Quantum cohomology of homogeneous spaces
Final Report Abstract
The research project under considerations was centered around the quantum cohomology of a projective homogeneous space X. We initially asked and solved the question about the computation of the minimal degree dX in the quantum product of two point classes in terms of Kostant’s cascade of orthogonal roots, while the uniqueness of minimal degrees in general was proved. We further completely solved the question of quasi-homogeneity of the moduli space of stable maps to X under the automorphism group of X for all minimal degrees. The relevance of quasi-homogeneity of this space becomes apparent in the light of a systematic quantum to classical principle for every X, which is a geometric way to compute Gromov-Witten invariants on X in terms of intersection cohomology on some auxiliary homogeneous space. In order to prove certain vanishing conjectures for Gromov-Witten invariants of degree > dX , we further studied noncommutative models of cohomology of G/B with the help of a specific Nichols algebra associated to reflections.
