# Studying the metacurvature by super calculus

To any Poisson manifold $(P,\pi)$ is associated an anchor map $\pi_\sharp:T^*P\longrightarrow TP$ given by $\beta(\pi_\sharp(\alpha))=\pi(\alpha,\beta),$ and a Lie bracket on $1$-forms $$[\alpha,\beta]= \mathscr{L}_{\pi_{\#}(\alpha)}\beta-\mathscr{L}_{\pi_{\#}(\beta)}\alpha-d(\pi(\alpha,\beta)).$$ A contravariant connection on $(P,\pi)$ is an $\mathbb{R}$-bilinear map $$\begin{array}{cccc} \mathcal{D} : &\Omega^1(P)\times\Omega^1(P)&\longrightarrow&\Omega^1(P) \cr &(\alpha,\beta)&\longmapsto&\mathcal{D}_\alpha\beta \end{array}$$ such that, for any $f\in C^\infty(M)$

$\mathcal{D}_{f\alpha}\beta=f\mathcal{D}_\alpha\beta$,

$\mathcal{D}_\alpha\left(f\beta\right)=f\mathcal{D}_\alpha\beta+\pi_\sharp(\alpha)(f)\beta$.

The torsion and the curvature of a contravariant connection ${\cal D}$ are formally identical to the usual definitions $$T(\alpha,\beta)={\cal D}_\alpha\beta-{\cal D}_\beta\alpha-[\alpha,\beta]\quad\mbox{and}\quad K(\alpha,\beta)={\cal D}_\alpha{\cal D}_\beta-{\cal D}_\beta{\cal D}_\alpha-{\cal D}_{[\alpha,\beta]}.$$ Eli Hawkins showed here that if $$0 \to \hbar\Omega^* \to \Omega^* \stackrel{\cal P}\to \Omega^*(M) \to 0$$ is a deformation of $\Omega^*(M)$, the differential graded algebra of differential forms, then we can define a generalized Poisson bracket on $\Omega^*(M)$ by $$\{\alpha,\beta\} = \mathcal{P}\left(\frac{i}{\hbar} [\hat\alpha,\hat\beta]\right).$$ This Poisson bracket defines a contravariant connection: $\{f,\alpha\}=\mathcal{D}_{df}\alpha$ and a metacurvature tensor a symetric $(2,3)$-tensor $\mathcal{M}$, given by $$\mathcal{M}(df,\alpha,\beta)=\{f,\{\alpha,\beta\}\}-\{\{f,\alpha\},\beta\}- \{\{f,\beta\},\alpha\}.$$ The generalized Poisson bracket staisfies the graded Jacobi identity if, and only if, the curvature and the metcurvature vanishe identically. My problem is to study the metacurvature tensor, for which there is no formula except for the symplectic case (Theorem 2.4). I was suggested to use technics from super calculus. I do not know yet what it is, may you suggest me some nice references? Thank you!

-
Sorry, I wasn't able to edit correctly my message. I managed to replace standard latex symbols (as \{, by \lbrace) but I don't understand why some parts are not properly compiled! –  amine Aug 11 '12 at 11:30
Thank you, fedja! –  amine Aug 11 '12 at 11:50