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

Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathrm{Fl}G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathrm{Fl}G$ to construct certain affine Gromov-Witten invariants of $\mathrm{Fl}G$, and to obtain a family of "affine quantum Chevalley" operators $\Lambda 0,\dots ,\Lambda n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $H\ast (\mathrm{Fl}G)$ with coefficients in $Z[q0,\dots ,qn]$. 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 $G=\mathrm{SL}n(C)$. The first quantum ring is a deformation of the subalgebra of $H\ast (\mathrm{Fl}G)$ generated by divisors. The second ring, denoted $\mathrm{QH}\mathrm{af}\ast (G/B)$, deforms the ordinary quantum cohomology ring $\mathrm{QH}\ast (G/B)$ by adding an affine quantum parameter $q0$. We prove that $\mathrm{QH}\mathrm{af}\ast (G/B)$ 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 $\mathrm{QH}\mathrm{af}\ast (G/B)$ 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 $G$.