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Consider the generalized flag manifold    G  /  B    and the corresponding affine flag manifoldF  ℓ   G   . In this paper we use curve neighborhoods for Schubert varieties inF  ℓ   G    to construct certain affine Gromov–Witten invariants ofF  ℓ   G   , and to obtain a family of ‘affine quantum Chevalley’ operatorsΛ  0   ,  …  ,   Λ  nindexed by the simple roots in the affine root system of   G  . These operators act on the cohomology ringH  ∗    (    F  ℓ   G   )with coefficients in    Z  [   q  0   ,  …  ,   q  n   ]   . 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 ofH  ∗    (    F  ℓ   G   )generated by divisors. The second ring, denotedQH   aff   ∗    (  G  /  B  )    , deforms the ordinary quantum cohomology ringQH  ∗    (  G  /  B  )by adding an affine quantum parameter    q  0   . We prove thatQH   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 ofQH   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  .

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