The hidden symmetry of Kontsevich's graph flows on the spaces of Nambu-determinant Poisson brackets

Abstract

Kontsevich's graph flows are -- universally for all finite-dimensional affine Poisson manifolds -- infinitesimal symmetries of the spaces of Poisson brackets. We show that the previously known tetrahedral flow and the recently obtained pentagon-wheel flow preserve the class of Nambu-determinant Poisson bi-vectors $P=[[\varrho (\mathbf{x}){\partial }_x\wedge {\partial }_y\wedge {\partial }_z,a]]$ on ${\mathbb{R}}^3\ni \mathbf{x}=(x,y,z)$ and $P=[[[[\varrho (\mathbf{y}){\partial }_{x^1}\wedge \dots \wedge {\partial }_{x^4},a_1]],a_2]]$ on ${\mathbb{R}}^4\ni \mathbf{y}$, including the general case $\varrho \not\equiv 1$. We detect that the Poisson bracket evolution $\dot{P}=Q_{\gamma }(P^{{\otimes }^{\#Vert(\gamma )}})$ is trivial in the second Poisson cohomology, $Q_{\gamma }=[[P,\vec{X}([\varrho ],[a])]]$, for the Nambu-determinant bi-vectors $P(\varrho ,[a])$ on ${\mathbb{R}}^3$. For the global Casimirs $\mathbf{a}=(a_1,\dots ,a_{d-2})$ and inverse density $\varrho$ on ${\mathbb{R}}^d$, we analyse the combinatorics of their evolution induced by the Kontsevich graph flows, namely $\dot{\varrho }=\dot{\varrho }([\varrho ],[\mathbf{a}])$ and $\dot{\mathbf{a}}=\dot{\mathbf{a}}([\varrho ],[\mathbf{a}])$ with differential-polynomial right-hand sides. Besides the anticipated collapse of these formulas by using the Civita symbols (three for the tetrahedron ${\gamma }_3$ and five for the pentagon-wheel graph cocycle ${\gamma }_5$), as dictated by the behaviour $\varrho ({\mathbf{x}}^{\prime })=\varrho (\mathbf{x})\cdot \det \Vert \partial {\mathbf{x}}^{\prime }/\partial \mathbf{x}\Vert$ of the inverse density $\varrho$ under reparametrizations $\mathbf{x}\rightleftarrows {\mathbf{x}}^{\prime }$, we discover another, so far hidden discrete symmetry in the construction of these evolution equations.

arXiv commentPublished version; 27+iii pages, 1 figure, 2 tables, 8 research problems; Keywords: Poisson geometry, Nambu-determinant Poisson bracket, Poisson cohomology, symmetry, Kontsevich's graph complex

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Cited by in corpus (1)

• The tower of Kontsevich deformations for Nambu-Poisson structures on ${\mathbb{R}}^d$: dimension-specific micro-graph calculus