Poisson geometry and sigma models

• J. Baez,   Higher Gauge Theory and the String Group

Abstract:

Higher gauge theory is a generalization of gauge theory that describes the parallel transport not just of particles, but also strings or higher-dimensional branes. To handle strings, we categorify familiar notions from gauge theory and consider "principal 2-bundles" with a given "structure 2-group". These are a slight generalization of nonabelian gerbes. After a quick introduction to these ideas, we focus on the 2-groups String_k(G) associated to any compact simple Lie group G. We describe how this 2-group is built using the level-k central extension of the loop group of G, and how the classifying space for String_k(G)-2-bundles is related to the "string group".


• A. Cattaneo,   Courant sigma model and applications

Abstract:

We recall the definition of the Courant sigma model with boundary via AKSZ. In the case of Lie bialgebroids we study its reduced phase space and its relation to double symplectic groupoids. The problem of quantization of Lie bialgebras, and its difficulties, will also be discussed.


• R. Fernandes,   Reduction and integrability

Abstract:

For a proper action of a Lie group on a Poisson manifold, by Poisson diffeomorphisms, I will describe the symplectic groupoid of the reduced Poisson manifold.


• K. Gawedzki,   Gerbes and WZW orientifolds

Abstract:

Constructing amplitudes of the 2-dimensional field theories with Wess-Zumino term on non-oriented worldsheets requires a choice of a gerbe on the target manifold together with a so called Jandl structure introduced by Schreiber, Schweigert and Waldorf. I shall discuss how, in the context of the WZW models of conformal field theory, the classification of such structures reduces to a problem in finite-group cohomology.


• G. Halbout,   Quasi-Poisson manifolds, examples and quantization

Abstract:

Let g be a Lie algebra acting freely on a manifold X. A quasi-Poisson structure is given by a 2-tensor satisfying Jacobi up to the action of a 3-tensor in g^3. We will define quantization of such manifolds. We will recall how the problem of quantization is related with the one of dynimacal r-matrices. Finaly, we will prove existence of quantization for a large class of quasi-Poisson manifods. Proof uses an generalized formality theorem, which will give, as a special case, a new proof of Kontsevich's theorem.


• N. Ikeda,   New topological field theories from dimensional reduction of nonlinear gauge theories

Abstract:

We construct a new topological field theory with superfields with the negative degree by the AKSZ formalism. The AKSZ action is constructed consistent with dimensional reduction and the defomation theory. We consider a reduction from the Cournat sigma model in three dimensions to two dimensions. We consider an application of this model to a generalized complex structure.


• C. Klimcik,   q -> infty limit of quasitriangular WZW model

Abstract:

We study the $q\to\infty$ limit of the $q$-deformation of the WZW model on a compact simple and simply connected target Lie group. We show that the commutation relations of the $q\to\infty$ current algebra are underlied by certain affine Poisson structure on the group of holomorphic maps from the disc into the complexification of the target group. The Lie algebroid corresponding to this affine Poisson structure can be integrated to a global symplectic groupoid which turns out to be nothing but the phase space of the $q\to\infty$ limit of the $q$-WZW model. We also show that this symplectic grupoid admits a chiral decomposition compatible with its (anomalous) Poisson-Lie symmetries. Finally, we dualize the chiral theory in a remarkable way and we evaluate the exchange relations for the $q\to\infty$ chiral WZW fields in both the original and the dual pictures.


• P. Mnev,   Discrete BF-theory

Abstract:

We describe the construction of discrete BF theory as effective theory on Whitney forms, associated to simplicial or cubical complexes, defined via BV integral. We will also discuss the interpretation of the construction as an example of "quantum homotopy algebra" and the application to topological invariants of manifolds.


• S. Monnier,   Quantization of Wilson loops in WZW models

Abstract:

We describe a non-perturbative quantization of classical Wilson loops in the WZW model. The quantized Wilson loop is an operator acting on the Hilbert space of closed strings and commuting either with the full Kac-Moody chiral algebra or with one of its subalgebras. We prove that under open/closed string duality, it is dual to a boundary perturbation of the open string theory. As an application, we show that such operators are useful tools for identifying fixed points of the boundary renormalization group flow.


• D. Roytenberg ,   Weak Lie 2-algebras

Abstract:

A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and certain "up to homotopy" structures on the complex; for strictly skew-symmetric Lie 2- algebras these are $L_\infty$-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer-Cartan equation in some differential graded Lie algebras and $L_\infty$-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Liebialgebroids and Courant algebroids provide large classes of examples.


• P. Severa ,   Flat connections and Morita equivalence

Abstract:

We will discuss some symplectic manifolds that arise as moduli spaces of flat connections on surfaces with appropriate boundary conditions, and Lagrangian relations coming from 3dim bodies. The simplest example is the double symplectic groupoid integrating a Lie bialgebra. The motivation is to provide a "symplectic explanation" of Morita equivalence of quantum tori and to provide its generalization to nonabelian groups.


• X. Tang,   Noncommutative Poisson Structures on Orbifolds

Abstract:

In this talk, we report some recent study of Poisson geometry on orbifolds from the view point of noncommutative geometry. We found that noncommutative Poisson geometry on orbifolds contains many interesting and new phenomena. In simple examples, we will explain that double affine Hecke algebras shows up naturally in our study. We will also show in examples that the deformation quantization of Poisson structures on orbifolds are closely related to the deformation of the underlying singular quotients.


• R. von Unge,   On the Generalized Kahler potential

Abstract:

We will discuss how using superspace we are able to find a function containing all the data about the metric and antisymmetric b-field on an arbitrary Generalized Kahler manifold.


• M. Zabzine,   Poisson sigma model on the sphere

Abstract:

I  will discuss the path integral of the Poisson sigma model on sphere and study the correlators of quantum observables. I will argue that for the path integral to be well-defined the corresponding  Poisson structure should be unimodular. The construction of the finite dimensional BV theory will be presented and I will argue that it is responsible for the leading semiclassical contribution. For a (twisted) generalized Kahler manifold I will discuss the gauge fixed action for the Poisson sigma model. Using the localization we prove that for the holomorphic Poisson structure the semiclassical result for the correlators is indeed the full quantum result.