Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?

Abstract

From the paper “Formality Conjecture” (Ascona 1996):I am aware of only one such a class, it corresponds to simplest good graph, the complete graph with 4 vertices (and 6 edges). This class gives a remarkable vector field on the space of bi-vector fields on ℝd. The evolution with respect to the time t is described by the following non-linear partial differential equation: …, where α = ∑i,j αij∂ / ∂ xi ∧ ∂ / ∂ xj is a bi-vector field on ℝd.It follows from general properties of cohomology that 1) this evolution preserves the class of (real-analytic) Poisson structures, …In fact, I cheated a little bit. In the formula for the vector field on the space of bivector fields which one get from the tetrahedron graph, an additional term is present. … It is possible to prove formally that if α is a Poisson bracket, i.e. if [α, α] = 0 ∈ T2(ℝd), then the additional term shown above vanishes.By using twelve Poisson structures with high-degree polynomial coefficients as explicit counter-examples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at Poisson bi-vectors. The counterexamples at hand suggest a correction to the formula for the “exotic” flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance 1 : 6 for which the flow does preserve the space of Poisson bi-vectors.