Chern Simons Theory

The existence of the Pontryagin and Euler forms in a Weyl-Cartan space on the basis of the variational method with Lagrange multipliers are established. It is proved that these forms can be expressed via the exterior derivatives of the corresponding Chern-Simons terms in a Weyl-Cartan space with torsion and nonmetricity.

We exploit the properties of the three-dimensional hyperbolic space to discuss a simplicial setting for open/closed string duality based on (random) Regge triangulations decorated with null twistorial fields. We explicitly show that the twistorial N-points function, describing Dirichlet correlations over the moduli space of open N-bordered genus g surfaces, is naturally mapped into the Witten-Kontsevich intersection theory over the moduli space of N-pointed closed Riemann surfaces of the same genus. We also discuss various aspects of the geometrical setting which connects this model to PSL(2,C) Chern-Simons theory.

We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally finite type with respect to algebraically split surgery and with respect to Torelli surgery. Torelli surgery is a mutual generalization of blink surgery of Garoufalidis and Levine and clasper surgery of Habiro.

The goal of this work is to describe a categorical formalism for (Extended) Topological Quantum Field Theories (TQFTs) and present them as functors from a suitable category of cobordisms with corners to a linear category, generalizing 2d open-closed TQFTs to higher dimensions. The approach is based on the notion of an n-fold category by C. Ehresmann, weakened in the spirit of monoidal categories (associators, interchangers, Mac Lane's pentagons and hexagons), in contrast with the simplicial (weak Kan and complete Segal) approach of Jacob Lurie. We show how different Topological Quantum Field Theories, such as gauge, Chern-Simons, Yang-Mills, WZW, Seiberg-Witten, Rozansky-Witten, and AKSZ theories, as well as sigma model, may be described as functors from the pseudo n-fold category of cobordisms to a pseudo n-fold category of sets.

Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions. Chern-Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly (non-perturbatively) and explicitly solved. Abelian Chern-Simons theory provides a field theoretic interpretation of the linking and self-linking numbers of a link. In non-Abelian theories, vacuum expectation values of Wilson link operators yield a class of polynomial link invariants; the simplest of them is the famous Jones polynomial. Other invariants obtained are more powerful than that of Jones. Powerful methods for completely analytical and non-perturbative computation of these knot and link invariants have been developed. In the process answers to some of the open problems in knot theory are obtained. From these invariants for unoriented an...

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. We observe that in the context of higher Chern-Weil theory in smooth infinity-groupoids this statement generalizes from Lie algebras to L-infinity-algebras and further to L-infinity-algebroids. It turns out that the symplectic

Using the earlier developed classical Hamiltonian framework as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic

Abstract: We establish the isomorphism between a nonlinear $\ sigma $-model and the abelian gauge theory on an arbitrary curved background, which allows us to derive integrable models and the corresponding Lax representations from gauge theoretical ...

Chern-Simons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and explicitly solved. Expectation values of Wilson link operators yield a class of link invariants, the simplest of them is the famous Jones polynomial. Other