Rational reductions of the 2D-Toda hierarchy and mirror symmetry - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of the European Mathematical Society Année : 2017

## Rational reductions of the 2D-Toda hierarchy and mirror symmetry

Andrea Brini
• Fonction : Auteur correspondant
• PersonId : 946791

Connectez-vous pour contacter l'auteur
Guido Carlet
Stefano Romano
• Fonction : Auteur
• PersonId : 1011807
Paolo Rossi
• Fonction : Auteur
• PersonId : 995747

#### Résumé

We introduce and study a two-parameter family of symmetry reductions of the two-dimensional Toda lattice hierarchy, which are characterized by a rational factorization of the Lax operator into a product of an upper diagonal and the inverse of a lower diagonal formal difference operator. They subsume and generalize several classical 1+1 integrable hierarchies, such as the bigraded Toda hierarchy, the Ablowitz-Ladik hierarchy and E. Frenkel's $q$-deformed Gelfand-Dickey hierarchy. We establish their characterization in terms of block Toeplitz matrices for the associated factorization problem, and study their Hamiltonian structure. At the dispersionless level, we show how the Takasaki-Takebe classical limit gives rise to a family of non-conformal Frobenius manifolds with flat identity. We use this to generalize the relation of the Ablowitz-Ladik hierarchy to Gromov-Witten theory by proving an analogous mirror theorem for the general rational reduction: in particular, we show that the dual-type Frobenius manifolds we obtain are isomorphic to the equivariant quantum cohomology of a family of toric Calabi-Yau threefolds obtained from minimal resolutions of the local orbifold line.

### Dates et versions

hal-01556415 , version 1 (05-07-2017)

### Identifiants

• HAL Id : hal-01556415 , version 1
• ARXIV :
• DOI :

### Citer

Andrea Brini, Guido Carlet, Stefano Romano, Paolo Rossi. Rational reductions of the 2D-Toda hierarchy and mirror symmetry. Journal of the European Mathematical Society, 2017, 19 (3), pp.835 - 880. ⟨10.4171/JEMS/681⟩. ⟨hal-01556415⟩

### Exporter

BibTeX TEI Dublin Core DC Terms EndNote Datacite

### Collections

169 Consultations
0 Téléchargements