The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin–Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

💡 Research Summary

The paper develops a comprehensive framework for calculus on non‑commutative jet spaces, motivated by the observation that the Leibniz rule for derivations is invariant under cyclic permutations of the co‑multiples appearing as arguments. This cyclic invariance is taken as a guiding principle for constructing a new class of non‑commutative bundles. The base of each bundle is the sheaf of algebras generated by “walks’’ on a tessellated affine manifold; these walks encode directed paths on a discrete lattice and inherit a natural associative multiplication. The fibres are taken to be free associative algebras, or later quotients of such algebras by the equivalence relation that identifies elements differing only by a cyclic shift. In this way, the fibre algebra respects the same cyclic symmetry that underlies the Leibniz rule.

Having defined the bundle, the authors pass to its infinite jet prolongation J^∞(E), where E denotes the non‑commutative bundle. Jet coordinates now consist of the usual independent variables x^i, dependent fields u^α, and, crucially, non‑commutative fibre generators ξ^a together with all their derivatives ξ^a_I. The total differential d_H and the vertical differential d_V are introduced in a manner that respects the cyclic symmetry of the walk algebra; d_H acts on the base variables and on the walk algebra simultaneously, while d_V differentiates only with respect to the fibre generators.

The next major step is to embed this construction into Kontsevich’s formal non‑commutative symplectic super‑geometry. The authors define a Batalin–Vilkovisky (BV) Laplacian Δ of degree –1 on the space of local functionals over J^∞(E). Δ is built from a regularized trace on the free associative algebra and a Green’s operator that solves the non‑commutative analogue of the divergence equation. Using the cyclic invariance, the authors prove Δ² = 0, showing that all intermediate terms cancel pairwise.

Concomitantly, a Schouten bracket