Khudaverdian H. Homotopy Poisson brackets and thick morphisms (abstract)

Speaker: Hovhannes Khudaverdian

Title: Homotopy Poisson brackets and thick morphisms

Abstract:
For an arbitrary manifold $M$, consider the supermanifolds $\mathrm{\Pi }TM$ and $\mathrm{\Pi }{T}^{*}M$, where $\mathrm{\Pi }$ is the parity reversion functor. The supermanifold $\mathrm{\Pi }TM$ has an odd vector field that can be identified with the de Rham differential $d$; functions on it can be identified with differential forms on $M$. The supermanifold $\mathrm{\Pi }{T}^{*}M$ has a canonical odd Poisson bracket $[,]$ (the antibracket); functions on it can be identified with multivector fields on $M$. An arbitrary even function $P$ on $\mathrm{\Pi }{T}^{*}M$ which obeys the master equation $[P,P]=0$ defines an even homotopy Poisson structure on the manifold $M$ and an odd homotopy Poisson structure (the ``higher Koszul brackets") on differential forms on $M$.

We construct a non-linear transformation from differential forms endowed with the higher Koszul brackets to multivector fields considered with the antibracket by using the new notion of a thick morphism of supermanifolds, a notion recently introduced.

Based on joint work with Th. Voronov.

Event: Local and Nonlocal Geometry of PDEs and Integrability, 8-12 October 2018, SISSA, Trieste, Italy.
The conference in honor of Joseph Krasil'shchik's 70th birthday.