06/C206

 

Simetrías clásicas y cuánticas en física teórica y matemática

Classical and quantum symmetries in theoretical physics and mathematics

 

Director: TRINCHERO, Roberto

E-mail: trincher@cab.cnea.gov.ar

 

Co-Director: MONTANI, Hugo

 

Integrantes: FERNANDEZ, Javier; GRILLO, Sergio; CAPRIOTTI, Santiago

 

Resumen Técnico

Estudio de la clasificación de teorías conformes racionales en dos dimensiones y su relación con las álgebras de Hopf débiles y las simetrías cuánticas de los gráficos asociados a partir de la geometría de caminos sobre éstos.

Derivación de grupoides cuánticos, a través del teorema reconstrucción, como objeto representante para una categoría modular como la que conforman los módulos de una teoría de campos conforme.

Reconstrucción de semigrupoides cuánticos, del tipo matrices cuánticas rectangulares, y derivación de estructuras análogas a la propiedad de cuasitriangularidad de los grupos cuánticos usuales. Construcción de modelos integrables cuánticos asociados.

Usando técnicas de geometría no-conmutativa se consideraran variedades con dimensión no-entera como realización de primeros principios de la regularización dimensional.

Estudio de los sistemas integrables asociados a grupos cuánticos trenzados.

Caracterización geométrica de la dualidad T de Poisson Lie y generalización a algebroides de Lie.

Construcción de sistemas con simetría de grupoides simplécticos, relación con estructuras de Poisson generalizadas y vínculos.

 

Summary

Symmetry is the fundamental idea underlying our present understanding of the basic physical laws of the Nature. Its theoretical formulation involves many mathematical structures which give also a deep insight on the way the symmetries are realized.

The main aim of this project is to study some problems related to de so called ¨fundamental symmetries¨, that means, symmetries related to the physics of fundamental interactions and how they can help to solve physical models. In doing so, integrable systems play a central roll both at the classical and quantum level.

We will study quantum symmetries related to integrable systems in quantum field theory and statistical mechanics for which the natural framework is that provided by the non commutative geometry. Our attention will be addressed to the understanding of the relation between conformal symmetries and quantum groupoids (Hopf and weak Hopf algebras, and other structures), looking for relation between ADE graphs and modular categories. Also, different algebraic aspects of statistical systems will be studied, such as the integrable gluing different systems.

On the other side, some of the relevant properties of the symmetries in physical models can be traced back to their classical formulations in a geometrical framework. There, symplectic and Poisson geometry comes to unveils many rich properties surviving the quantization procedure. This is the case of the infinite dimensional symmetry underlying the sigma model. So, we address the problem of explaining these kind of symmetries in terms of momentum maps associated Poisson Lie structures, the roll of T-duality and the generalization to symplectic groupoids. Other problems will be involved as constrained systems, quasi Poisson and Dirac structures, etc, that will be also addressed