Linearization of Virasoro symmetries associated with semisimple Frobenius manifolds.(English)Zbl 07966475
Summary: For any semisimple Frobenius manifold, we prove that a tau-symmetric bihamiltonian deformation of its Principal Hierarchy admits an infinite family of linearizable Virasoro symmetries if and only if all the central invariants of the corresponding deformation of the bihamiltonian structure are equal to \(\frac{1}{24}\). As an important application of this result, we prove that the Dubrovin-Zhang hierarchy associated with the semisimple Frobenius manifold possesses a bihamiltonian structure which can be represented in terms of differential polynomials.
MSC:
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 37K30 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures |
Keywords:
Frobenius manifold;principal hierarchy;bihamiltonian structure;Virasoro symmetry;loop equationCite
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