Frobenius manifolds (Notes by David Calderbank). (English) Zbl 0867.53027

Hurtubise, Jacques (ed.) et al., Gauge theory and symplectic geometry. Proceedings of the NATO Advanced Study Institute and séminaire de mathématiques supérieures, Montréal, Canada, July 3–14, 1995. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 488, 69-112 (1997).
In these lectures, some themes in the work of Boris Dubrovin on Frobenius manifolds are discussed. The author focusses principally on those aspects that have a symplectic flavour, including Hamiltonian flows on coadjoint orbits, Poisson structures on loop spaces, and the symplectic geometry of flat connections on a punctured sphere. A major theme is to study the problem of solving the differential equations for a Frobenius manifold. These are nonlinear equations that appear in disguise in many other branches of mathematics. The author shows how to reformulate the equations in terms of the problem of determining flat connections on surfaces with given holonomy, the classical subject of isomonodromic deformations.
For the entire collection see [Zbl 0861.00016].
Reviewer: S.Nenov (Sofia)


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems