We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X * ) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.

An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.1 Plan of the paper. In Section 2, we recall nonabelian canonical BF theories and give a very formal, but intuitively clear, definition of Wilson surfaces, see (2.4) and (2.5). We discuss their formal properties and, in particular, we clarify why we expect their expectation values to yield invariants of higher-dimensional knots. (In this Section by invariant we mean a Diff 0 (N) × Diff 0 (M)-invariant function on the space of imbeddings N ֒→ M.)1 The necessity of considering imbeddings in the nonabelian theory, instead of more general smooth maps, arises at the quantum level (just like in the nonabelian Chern-Simons theory) in order to avoid singularities which make the observables ill-defined.

Abstract. For a complex manifold the Hochschild-Kostant-Rosenberg map does not respect the cup product on cohomology, but one can modify it using the square root of the Todd class in such a way that it does. This phenomenon is very similar to what happens in Lie theory with the Duflo-Kirillov modification of the Poincaré-Birkhoff-Witt isomorphism.In these lecture notes (lectures were given by the first author at ETH-Zürich in fall 2007) we state and prove Duflo-Kirillov theorem and its complex geometric analogue. We take this opportunity to introduce standard mathematical notions and tools from a very down-to-earth viewpoint.

ABSTRACT. This paper analyzes in details the Batalin-Vilkovisky quantization procedure for BF theories on an n-dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables. In particular, generalized Wilson loops for BF -theories with additional polynomial B-interactions are discussed in any dimensions. The paper also contains the explicit proofs to the Theorems stated in [16].

Abstract. In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality L∞-quasi-isomorphism for Hochschild chains is compatible with capproducts. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality L∞-quasi-isomorphism for Hochschild cochains.As in the cohomological situation our proof relies on a homotopy argument involving a variant of Kontsevich eye. In particular we clarify the rôle played by the I-cube introduced in [4].Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an A∞-algebra. In particular we prove that they naturally inherit the structure of an A∞-algebra with an A∞-(bi)module.