Geometry of iterated variations in the problem of deformation quantisation of field models

The Batalin-Vilkovisky supergeometry relies on the introduction of canonical conjugate pairs of dependent variables called the (anti)fields and (anti)ghosts. Understanding their variations as singular linear integral operators – acting on some suitable spaces of base functionals such as the action functional – is very profitable. In the first part of this talk I shall explain the geometry of integrations by parts (e.g., in the course of derivation of the Euler–Lagrange equations of motion). This will reveal why, whenever understood properly, the iterated variations are ($\mathbb Z/2 \mathbb Z$-graded) permutable. We then conclude that the definitions of such structures as the parity-odd variational Laplacian and variational Schouten bracket are operational; indeed, they amount to re-attachment algorithms for couplings of the BV-fields' normalised shifts (i.e., “variations”) and functionals' differentials. In the frames of this approach, a rigorous proof of several important identities became possible for these structures (those identities were traditionally accepted ad hoc in the past, see [1312.1262] and [1210.0726 v3] for details).
In the second part of this talk we consider the deformation quantisation problem for field theory models and we show how the geometry of iterated variations works in that problem's solution. In particular, I shall derive and substantiate the variational analogue of noncommutative but associative Moyal's $\star$-product. Let us recall that Kontsevich's deformation quantisation formula for the product in the algebra of smooth functions on a given finite-dimensional Poisson manifold is an immediate, well-known generalisation of Moyal's set-up to the case of Poisson bi-vectors with non-constant coefficients (see [q-alg/9709040]). The aim of this talk is to show that Kontsevich's original formula – involving a summation over weighted graphs in the explicit construction of $\star$-product – works nontrivially but literally in the variational geometry. This solves the problem of associative deformation quantisation for multiplicative structures in the algebras of local functionals. We shall see why the variational Poisson structures (encoded by the Hamiltonian total differential operators) mark points in the moduli spaces of deformation quantisations for field theory models.