is applied to various enumerativeproblems, among which characteristic numbers in any projective homogeneous variety,characteristic numbers for curves with cusp, prescribed triple contact, or double points.