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Deformation approach to quantisation of field models.

Le : 03/05/2017 14h00
Par : Arthemy Kiselev (JBI Groningen, IHES)
Lieu : i 103
Lien web :
Résumé : 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 $\times\mapsto\star=\times+\hbar\,\{\cdot,\cdot\}_{\mathcal{P}}+\bar{o}(\hbar)$ 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}].