University of Toronto's Symplectic Geometry Seminar

Jan. 19, 2004, 2:10 - 3

Augustin-Liviu Mare

University of Toronto

"Quantum cohomology of flag manifolds: the role of the Dubrovin connection"

Abstract: The Dubrovin connection has been an important ingredient in the study of the (small) quantum cohomology ring of the generalized flag manifold $G/B$. For instance B. Kim used the flatness of the Dubrovin connection when he proved that this ring is closely (and misteriously) related to the Hamiltonian system of Toda lattice type associated to $G$. I have proved recently -- see math.DG/0311320 -- that the quantum cohomology ring of $G/B$ consists of the space $H^*(G/B, R)\otimes R[q_1, ..., q_l]$ equipped with a product which is uniquely determined by the fact that it is a deformation of the cup product on $H^*(G/B, R)$, it is commutative, associative, graded with respect to $\deg(q_i)=4$, it satisfies a certain relation (of degree two), and the corresponding Dubrovin type connection is flat. In the first part of the talk I will sketch the idea of the proof. There are two consequences of this result which I will also address in my talk: (1) the flatness of the Dubrovin connection characterizes essentially the solutions of the ``quantum Giambelli problem" for $G/B$ (this fact is suggested in a recent paper by M. Guest); (2) one can give a conceptually new proof of the ``quantum Chevalley formula" (announced by D. Peterson and proved by W. Fulton and C. Woodward).