Alberto S. Cattaneo

A Path Integral Approach¶to the Kontsevich Quantization Formula

Abstract: We give a quantum field theory interpretation of Kontsevichs 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.

Poisson sigma models and symplectic groupoids

We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.

From local to global deformation quantization of Poisson manifolds

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifold, based on M. Kontsevichs local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.

Relative formality theorem and quantisation of coisotropic submanifolds

We prove a relative version of Kontsevichs formality theorem. This theorem involves a manifold M and a submanifold C and reduces to Kontsevichs theorem if C=M. It states that the DGLA of multivector fields on an infinitesimal neighbourhood of C is L∞-quasiisomorphic to the DGLA of multidifferential operators acting on sections of the exterior algebra of the conormal bundle. Applications to the deformation quantisation of coisotropic submanifolds are given. The proof uses a duality transformation to reduce the theorem to a version of Kontsevichs theorem for supermanifolds, which we also discuss. In physical language, the result states that there is a duality between the Poisson sigma model on a manifold with a D-brane and the Poisson sigma model on a supermanifold without branes (or, more properly, with a brane which extends over the whole supermanifold).

Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model

Abstract.General boundary conditions (‘branes’) for the Poisson sigma model are studied. They turn out to be labeled by coisotropic submanifolds of the given Poisson manifold. The role played by these boundary conditions both at the classical and at the perturbative quantum level is discussed. It turns out to be related at the classical level to the category of Poisson manifolds with dual pairs as morphisms and at the perturbative quantum level to the category of associative algebras (deforming algebras of functions on Poisson manifolds) with bimodules as morphisms. Possibly singular Poisson manifolds arising from reduction enter naturally into the picture and, in particular, the construction yields (under certain assumptions) their deformation quantization.

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 Kontsevichs deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds.

Integral invariants of 3-manifolds. II

This note is a sequel to our earlier paper of the same title [4] and describes invariants of rational homology 3-spheres associated to acyclic orthogonal local systems. Our work is in the spirit of the Axelrod–Singer papers [1], generalizes some of their results, and furnishes a new setting for the purely topological implications of their work.

POISSON SIGMA MODELS AND DEFORMATION QUANTIZATION

This is a review aimed at the 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 noncommutativity of the string endpoint coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevichs star product. Finally we comment on the relation between the two approaches.

Configuration spaces and Vassiliev classes in any dimension

The real cohomology of the space of imbeddings of S 1 into R n , n> 3, is studied by using conguration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes dened on the corresponding spaces of immersions. AMS Classication 58D10; 55R80, 81Q30

Topological BF theories in 3 and 4 dimensions

In this paper we discuss topological BF theories in 3 and 4 dimensions. Observables are associated to ordinary knots and links (in 3 dimensions) and to 2‐knots (in 4 dimensions). The vacuum expectation values of such observables give a wide range of invariants. Here we consider mainly the 3 dimensional case, where these invariants include Alexander polynomials, HOMFLY polynomials and Kontsevich integrals.