

Complex analysis and mathematical physics
April 6, 2015 16:00, Moscow, Steklov Mathematical Institute, Room 430 (8 Gubkina)






Geometry of iterated variations in the problem of deformation
quantisation of field models.
A. V. Kiselev^{} ^{} Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen

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Abstract:
The BatalinVilkovisky 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 parityodd variational Laplacian and
variational Schouten bracket are operational; indeed, they amount to
reattachment algorithms for couplings of the BVfields' 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
finitedimensional Poisson manifold is an immediate, wellknown
generalisation of Moyal's setup to the case of Poisson bivectors
with nonconstant coefficients (see [qalg/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.

