An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice

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Consider the generalized flag manifold G/B and the corresponding affine flag manifold FlG. In this paper we use curve neighborhoods for Schubert varieties in FlG to construct certain affine Gromov-Witten invariants of FlG, and to obtain a family of "affine quantum Chevalley" operators Λ0,…,Λn indexed by the simple roots in the affine root system of G. These operators act on the cohomology ring H∗(FlG) with coefficients in Z[q0,…,qn]. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for G=SLn(C). The first quantum ring is a deformation of the subalgebra of H∗(FlG) generated by divisors. The second ring, denoted QHaf∗(G/B), deforms the ordinary quantum cohomology ring QH∗(G/B) by adding an affine quantum parameter q0. We prove that QHaf∗(G/B) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of QHaf∗(G/B) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of G.

math.AG, math.CO, math.QA, 14N35 (Primary), 14M15, 17B67, 37K10, 37N20 (Secondary)