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

Arthemy V. Kiselev

Published 2014-10-01Version 1

By studying one example of three variational multi-vectors in two very different ways, we inspect the mechanism(s) for validity of Jacobi's identity for the variational Schouten bracket (e.g., in the geometry of Poisson bi-vectors or Batalin-Vilkovisky formalism). Namely, we illustrate and contrast the logics of "genuine" and "naive" geometries of iterated variations which are contained in the well-known identities satisfied by the Schouten bracket and BV-Laplacian. Whereas the true picture from [arXiv:1312.1262] and [arXiv:1312.4140] keeps track of several copies of the integration manifold in both the functionals and pairs of variations' parity-even and odd components, too early does the traditional approach merge the integration domains. Using an elementary counterexample, we point at an inconsistency in the traditional paradigm. PACS: 02.40.-k, 11.10.-z; also 02.30.Ik, 02.30.Jr, 11.15.-q, 11.30.-j