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 and the corresponding affine flag manifold . In this paper we use curve neighborhoods for Schubert varieties in to construct certain affine Gromov–Witten invariants of , and to obtain a family of ‘affine quantum Chevalley’ operators indexed by the simple roots in the affine root system of . These operators act on the cohomology ring with coefficients in . 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 . The first quantum ring is a deformation of the subalgebra of generated by divisors. The second ring, denoted , deforms the ordinary quantum cohomology ring by adding an affine quantum parameter . We prove that 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 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 .