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

Abstract

We introduce and study a two-parameter family of symmetry reductions of the 2D 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 GelfandDickey hierarchy. We establish their characterization in terms of block Töplitz 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 AblowitzLadik 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.

Crepant resolutions and open strings

“…The ultimate justification of this statement resides in the relation of the Gromov-Witten theory of X and Y with integrable systems, and notably the two-dimensional Toda hierarchy; the details of this connection can be found in [12]. For the purposes of this paper, it is enough to offer a direct proof of the existence of said local isomorphisms.…”

“…that there exists a global quantum D-module on the A-model moduli space which locally agrees with the Frobenius structure given by quantum cohomology. In order to verify his proposal in the equivariant setting relevant for this paper, we give a new construction of this global structure: motivated by the connection of the Gromov-Witten theory of A n -surface singularities to certain integrable systems of Toda-type [12], we realize the global A-model quantum D-module as a system of one-dimensional Euler-Pochhammer hypergeometric periods. This mirror picture possesses several remarkable properties which enable us to verify in detail our proposals for the open CRC, as well as proving along the way various results of independent interest on Ruan's conjecture as well as its refinements (Iritani, Coates-Iritani-Tseng) and extensions (the higher genus CRC).…”

Tau-Structure for the Double Ramification Hierarchies

“…We have the following result (see also [BCRR17] for an application to the Gromov-Witten theory of local P 1 -orbifolds).…”

Frobenius manifolds and a new class of extended affine Weyl groups of A-type

“…• Is it possible to obtain an explicit realization of the integrable hierarchies associated with the Frobenius manifolds on the orbit space of W (k,k+1) (A l )? Perhaps this problem is related to the works in [23] or [24] about rational reductions of the 2D-Toda hierarchy, or in [25,26]…”