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

Abstract

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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Cited by in corpus (10)

• $F$-manifolds, multi-flat structures and Painlevé transcendents
• From Darboux-Egorov system to bi-flat $F$-manifolds
• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains
• Some generalizations of the variety of transposed Poisson algebras
• Symmetries of F-manifolds with eventual identities and special families of connections
• Grothendieck-Verdier duality patterns in quantum algebra
• Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials
• Regular F -manifolds with eventual identities
• Hermitian metrics on F-manifolds
• (TE)-structures over the irreducible 2-dimensional globally nilpotent F-manifold germ