An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice
| dc.contributor.author | Mare, A.-L. | en |
| dc.contributor.author | Mihalcea, L. C. | en |
| dc.contributor.department | Mathematics | en |
| dc.date.accessioned | 2017-01-02T18:36:26Z | en |
| dc.date.available | 2017-01-02T18:36:26Z | en |
| dc.date.issued | 2016-06-23 | en |
| dc.description.abstract | Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $\Lambda_0, \ldots, \Lambda_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. 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 $\mathrm{H}^*(\mathcal{Fl}_G)$ generated by divisors. The second ring, denoted $\mathrm{QH}^*_{\mathrm{af}}(G/B)$, deforms the ordinary quantum cohomology ring $\mathrm{QH}^*(G/B)$ by adding an affine quantum parameter $q_0$. We prove that $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ 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 $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ 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 $G$. | en |
| dc.description.notes | v3: we strengthen the main result so it holds across all Lie types | en |
| dc.identifier.uri | http://hdl.handle.net/10919/73919 | en |
| dc.relation.uri | http://arxiv.org/abs/1409.3587v3 | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject | math.AG | en |
| dc.subject | math.CO | en |
| dc.subject | math.QA | en |
| dc.subject | 14N35 (Primary) | en |
| dc.subject | 14M15 | en |
| dc.subject | 17B67 | en |
| dc.subject | 37K10 | en |
| dc.subject | 37N20 (Secondary) | en |
| dc.title | An affine deformation of the quantum cohomology ring of flag manifolds and periodic Toda lattice | en |
| dc.type | Article - Refereed | en |
| pubs.organisational-group | /Virginia Tech | en |
| pubs.organisational-group | /Virginia Tech/All T&R Faculty | en |
| pubs.organisational-group | /Virginia Tech/Science | en |
| pubs.organisational-group | /Virginia Tech/Science/COS T&R Faculty | en |
| pubs.organisational-group | /Virginia Tech/Science/Mathematics | en |