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 $P=[\![ \varrho(\boldsymbol{x})\,\partial_x\wedge\partial_y\wedge\partial_z,a]\!]$ on $\mathbb{R}^3\ni\boldsymbol{x}=(x,y,z)$ and $P=[\![ [\![\varrho(\boldsymbol{y})\,\partial_{x^1}\wedge\ldots\wedge\partial_{x^4},a_1]\!],a_2]\!]$ on $\mathbb{R}^4\ni\boldsymbol{y}$, 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 $\mathbf{a} = (a_1,\ldots,a_{d-2})$ 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 $\varrho(\mathbf{x}') = \varrho(\mathbf{x}) \cdot \det \| \partial \mathbf{x}' / \partial \mathbf{x} \|$ of the inverse density $\varrho$ under reparametrizations $\mathbf{x} \rightleftarrows \mathbf{x}'$, we discover another, so far hidden discrete symmetry in the construction of these evolution equations.