Dirac approach to constrained submanifolds in a double loop group: From Wess-Zumino-Novikov-Witten to Poisson-Lie σ-model

• Autores: Montani, Hugo Santos; Zuccalli, Marcela
• Año de publicación: 2014
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: DIRAC METHOD ON DOUBLE LIE GROUPS; CENTRAL EXTENSIONS AND LOOP GROUPS; WZNW MODEL; POISSON-LIE SIGMA MODEL
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

We study the restriction to a family of second class constrained submanifolds in the cotangent bundle of a double Lie group equipped with a 2-cocycle extended symplectic form to build the corresponding Dirac brackets. It is shown that, for 2-cocycle vanishing on each isotropic subspace of the associated Manin triple, the Dirac bracket contains no traces of the cocycle. We also investigate the restriction of the left translation action of the double Lie group on its cotangent bundle, where it fails to be a canonical transformation. However, the Hamiltonian symmetry is restored on some special submanifolds. The main application is to loop groups, showing that a WZNW-type model on the double Lie group with a quadratic Hamilton function in the momentum maps associated with the left translation action on the cotangent bundle with the canonical symplectic form, restricts to a collective system on some special submanifolds. There, the Lagrangian version coincides with the so-called Poisson-Lie σ-model.
Fil: Montani, Hugo Santos. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de la Patagonia Austral. Unidad Académica Caleta Olivia. Departamento de Ciencias Exactas y Naturales; Argentina
Fil: Zuccalli, Marcela. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina