139 research outputs found
The construction of Frobenius manifolds from KP tau-functions
Frobenius manifolds (solutions of WDVV equations) in canonical coordinates
are determined by the system of Darboux-Egoroff equations. This system of
partial differential equations appears as a specific subset of the
-component KP hierarchy. KP representation theory and the related Sato
infinite Grassmannian are used to construct solutions of this Darboux-Egoroff
system and the related Frobenius manifolds. Finally we show that for these
solutions Dubrovin's isomonodromy tau-function can be expressed in the KP
tau-function.Comment: 29 pages, latex2e, no figure
Regular F-manifolds: initial conditions and Frobenius metrics
A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E),
such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any
p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up
to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow
T_{p}M. We obtain that any regular F-manifold admits a preferred system of
local coordinates and we find conditions, in these coordinates, for a metric to
be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular
F-manifolds. We show that any regular F-manifold is locally isomorphic to the
parameter space of a Malgrange universal connection. We prove an initial
condition theorem for Frobenius metrics on regular F-manifolds.Comment: 35 pages; with respect to the previous version, Section 4 is
reorganised; reference [17] is added; other minor correction
Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures
Submanifolds of Frobenius manifolds are studied. In particular, so-called
natural submanifolds are defined and, for semi-simple Frobenius manifolds,
classified. These carry the structure of a Frobenius algebra on each tangent
space, but will, in general, be curved. The induced curvature is studied, a
main result being that these natural submanifolds carry a induced pencil of
compatible metrics. It is then shown how one may constrain the bi-Hamiltonian
hierarchies associated to a Frobenius manifold to live on these natural
submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.Comment: 27 Pages, LaTeX, 1 figur
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