For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov–Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one- dimensional Frobenius manifold the Hodge hierarchy is an integrable deformation of the Korteweg–de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.

Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs / Dubrovin, Boris; Liu, Si Qi; Yang, Di; Zhang, Y.. - In: ADVANCES IN MATHEMATICS. - ISSN 0001-8708. - 293:April(2016), pp. 382-435. [10.1016/j.aim.2016.01.018]

Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

Dubrovin, Boris;
2016-01-01

Abstract

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov–Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one- dimensional Frobenius manifold the Hodge hierarchy is an integrable deformation of the Korteweg–de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.
2016
293
April
382
435
https://arxiv.org/abs/1409.4616
Dubrovin, Boris; Liu, Si Qi; Yang, Di; Zhang, Y.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/11957
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