Frobenius Manifolds

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

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.

The State of the Art in the Theory of Frobenius Manifolds

Frobenius Mannigfaltigkeiten - ein aktuelles Gebiet, das Algebraische Geometrie und Quanten Kohomologie verbindet, motiviert wurde es durch die Physik. Dieses Buch ist augenblicklich das einzige, das alle wichtigen Experten zusammenbringt und die verschieden thematischen Schwerpunkte zusammenstellt, es gibt einen hervorragenden Überblick über "the state of the art" dieses Forschungsgebietes.

Gauss-Manin systems, Brieskorn lattices and Frobenius structures (II).- Opposite filtrations, variations of Hodge structure, and Frobenius modules.- The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants.- Symplectic geometry of Frobenius structures.- Unfoldings of meromorphic connections and a construction of Probenius manifolds.- Discrete torsion, symmetric products and the Hubert scheme.- Relations among universal equations for Gromov-Witten invariants.- Extended modular operad.- Operads, deformation theory and F-manifolds.- Witten's top Chern class on the moduli space of higher spin curves.- Uniformization of the orbifold of a finite reflection group.- The Laplacian for a Frobenius manifold.- Virtual fundamental classes, global normal cones and Fulton's canonical classes.- A note on BPS invariants on Calabi-Yau 3-folds.- List of Participants.