Deformation Quantization and Reduction

This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and pre-Poisson submanifolds, their appearance as branes of the PSM, quantization in terms of L-infinity and A-infinity algebras, and bimodule structures are recalled. As an application, an "almost" functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson-Lie groups.Comment: 23 pages, 3 figures; added references, corrected typo

Cabled Wilson Loops in BF Theories

A generating function for cabled Wilson loops in three-dimensional BF theories is defined, and a careful study of its behavior for vanishing cosmological constant is performed. This allows an exhaustive description of the unframed knot invariants coming from the pure BF theory based on SU(2), and in particular, it proves a conjecture relating them to the Alexander-Conway polynomial.Comment: 30 pages, LaTe

On the AKSZ formulation of the Poisson sigma model

We review and extend the Alexandrov-Kontsevich-Schwarz-Zaboronsky construction of solutions of the Batalin-Vilkovisky classical master equation. In particular, we study the case of sigma models on manifolds with boundary. We show that a special case of this construction yields the Batalin-Vilkovisky action functional of the Poisson sigma model on a disk. As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds.Comment: 19 page

A path integral approach to the Kontsevich quantization formula

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin-Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra.Comment: 22 pages, 2 figures, references added. Conjecture on the center made more precis

Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time

The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.Comment: 13 pages, 3 figures. in v2: Fixed few misprint

Poisson sigma models and deformation quantization

This is a review aimed at a physics audience on the relation between Poisson sigma models on surfaces with boundary and deformation quantization. These models are topological open string theories. In the classical Hamiltonian approach, we describe the reduced phase space and its structures (symplectic groupoid), explaining in particular the classical origin of the non-commutativity of the string end-point coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches.Comment: 11 page

The reduced phase space of Palatini-Cartan-Holst theory

General relativity in four dimensions can be reformulated as a gauge theory, referred to as Palatini-Cartan-Holst theory. This paper describes its reduced phase space using a geometric method due to Kijowski and Tulczyjew and its relation to that of the Einstein-Hilbert approach.Comment: Revised version comprising new results, a correction of Th 4.22 and the arguments leading to it. Manuscript accepted for publication in AHP. 31 page

Formality and Star Products

These notes, based on the mini-course given at the PQR2003 Euroschool held in Brussels in 2003, aim to review Kontsevich's formality theorem together with his formula for the star product on a given Poisson manifold. A brief introduction to the employed mathematical tools and physical motivations is also given.Comment: 49 pages, 9 figures; proceedings of the PQR2003 Euroschool. Version 2 has minor correction

BV-BFV approach to General Relativity: Palatini-Cartan-Holst action

We show that the Palatini--Cartan--Holst formulation of General Relativity in tetrad variables must be complemented with additional requirements on the fields when boundaries are taken into account for the associated BV theory to induce a compatible BFV theory on the boundary.Comment: 22 pages. Corrected typos in some formulae. Minor aesthetic fixe