Let us quickly recall what I mean by a QS and an odd Jacobi manifold.
Definition A supermanifold equipped with a Schouten structure S and a homological vector field Q such that
\(\{ S, \mathcal{Q} \} =0 \),
where \(\mathcal{Q}\) is the symbol of the homological vector field is said to be a QS-manifold.
This definition allows us to write everything in terms of an odd function quadratic in momenta and an odd function linear in momenta, ie. functions on the total space of the cotangent bundle of our supermanifold. The bracket in the above is the canonical Poisson bracket. (The example I will give will make this clearer.)
Definition A supermanifold equipped with an almost Schouten structure S and a homological vector field Q such that
\(\{ S, S \} ={-} 2 \mathcal{Q} S\),
\(\{ S,\mathcal{Q} \} =0\),
where \(\mathcal{Q}\) is the symbol of the homological vector field is said to be an odd Jacobi manifold.
Both these species of supermanifold are very similar. QS-manifolds have a genuine Schouten structure, that is an odd function quadratic in momenta such that it Poisson self-commutes and Poisson commutes with the symbol of the homological vector field. An odd Jacobi manifold consists of an almost Schouten structure that has a very specific Poisson self-commutator and Poisson commutes with the symbol of the homological vector field.
On to our example…
Consider the supermanifold \(\mathbb{R}^{1|1}\), which we equip with local coordinates \((t, \xi)\). Here \(t\) is the commuting coordinate and \(\xi \) is the anticommuting coordinate. This supermanifold comes equipped with a canonical Schouten structure
\(S = {-}\pi p\),
where we employ fibre coordinates \((p, \pi)\) on the cotangent bundle. As the above structure does not contain conjugate variables is it cleat that
\(\{S,S \}=0\).
We can go a little further than this as we also have a canonical homological vector field, which indeed gives rise to a symbol that Poisson commutes with the Schouten structure:
\(\mathcal{Q} = {-}\pi\).
So \(\mathbb{R}^{1|1}\) is a QS-manifold, canonically. The associated Schouten bracket is given by
\([f,g]_{S} = ({-}1)^{\widetilde{f}}\frac{\partial f}{\partial \xi} \frac{\partial g}{\partial t} {-} \frac{\partial f}{\partial t}\frac{\partial g}{\partial \xi}\),
for all \(f,g \in C^{\infty}(\mathbb{R}^{1|1})\).
Interestingly, we can also consider these structures as being odd Jacobi. Explicitly one can calculate the Poisson self-commutator of the Schouten structure and arrive at
\(\{ S, S\} = {-} 2 \left( {-} \pi\right)\left( {-}\pi p\right)\),
which is of course zero as \(\pi^{2}=0\). But also notice that this defines an odd Jacobi structure! We then can assign an odd Jacobi bracket as
\([f,g]_{J} = ({-}1)^{\widetilde{f}}\frac{\partial f}{\partial \xi} \frac{\partial g}{\partial t} {-} \frac{\partial f}{\partial t}\frac{\partial g}{\partial \xi}{-}({-}1)^{\widetilde{f}}\left( \frac{\partial f}{\partial \xi}\right)g {-}f\left( \frac{\partial g}{\partial \xi}\right) \).
The Schouten bracket satisfies a strict Leibniz rule as where the odd Jacobi bracket does not, we have an “anomaly” term in the derivation property. Both satisfy the appropriate graded version of the Jacobi identity.
Interestingly, the Schouten structure on \(\mathbb{R}^{1|1}\) is in fact non-degenerate so we have an odd symplectic supermanifold. One can also consider \(\mathbb{R}^{1|1}\) as an even contact manifold, but I will delay talking about that for now.
One could of course “compactify” \(\mathbb{R}\) and consider the supercircle \(\mathbb{S}^{1|1}\), and this naturally also can be considered as QS and odd Jacobi. Again we have a natural contact structure here and this has been studied in relation to super versions of the Schwarzian derivative. This is really another story…
More details can be found in an older post of mine here. A preprint about odd Jacobi structures can be found on the arXiv here.