Deformation approach to quantisation of field models

Physique Mathématique

Salle Kampé de Fériet M2
Arthemy Kiselev
JBI Groningen, MPIM Bonn, IHES
Mardi, 25 Avril, 2017 - 11:00 - 12:00

Let $(N^n,\mathcal{P})$ be a finite-dimensional affine Poisson manifold
and $A=C^\infty(N^n)$ be the ring of functions on it.
Kontsevich proved $[\texttt{q-alg/9709040}] $ that
the usual product $\times$ in $A$ can always be quantized via
towards a given Poisson bracket $\{\cdot,\cdot\}_{\mathcal{P}}$
so that $\star$ stays associative.
The higher-order terms at $\hbar^k$ in $\star$ are constructed by using
a pictorial language of oriented graphs.

We extend the deformation quantisation procedure $\times\mapsto\star$ to
a field-theoretic set-up of affine bundles $\pi$ with fibres $N^n$
over points of an affine base manifold $M^m$, of local functionals
taking $\Gamma(\pi)\to\Bbbk$, and variational Poisson bi-vectors $\boldsymbol{\mathcal{P}}$.
An extension of the Kontsevich graph technique is done by using
the geometry of iterated variations $[\texttt{1210.0726}]$.