On Hypergeometric Functions Connected with Quantum Cohomology of Flag Spaces

In the Givental’s work on the Gromov-Witten invariants for projective complete intersections, [G1], the principal role is played by certain formal power series connected with the quantum cohomology of a manifold. One has a manifold X with a natural torus action, with the finite number of fixed points, and one has a power series Zw associated with each fixed point xw. The coefficients of these series are certain integrals over the spaces of stable maps of genus 0 curves with two marked points to X . These series form a fundamental system of solutions of a certain lisse D-module on a power of the punctured disk. The small quantum cohomology of X coincides with the algebra of functions on its characteristic variety. The series Zw are uniquely determined by certain recursion relations relating Zw with all Zw′ if xw is connected with xw′ by a fixed line.