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A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle $$T^3M$$ of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle $$T^2A$$ for a vector bundle A using a connection in A to define a grid in $$T^2A$$ . In this case the curvature emerges from the warp theorem.

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n + 1. In previous work with Baez and Hoffnung, we described how the ‘higher analogs’ of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L ∞-algebras: graded vector spaces equipped with a collection of skew-symmetric multi-brackets that satisfy a generalized Jacobi identity. Here, we generalize our previous result. Given an n-plectic manifold, we explicitly construct a corresponding Lie n-algebra on a complex consisting of differential forms whose multi-brackets are specified by the n-plectic structure. We also show that any n-plectic manifold gives rise to another kind of algebraic structure known as a differential graded Leibniz algebra. We conclude by describing the similarities between these two structures within the context of an open problem in the theory of strongly homotopy algebras. We also mention a possible connection with the work of Barnich, Fulp, Lada, and Stasheff on the Gelfand–Dickey–Dorfman formalism.

It is proved that a special case of Feynman identity is a consequence of the generalized Witt identity of some Lie algebra.

We explain how to use a certain new ‘Jacobi identity’ for vertex operator algebras, announced in a previous paper (math.QA/9909178), to interpret and generalize S. Bloch's recent work relating values of the Riemann zeta function at negative integers to a certain Lie algebra of operators.

The Hamiltonian formalism for evolution equations and, in particular, the Olver method for the verification of the Jacobi identity, are extended to systems of coupled bosonic and fermionic classical fields. A superspace Hamiltonian formulation is also presented.