Bimodules and branes in deformation quantization

Abstract

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 $\mathrm{S}(X^{\ast })$ and $\wedge (X)$ 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.

40 pages, 15 figures; a small change of notations in the definition of the 4-colored propagators; an Addendum about the appearance of loops in the $L_{\infty }$-quasi-isomorphism and a corresponding change in the proof of Theorem 7.2; several changes regarding completions, when dealing with general $A_{\infty }$-structures

References in corpus (5)

• A Counterexample to the Quantizability of Modules
• Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
• Infinity inner products on A-infinity algebras
• Poisson Sigma Model with branes and hyperelliptic Riemann surfaces
• Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Cited by in corpus (12)

• Biquantization of symmetric pairs and the quantum shift
• Deformation quantization with generators and relations
• Two-colored noncommutative Gerstenhaber formality and infinity Duflo isomorphism
• Gravity algebra structure on the negative cyclic homology of Calabi-Yau algebras
• A note on the Koszul complex in deformation quantization
• Operads, configuration spaces and quantization
• Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras
• Characteristic classes in deformation quantization
• The explicit equivalence between the standard and the logarithmic star product for Lie algebras
• Deformation Quantization of A-infinity Equivalences
• The Chevalley--Eilenberg complex and deformation quantization in presence of two branes
• Quantizing Derived Mapping Stacks