Abstract. 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.
The structures in target space geometry that correspond to conformally
invariant boundary conditions in WZW theories are determined both by studying
the scattering of closed string states and by investigating the algebra of open
string vertex operators. In the limit of large level, we find branes whose
world volume is a regular conjugacy class or, in the case of symmetry breaking
boundary conditions, a `twined' version thereof. In particular, in this limit
one recovers the commutative algebra of functions over the brane world volume,
and open strings connecting different branes disappear. At finite level, the
branes get smeared out, yet their approximate localization at (twined)
conjugacy classes can be detected unambiguously.
As a by-product, it is demonstrated how the pentagon identity and tetrahedral
symmetry imply that in any rational conformal field theory the structure
constants of the algebra of boundary operators coincide with specific entries
of fusing matrices.Comment: 29 pages, LaTeX2e; Reference to [9] clarified, where the relation
between boundary structure constants and fusing matrices has appeared
independentl
We prove a relative version of Kontsevich's formality theorem. This theorem
involves a manifold M and a submanifold C and reduces to Kontsevich's theorem
if C=M. It states that the DGLA of multivector fields on an infinitesimal
neighbourhood of C is L-infinity-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 Kontsevich's 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).Comment: 31 pages. Details on globalization adde
We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.* Supported by the Swiss National Foundation (SNF)
ABSTRACT. 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.
The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eightvertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3, Z) _ Z 3 associated to a non-trivial class in H 3
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