Ebook: Frobenius Manifolds: Quantum Cohomology and Singularities

Frobenius manifolds are complex manifolds with a multiplication and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in 1991 by Dubrovin, motivated by physics results. Another source of Frobenius manifolds is singularity theory. Duality between string theories lies behind the phenomenon of mirror symmetry. One mathematical formulation can be given in terms of the isomorphism of certain Frobenius manifolds. A third source of Frobenius manifolds is given by integrable systems, more precisely, bihamiltonian hierarchies of evolutionary PDE's. As in the case of quantum cohomology, here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be determined implicitly by the underlying Frobenius manifolds. Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems are flourishing areas since the early 90's. An activity was organized at the Max-Planck-Institute for Mathematics in 2002, with the purpose of bringing together the main experts in these areas. This volume originates from this activity and presents the state of the art in the subject.

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Frobenius manifolds are complex manifolds with a multiplication and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in 1991 by Dubrovin, motivated by physics results. Another source of Frobenius manifolds is singularity theory. Duality between string theories lies behind the phenomenon of mirror symmetry. One mathematical formulation can be given in terms of the isomorphism of certain Frobenius manifolds. A third source of Frobenius manifolds is given by integrable systems, more precisely, bihamiltonian hierarchies of evolutionary PDE's. As in the case of quantum cohomology, here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be determined implicitly by the underlying Frobenius manifolds. Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems are flourishing areas since the early 90's. An activity was organized at the Max-Planck-Institute for Mathematics in 2002, with the purpose of bringing together the main experts in these areas. This volume originates from this activity and presents the state of the art in the subject.

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Frobenius manifolds are complex manifolds with a multiplication and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in 1991 by Dubrovin, motivated by physics results. Another source of Frobenius manifolds is singularity theory. Duality between string theories lies behind the phenomenon of mirror symmetry. One mathematical formulation can be given in terms of the isomorphism of certain Frobenius manifolds. A third source of Frobenius manifolds is given by integrable systems, more precisely, bihamiltonian hierarchies of evolutionary PDE's. As in the case of quantum cohomology, here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be determined implicitly by the underlying Frobenius manifolds. Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems are flourishing areas since the early 90's. An activity was organized at the Max-Planck-Institute for Mathematics in 2002, with the purpose of bringing together the main experts in these areas. This volume originates from this activity and presents the state of the art in the subject.
Content:
Front Matter....Pages i-xii
Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)....Pages 1-18
Opposite filtrations, variations of Hodge structure, and Frobenius modules....Pages 19-43
The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants....Pages 45-89
Symplectic geometry of Frobenius structures....Pages 91-112
Unfoldings of meromorphic connections and a construction of Frobenius manifolds....Pages 113-144
Discrete torsion, symmetric products and the Hubert scheme....Pages 145-167
Relations among universal equations for Gromov-Witten invariants....Pages 169-180
Extended Modular Operad....Pages 181-211
Operads, deformation theory and F-manifolds....Pages 213-251
Witten’s top Chern class on the moduli space of higher spin curves....Pages 253-264
Uniformization of the orbifold of a finite reflection group....Pages 265-320
The Laplacian for a Frobenius manifold....Pages 321-339
Virtual fundamental classes, global normal cones and Fulton’s canonical classes....Pages 341-358
A Note on BPS Invariants on Calabi-Yau 3-folds....Pages 359-375
Back Matter....Pages 377-379

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Frobenius manifolds are complex manifolds with a multiplication and a metric on the holomorphic tangent bundle, which satisfy several natural conditions. This notion was defined in 1991 by Dubrovin, motivated by physics results. Another source of Frobenius manifolds is singularity theory. Duality between string theories lies behind the phenomenon of mirror symmetry. One mathematical formulation can be given in terms of the isomorphism of certain Frobenius manifolds. A third source of Frobenius manifolds is given by integrable systems, more precisely, bihamiltonian hierarchies of evolutionary PDE's. As in the case of quantum cohomology, here Frobenius manifolds are part of an a priori much richer structure, which, because of strong constraints, can be determined implicitly by the underlying Frobenius manifolds. Quantum cohomology, the theory of Frobenius manifolds and the relations to integrable systems are flourishing areas since the early 90's. An activity was organized at the Max-Planck-Institute for Mathematics in 2002, with the purpose of bringing together the main experts in these areas. This volume originates from this activity and presents the state of the art in the subject.
Content:
Front Matter....Pages i-xii
Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)....Pages 1-18
Opposite filtrations, variations of Hodge structure, and Frobenius modules....Pages 19-43
The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants....Pages 45-89
Symplectic geometry of Frobenius structures....Pages 91-112
Unfoldings of meromorphic connections and a construction of Frobenius manifolds....Pages 113-144
Discrete torsion, symmetric products and the Hubert scheme....Pages 145-167
Relations among universal equations for Gromov-Witten invariants....Pages 169-180
Extended Modular Operad....Pages 181-211
Operads, deformation theory and F-manifolds....Pages 213-251
Witten’s top Chern class on the moduli space of higher spin curves....Pages 253-264
Uniformization of the orbifold of a finite reflection group....Pages 265-320
The Laplacian for a Frobenius manifold....Pages 321-339
Virtual fundamental classes, global normal cones and Fulton’s canonical classes....Pages 341-358
A Note on BPS Invariants on Calabi-Yau 3-folds....Pages 359-375
Back Matter....Pages 377-379
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