Integrable hierarchies and F-manifolds with compatible connection

Abstract

Building on the interplay between geometry and integrability, we show that F-manifolds with compatible connection $(\nabla,\circ,e)$ are the geometric counterpart of integrable systems of quasilinear first order evolutionary PDEs. We consider F-manifolds equipped with an Euler vector field and assume that the operator $L=E\circ$ is regular. This generalises previous results in the semisimple context. As an example we study regular F-manifolds with compatible connection $(\nabla,\circ,e,E)$ associated with integrable hierarchies obtained from the solutions of the equation $d\cdot d_L \,a_0=0$ by applying the construction of [27]. We show that $n$-dimensional F-manifolds associated to operators $L$ with $r\le n$ Jordan blocks $L_\alpha$ of size $m_\alpha$ are classified by $n$ arbitrary functions of a single variable, where each block $L_\alpha$ contributes with $m_\alpha$ functions of the variable appearing in the diagonal of the block. In the case of a single Jordan block of arbitrary size we show that flat connections $\nabla$ correspond to linear solutions $a_0$. This generalises part of the construction of [31] where special linear solutions were considered. We illustrate the construction in dimensions $2,3,$ and $4$ for any choice of Jordan canonical form and any choice of the corresponding solution $a_0$. In these dimensions we have that linear solutions define bi-flat F-manifolds, and that the special linear solutions studied in [31] are related to Riemannian F-manifolds with Killing unit vector field. We conjecture that this is true in general.