Kontsevich graphs act on Nambu–Poisson brackets, IV. When the invisible becomes crucial

Abstract Kontsevich’s graphs allow encoding multi-vectors whose coefficients are differential- polynomial in the coefficients of a given Poisson bracket on an affine real manifold. Encoding formulas by directed graphs adapts to the class of Nambu-determinant Poisson brackets, yet the graph topology becomes dimension-specific. To inspect whether a given Kontsevich graph cocycle γ acts (non)trivially – in the second Poisson cohomology – on the space of Nambu brackets, taking a vector field solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mover accent="true"> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mi>d</mml:mi> <mml:mi>γ</mml:mi> </mml:msubsup> </mml:math> from dimension d does not work in d + 1. For 2 ≤ d ≤ 4, the action of tetrahedron γ 3 on Nambu brackets is known to be a Poisson coboundary, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mi>P</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mo>〚</mml:mo> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mrow> <mml:mover accent="true"> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> <mml:mrow> <mml:mi>d</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>〛</mml:mo> </mml:mrow> </mml:math> . We explore which minimal (sub)sets of graphs, encoding (non)vanishing objects over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>ℝ</mml:mi> <mml:mrow> <mml:mi>a</mml:mi> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msubsup> </mml:math> , generate the topological data that suffice for a solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mover accent="true"> <mml:mi>X</mml:mi> <mml:mo>→</mml:mo> </mml:mover> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> to appear. We detect that there can be no solution in higher dimension without invisible graphs that vanish as formulas in d = 3, but whose descendants do not all vanish over d = 4.