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

Abstract

Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold ${\mathcal{F}l}_G$. In this paper we use curve neighborhoods for Schubert varieties in ${\mathcal{F}l}_G$ to construct certain affine Gromov-Witten invariants of ${\mathcal{F}l}_G$, 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 ${\mathrm{H}}^{\ast }({\mathcal{F}l}_G)$ with coefficients in $\mathbb{Z}[q_0,\dots ,q_n]$. 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 $G={\mathrm{SL}}_n(\mathbb{C})$. The first quantum ring is a deformation of the subalgebra of ${\mathrm{H}}^{\ast }({\mathcal{F}l}_G)$ 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 $q_0$. 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 $G$.

v3: we strengthen the main result so it holds across all Lie types

References in corpus (1)

• Quantum Integrability and Generalised Quantum Schubert Calculus