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A Poisson bracket on the space of Poisson structures

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 نشر من قبل Thomas Machon
 تاريخ النشر 2020
  مجال البحث فيزياء
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 تأليف Thomas Machon




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Let $M$ be a smooth closed orientable manifold and $mathcal{P}(M)$ the space of Poisson structures on $M$. We construct a Poisson bracket on $mathcal{P}(M)$ depending on a choice of volume form. The Hamiltonian flow of the bracket acts on $mathcal{P}(M)$ by volume-preserving diffeomorphism of $M$. We then define an invariant of a Poisson structure that describes fixed points of the flow equation and compute it for regular Poisson 3-manifolds, where it detects unimodularity. For unimodular Poisson structures we define a further, related Poisson bracket and show that for symplectic structures the associated invariant counting fixed points of the flow equation is given in terms of the $d d^Lambda$ and $d+ d^Lambda$ symplectic cohomology groups defined by Tseng and Yau.



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