Deformation quantization of field models: iterated variations in the star-products

In field theory models, the variations of unknowns are singular
linear integral operators that act on spaces of differentials of local
functionals -- such as a model's action functional that takes (gauge)
field configurations to numbers. It is then readily seen that the
iterated variations are (graded-) permutable; this resolved the
obstructions to rigorous proof of several important identities for the
Batalin-Vilkovisky Laplacian and variational Schouten bracket (those
identities had traditionally been accepted ad hoc in the past, see
[1312.1262] and [1210.0726 v3] for details).

   In 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, we derive
the variational analogue of noncommutative but associative Moyal's
star-product.

   Preserving the associativity, Kontsevich's quantization formula
deforms the product in algebras of smooth functions on
finite-dimensional Poisson manifolds; the formula is a renowned
generalization 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 summation over weighted graphs in
the explicit construction of star-products does work nontrivially but
literally for the algebras of local functionals. This reveals why the
variational Poisson structures (either encoded by Hamiltonian
differential operators or, after the Fourier transform, realized via
the Virasoro or W-algebra generators) mark points in the moduli spaces
of deformation quantizations for field theory models.