Supertransvectants and symplectic geometry

TL;DR

This paper explores supertransvectants, invariant bilinear operations on weighted densities on the supercircle, showing their relation to Poisson brackets and constructing star-products, thus advancing the understanding of symplectic geometry in supergeometry.

Contribution

It establishes the equivalence of supertransvectants with iterated Poisson brackets and constructs star-products involving these operations.

Findings

Supertransvectants coincide with iterated Poisson and ghost Poisson brackets.

Construction of star-products using supertransvectants.

Enhanced understanding of symplectic structures in supergeometry.

Abstract

We consider the $osp(1|2)$-invariant bilinear operations on weighted densities on the supercircle $S^{1|1}$ called the supertransvectants. These operations are analogues of the famous Gordan transvectants (or Rankin-Cohen brackets). We prove that these operations coincide with the iterated Poisson and ghost Poisson brackets on ${\mathbb R}^{2|1}$ and apply this result to construct star-products involving the supertransvectants.