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arXiv · 1207.5731

Reciprocal $F$-manifolds

Alessandro Arsie·Paolo Lorenzoni
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Abstract

We consider the action of a special class of reciprocal transformation on the principal hierarchy associated to a semisimple $F$-manifold with compatible flat structure $(M,\circ,\nabla,e)$. Under some additional assumptions, the hierarchy obtained applying these reciprocal transformations is also associated to an $F$-manifold with compatible flat structure that we call reciprocal $F$-manifold. We also consider the special case of bi-flat $F$-manifolds $(M,\circ,*,\nabla^{(1)},\nabla^{(2)},e,E)$ and we study reciprocal transformations preserving flatness of both the connections $\nabla^{(1)}$ and $\nabla^{(2)}$ and how they act on corresponding solutions of an augmented Darboux-Egorov system.

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Alessandro Arsie, Paolo Lorenzoni. 2012-07-24. Reciprocal $F$-manifolds. https://doi.org/10.1016/j.geomphys.2013.03.029

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