Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants

Abstract

Nambu-determinant brackets on $R^d\ni x=(x^1,...,x^d)$, $\{f,g\}_d(x)=\rho(x) \det(\partial(f,g,a_1,...,a_{d-2})/\partial(x^1,...,x^d))$, with $a_i\in C^\infty(R^d)$ and $\rho\partial_x\in\mathfrak{X}^d(R^d)$, are a class of Poisson structures with (non)linear coefficients, e.g., polynomials of arbitrarily high degree. With good cocycles in the graph complex, Kontsevich associated universal -- for all Poisson bivectors $P$ on affine $R^d_{aff}$ -- elements $\dot{P}=Q^\gamma(P)\in H^2_{P}(R^d_{aff})$ in the Lichnerowicz-Poisson second cohomology groups; we note that known graph cocycles $\gamma$ preserve the Nambu-Poisson class $\{P(\rho,a)\}$, and we express, directly from $\gamma$, the evolution $\dot{\rho}$,$\dot{a}$ that induces $\dot{P}$. Over all $d\geq2$ at once, there is no universal mechanism for the bivector cocycles $Q^\gamma_d$ to be trivial, $Q^\gamma_d=[\![P,\vec{X}^\gamma_d(P)]\!]$, w.r.t. vector fields defined uniformly for all dimensions $d$ by the same graph formula. While over $R^2$, the graph flows $\dot{P} = Q^{\gamma_i}_{2D}(P(\rho))$ for $\gamma\in\{\gamma_3,\gamma_5,\gamma_7,...\}$ are trivialized by vector fields $\vec{X}^{\gamma_i}_{2D}=(dx\wedge dy)^{-1}d_{dR}(Ham^{\gamma_i}(P))$ of peculiar shape, we detect that in $d\geq3$, the 1-vectors from 2D, now with $P(\rho,a_1,...,a_{d-2})$ inside, do not solve the problems $Q^{\gamma_i}_{d\geq3}=[\![P,{\vec{X}^{\gamma_i}_{d\geq3}}(P(\rho,a))]\!]$, yet they do yield good Ansatz where we find solutions $\vec{X}^{\gamma_i}_{d=3,4}(P(\rho,a))$. In the study of the step $d\mapsto d+1$, by adapting the Kontsevich graph calculus to the Nambu-Poisson class of brackets, we discover more identities for the Jacobian determinants within $P(\rho,a)$, i.e. for multivector-valued $GL(d)$-invariants on $R^d_{aff}$.