Dubrovin's duality for $F$-manifolds with eventual identities

Details Dubrovin's duality for $F$-manifolds with eventual identities Dubrovin's duality for $F$-manifolds with eventual identities

2010-06-03

arxiv.org/abs/1006.0652v1

Mathematics

Differential Geometry

Scientific paper

A vector field E on an F-manifold (M, o, e) is an eventual identity if it is invertible and the multiplication X*Y := X o Y o E^{-1} defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin. We develop a duality between F-manifolds with eventual identities and we show that is compatible with the local irreducible decomposition of F-manifolds and preserves the class of Riemannian F-manifolds. We find necessary and sufficient conditions on the eventual identity which insure that harmonic Higgs bundles and DChk-structures are preserved by our duality. We use eventual identities to construct compatible pair of metrics.

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