[Paper Review] Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids

Andrew James Bruce|arXiv (Cornell University)|Jun 28, 2012

Homotopy and Cohomology in Algebraic Topology26 references3 citations

TL;DR

This paper introduces odd Jacobi manifolds as a Grassmann-odd analogue of classical Jacobi structures on supermanifolds, defined via an odd Schouten structure and a homological vector field satisfying conditions analogous to the classical Jacobi identity. The key contribution is showing that Jacobi algebroids arise naturally as quasi Q-manifolds—curved Q-structures—offering a new geometric framework for generalized Lie algebroids and extending the link between odd symplectic geometry and gauge theory formalisms.

ABSTRACT

In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown that Jacobi algebroids can be understood in terms of a kind of curved Q-manifold, which we will refer to as a quasi Q-manifold.

Motivation & Objective

• To develop a Grassmann-odd analogue of classical Jacobi structures on supermanifolds, filling a gap in the theory of odd geometric structures.
• To clarify the geometric nature of Jacobi algebroids by reinterpreting them as quasi Q-manifolds—curved Q-structures with specific compatibility conditions.
• To extend the understanding of Lie algebroids in the presence of a 1-cocycle by embedding them in the framework of odd Jacobi geometry.
• To provide a new perspective on generalized Lie algebroids using odd supergeometric structures, particularly in the context of graded manifolds with Z-grading.
• To lay the foundation for higher structures such as $L_∞$-algebras and non-linear Jacobi algebroids through odd Jacobi geometry.

Proposed method

• Define odd Jacobi structures on supermanifolds using a Grassmann-odd, fibre-wise polynomial of degree two in the cotangent bundle, denoted as an 'almost Schouten structure'.
• Introduce a homological vector field $Q$ of weight one that satisfies $[Q, Q] = 0$ modulo correction terms, generalizing the notion of a Q-structure.
• Establish the compatibility of the odd Schouten structure and the homological vector field via conditions analogous to the classical Jacobi identity: $[[S,S]] = 2E \wedge S$ and $\mathcal{L}_E S = 0$, where $E$ is associated with $Q$.
• Use the parity reversion functor $\Pi$ to relate the total space of $\Pi E^*$ to the dual of a vector bundle, enabling the construction of canonical double vector bundle morphisms.
• Construct a symplectomorphism $R: T^*(\Pi E^*) \to T^*(\Pi E)$ using canonical coordinates and weight assignments, proving that the morphism preserves the symplectic structure.
• Reinterpret Jacobi algebroids as quasi Q-manifolds by showing that the underlying geometric data—Schouten structure and homological vector field—satisfy the required curved Q-structure axioms.

Experimental results

Research questions

• RQ1 How can a Grassmann-odd analogue of classical Jacobi structures be consistently defined on supermanifolds?
• RQ2 What is the geometric characterization of Jacobi algebroids in terms of odd supergeometric structures such as quasi Q-manifolds?
• RQ3 Can the notion of a Lie algebroid in the presence of a 1-cocycle be naturally embedded within the framework of odd Jacobi geometry?
• RQ4 What is the role of the canonical double vector bundle morphism in relating different odd supergeometric structures on $T^*(\Pi E^*)$ and $T^*(\Pi E)$?
• RQ5 How might odd Jacobi structures generalize to non-linear or higher homotopy structures such as $L_\infty$-algebras?

Key findings

• Odd Jacobi manifolds are defined by a Grassmann-odd Schouten structure $S$ and a homological vector field $Q$ satisfying $[Q, Q] = 0$ up to a correction term involving a vector field $E$, generalizing classical Jacobi structures to the odd super setting.
• The associated odd Jacobi bracket on functions satisfies the Schouten bracket axioms but fails to be a derivation unless $E = 0$, mirroring the classical Jacobi bracket's failure of the Leibniz rule.
• Jacobi algebroids are shown to be equivalent to quasi Q-manifolds—supermanifolds with a homological vector field $Q$ and a compatible odd Schouten structure $S$ satisfying $[[S,S]] = 2E \wedge S$ and $\mathcal{L}_E S = 0$.
• The canonical double vector bundle morphism $R: T^*(\Pi E^*) \to T^*(\Pi E)$ is proven to be a symplectomorphism, preserving the symplectic structures $dp_A dx^A + d\pi^\alpha d\eta_\alpha$ and $dp_A dx^A + d\pi_\alpha d\xi^\alpha$.
• The weight and Grassmann parity assignments on coordinates—e.g., $\operatorname{w}(\eta_\alpha) = 1$, $\widetilde{\eta}_\alpha = \widetilde{\alpha} + 1$—are essential for maintaining consistency in the transformation laws and symplectic structure.
• The framework provides a natural setting for generalised Lie algebroids, showing that they arise as the odd supergeometric realizations of Jacobi algebroids via the quasi Q-structure.