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Frobenius Manifolds

Quantum Cohomology and Singularities

  • Book
  • © 2004

Overview

Editors:
  1. Klaus Hertling

    1. Institut für Mathematik A 5, 6, Universität Mannheim, Mannheim, Germany

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  2. Matilde Marcolli

    1. Max-Planck-Institut für Mathematik, Bonn, Germany

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  • Aktuelles aus der mathematischen Forschung

Part of the book series: Aspects of Mathematics (ASMA, volume 36)

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About this book

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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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xii
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  2. Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II)

    • Antoine Douai, Claude Sabbah
    Pages 1-18

  3. Opposite filtrations, variations of Hodge structure, and Frobenius modules

    • Javier Fernandez, Gregory Pearlstein
    Pages 19-43

  4. The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants

    • Ezra Getzler
    Pages 45-89

  5. Symplectic geometry of Frobenius structures

    • Alexander B. Givental
    Pages 91-112

  6. Unfoldings of meromorphic connections and a construction of Frobenius manifolds

    • Claus Hertling, Yuri Manin
    Pages 113-144

  7. Discrete torsion, symmetric products and the Hubert scheme

    • Ralph M. Kaufmann
    Pages 145-167

  8. Relations among universal equations for Gromov-Witten invariants

    • Xiaobo Liu
    Pages 169-180

  9. Extended Modular Operad

    • A. Losev, Yu Manin
    Pages 181-211

  10. Operads, deformation theory and F-manifolds

    • S. A. Merkulov
    Pages 213-251

  11. Witten’s top Chern class on the moduli space of higher spin curves

    • Alexander Polishchuk
    Pages 253-264

  12. Uniformization of the orbifold of a finite reflection group

    • Kyoji Saito
    Pages 265-320

  13. The Laplacian for a Frobenius manifold

    • Ikuo Satake
    Pages 321-339

  14. Virtual fundamental classes, global normal cones and Fulton’s canonical classes

    • Bernd Siebert
    Pages 341-358

  15. A Note on BPS Invariants on Calabi-Yau 3-folds

    • Atsushi Takahashi
    Pages 359-375

  16. Back Matter

    Pages 377-379
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Editors and Affiliations

  • Institut für Mathematik A 5, 6, Universität Mannheim, Mannheim, Germany

    Klaus Hertling

  • Max-Planck-Institut für Mathematik, Bonn, Germany

    Matilde Marcolli

About the editors

Prof. Dr. Claus Hertling, Institut für Mathematik, Universität Mannheim, Germany
Prof. Dr. Matilde Marcolli, Max-Planck-Institute for Mathematics, Bonn, Germany

Bibliographic Information

  • Book Title: Frobenius Manifolds

  • Book Subtitle: Quantum Cohomology and Singularities

  • Editors: Klaus Hertling, Matilde Marcolli

  • Series Title: Aspects of Mathematics

  • DOI: https://doi.org/10.1007/978-3-322-80236-1

  • Publisher: Vieweg+Teubner Verlag Wiesbaden

  • eBook Packages: Springer Book Archive

  • Copyright Information: Friedr. Vieweg & Sohn Verlag/GWV Fachverlage GmbH, Wiesbaden 2004

  • Softcover ISBN: 978-3-322-80238-5Published: 10 January 2012

  • eBook ISBN: 978-3-322-80236-1Published: 06 December 2012

  • Series ISSN: 0179-2156

  • Edition Number: 1

  • Number of Pages: XII, 378

  • Topics: Differential Geometry, Geometry

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