# University of Toronto's Symplectic Geometry Seminar

Jan. 19, 2004, 2:10 - 3

SS5017A

## 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).