Brini, A. orcid.org/0000-0002-3758-827X, Carlet, G., Romano, S. et al. (1 more author) (2017) Rational reductions of the 2D-Toda hierarchy and mirror symmetry. Journal of the European Mathematical Society, 19 (3). pp. 835-880. ISSN 1435-9855
Abstract
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 di erence 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 T oeplitz 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 European Mathematical Society. This is an author-produced version of a paper subsequently published in Journal of the European Mathematical Society. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Rational reductions; Gromov–Witten; integrable hierarchies; mirror symmetry; 2D-Toda; Ablowitz–Ladik |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 15 Oct 2019 11:36 |
Last Modified: | 17 Oct 2019 05:01 |
Status: | Published |
Publisher: | EMS Publishing House |
Refereed: | Yes |
Identification Number: | 10.4171/jems/681 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:152162 |