Analyse des EDP, géométrie et physique

Formality theorem for differential graded manifolds

Non programmé

50m

Amphis Laurent Schwartz (Toulouse)

Orateur

Prof. Mathieu Stiénon (Penn State University)

Description

The Atiyah class of a dg manifold $(\mathcal{M},Q)$ is the obstruction to the existence of an affine connection on the graded manifold $\mathcal{M}$ that is compatible with the homological vector field $Q$. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. Using Kontsevich's famous formality theorem, Liao, Xu and I established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold $(\mathcal{M},Q)$, there exists an $L_{\mathrm{\infty }}$ quasi-isomorphism of differential graded Lie algebras from the space of polyvector fields on $\mathcal{M}$ endowed with the Schouten bracket $[-,-]$ and the differential $[Q,-]$ to the space of polydifferential operators on $\mathcal{M}$ endowed with the Gerstenhaber bracket $[-,-]$ and the differential $[m+Q,-]$, whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold $(\mathcal{M},Q)$ with the Hochschild--Kostant--Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we proved the Kontsevich--Shoikhet conjecture: a Kontsevich--Duflo type theorem holds for all finite-dimensional smooth dg manifolds. This last result shows that, when understood in the unifying framework of dg manifolds, the classical Duflo theorem of Lie theory and the Kontsevich--Duflo theorem for complex manifolds are really just one and the same phenomenon.