Dynamics of Wess-Zumino-Witten and Chern-Simons Theories

TL;DR

This thesis explores the Hamiltonian structure of WZW and Chern-Simons supergravity theories, addressing their algebraic extensions, irregularities, and specific dynamics in five dimensions, revealing new invariant actions and BPS states.

Contribution

It constructs a super WZW model coupled to supergravity with invariant actions, analyzes irregularities in higher-dimensional CS theories, and studies the dynamics and BPS states in D=5 CS supergravity.

Findings

Extended Kac-Moody and Virasoro algebras to superalgebras.

Identified conditions for regularization of irregular systems.

Found BPS backgrounds and derived charges satisfying supersymmetric WZW(4) algebra.

Abstract

This thesis is devoted to the study of three problems on the Wess-Zumino-Witten (WZW) and Chern-Simons (CS) supergravity theories in the Hamiltonian framework: 1) The two-dimensional super WZW model coupled to supergravity is constructed. The canonical representation of Kac-Moody algebra is extended to the super Kac-Moody and Virasoro algebras. Then, the canonical action is constructed, invariant under local supersymmetry transformations. The metric tensor and Rarita-Schwinger fields emerge as Lagrange multipliers of the components of the super energy-momentum tensor. 2) In higher dimensions, CS theories are irregular systems, that is, they have constraints which are functionally dependent in some sectors of phase space. In these cases, the standard Dirac procedure must be redefined, as it is shown in the simplified case of finite number of degrees of freedom. Irregular systems fall into two classes depending on their behavior in the vicinity of the constraint surface. In one case, it is possible to regularize the system without ambiguities, while in the other, regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. Irregularities have important consequences in the linearized approximation of nonlinear theories. 3) The dynamics of CS supergravity theory in D=5, based on the supersymmetric extension of the AdS algebra, su(2,2|4), is analyzed. A class of backgrounds is found, providing a regular and generic effective theory. Some of these backgrounds are shown to be BPS states. The charges for the simplest choice of asymptotic conditions are obtained, and they satisfy a supersymmetric extension of the classical WZW(4) algebra, associated to su(2,2|4).