Integrable hierarchies and F-manifolds with compatible connection
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910593162674176 |
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| author | Lorenzoni, Paolo Perletti, Sara van Gemst, Karoline |
| author_facet | Lorenzoni, Paolo Perletti, Sara van Gemst, Karoline |
| contents | 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 $L_α$ of size $m_α$ are classified by $n$ arbitrary functions of a single variable, where each block $L_α$ contributes with $m_α$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02585 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integrable hierarchies and F-manifolds with compatible connection Lorenzoni, Paolo Perletti, Sara van Gemst, Karoline Mathematical Physics Differential Geometry 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 $L_α$ of size $m_α$ are classified by $n$ arbitrary functions of a single variable, where each block $L_α$ contributes with $m_α$ 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. |
| title | Integrable hierarchies and F-manifolds with compatible connection |
| topic | Mathematical Physics Differential Geometry |
| url | https://arxiv.org/abs/2408.02585 |