The right-hand side of the Jacobi identity: to be naught or not to be?

TL;DR

This paper investigates the conditions under which the variational Schouten bracket satisfies the Jacobi identity, clarifying the role of the BV-Laplacian and introducing a geometric calculus approach to address analytic difficulties in gauge theory quantization.

Contribution

It demonstrates that the graded Jacobi identity for the variational Schouten bracket holds when using Gel'fand's calculus, contrasting with previous methods that failed to ensure the BV-Laplacian's derivation property.

Findings

The Jacobi identity holds with Gel'fand's calculus approach.

Old methods can produce non-zero cohomology representatives.

Counterexample shows failure of BV-Laplacian as a derivation without Gel'fand's calculus.

Abstract

The geometric approach [1312.1262] to iterated variations of local functionals -- e.g., of the (master-)action functional -- resulted in an extension of the deformation quantisation technique to the set-up of Poisson models of field theory [IHES/M/15/13]. It also allowed of a rigorous proof ([1312.1262],[1210.0726]) for the main inter-relations between the Batalin-Vilkovisky (BV) Laplacian $\Delta$ and variational Schouten bracket. The ad hoc use of these relations had been a known analytic difficulty in the BV-formalism for quantisation of gauge systems; now achieved, the proof does actually not require the assumption of graded-commutativity [1210.0726]. Explained in our previous work, geometry's self-regularisation is rendered by Gel'fand's calculus of singular linear integral operators supported on the diagonal. We now illustrate that analytic technique by inspecting the validity mechanism [1312.4140] for the graded Jacobi identity which the variational Schouten bracket does satisfy (whence $\Delta^2=0$, i.e., the BV-Laplacian is a differential acting in the algebra of local functionals). By using one tuple of three variational multi-vectors twice, we contrast the new logic of iterated variations -- when the right-hand side of Jacobi's identity vanishes altogether -- with the old method: interlacing its steps and stops, it could produce some non-zero representative of the trivial class in the top-degree horizontal cohomology. But we then show at once by an elementary counterexample why, in the frames of the old approach that did not rely on Gel'fand's calculus, the BV-Laplacian failed to be a graded derivation of the variational Schouten bracket. Keywords: Variational multi-vectors, Schouten bracket, Jacobi identity, Batalin-Vilkovisky Laplacian, symbolic computations. PACS: 02.40.-k, 11.10.-z; also 02.30.Ik, 02.30.Jr, 11.15.-q, 11.30.-j