Buring R.¹, Kiselev A.² (¹Mathematical Institute, Johannes Gutenberg University of Mainz, Germany; ²University of Groningen, The Netherlands; IHES, France)

The tower of Kontsevich deformations for Nambu-Poisson structures on R^d: dimension-specific micro-graph calculus

Abstract: In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by copies of a given Poisson structure; in turn, the Nambu-determinant Poisson brackets themselves are differentialpolynomial in the Casimir(s) and a density $\varrho$ times the Civita symbol. We now resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Civita symbol (times $\varrho$). Using this micro-graph calculus, we establish that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we obtain a micro-graph realization $X^\gamma$ of the trivializing vector field $\vec{X}$ over $\mathbb{R}^3$. This $\vec{X}$ does project to the known trivializing vector field for the $\gamma_3$-flow over $\mathbb{R}^2$. We conjecture that over all $\mathbb{R}^{d\geqslant 3}$, Kontsevich's $\gamma_3$-flows of Nambu-Poisson brackets are coboundaries; the trivializing vector fields then project down under $\mathbb{R}^{d} \to \mathbb{R}^{d-1}$.