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

Ricardo Buring, Arthemy V. Kiselev

Abstract

In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density $\varrho$ times Levi-Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi-Civita symbol${}\times\varrho$. Using this micro-graph calculus, we show that Kontsevich's tetrahedral $γ_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\smash{\mathbb{R}^3}$ using micro-graphs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $γ_3$-flow over $\smash{\mathbb{R}^2}$.

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

Abstract

In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density times Levi-Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi-Civita symbol. Using this micro-graph calculus, we show that Kontsevich's tetrahedral -flow on the space of Nambu-determinant Poisson brackets over is a Poisson coboundary: we realize the trivializing vector field over using micro-graphs. This projects to the known trivializing vector field for the -flow over .
Paper Structure (3 sections, 9 theorems, 8 equations)

This paper contains 3 sections, 9 theorems, 8 equations.

Table of Contents

  1. Preliminaries: the tetrahedral flow ${\dot{P}=Q_{\gamma_3}(P)}$ over twofolds
  2. Nambu-determinant Poisson brackets: the $\gamma_3$-flow over $\mathbb{R}^3$
  3. (Non)triviality of $\gamma_3$-flow for Nambu--Poisson brackets on $\mathbb{R}^4$

Key Result

Proposition 1

1. In dimension $d=2$ (where every bi-vector $P=\varrho(x,y)\cdot(\partial_x\otimes$$\partial_y - \partial_y\otimes\partial_x)$ is Poisson,For the same reason, the Poisson condition -- trivial in $d=2$ -- is preserved by any Kontsevich bi-vector graph (not necessarily obtained from a cocycle, on $n with respect to the class of $1$-vector ${\vec{X}} = \partial_j \bigl( \partial_k \partial_m(P^{ij}

Theorems & Definitions (27)

  • Example 1: Ascona96 and f16
  • Proposition 1: f16
  • proof
  • Remark 1
  • Remark 2: on $GL(\infty)$-invariants parent to $GL(d)$-invariants
  • Definition 1
  • Remark 3
  • Theorem 2: skew21
  • Definition 2
  • Example 2
  • ...and 17 more