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

Publisher: John Wiley & Sons Inc

E-ISSN: 1460-244x|116|1|135-181

ISSN: 0024-6115

Source: Proceedings of the London Mathematical Society, Vol.116, Iss.1, 2018-01, pp. : 135-181

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Abstract

AbstractConsider the generalized flag manifold G/B and the corresponding affine flag manifold FG. In this paper we use curve neighborhoods for Schubert varieties in FG to construct certain affine Gromov–Witten invariants of FG, 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(FG) 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= SL n(C). The first quantum ring is a deformation of the subalgebra of H(FG) generated by divisors. The second ring, denoted QH aff (G/B), deforms the ordinary quantum cohomology ring QH(G/B) by adding an affine quantum parameter q0. We prove that QH aff (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 QH aff (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.