Aspects of integrable nonlinear sigma models

• Main Author: Hassan, M. U.
• Published: University of Cambridge 2000
• Subjects: 519
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.603844

In the first chapter we introduce the subject and review some existing ideas about conservation laws and dual symmetry in nonlinear sigma models. We also give a brief introduction to supersymmetry in two-dimensional spacetime. In chapter 2 we construct local conserved charges for each PCM (with and without a Wess-Zumino term), which are based on symmetric invariant tensors of the underlying Lie algebra. We then study their Poisson bracket algebra and we find that there exists an infinite family of commuting local charges with spins equal to the exponents modulo the Coxeter number of the algebra. These local charges are shown to be in involution with the nonlocal charges of the PCM (with and without a Wess-Zumino term). Finally we discuss conservation of some of these charges in the quantum PCMs. In chapter 3 we construct two families, each with finitely many members, of commuting local charges with spins equal to the exponents of the algebra (but with no repetition modulo the Coxeter number). We briefly discuss the effect of adding a Wess-Zumino term in the SPCM and its behaviour at critical (WZW) points. The local charges in the PCMs and SPCMs are only indirectly related. At the end of the chapter we comment on quantum conservation of some higher spin superfield currents in the SPCMs. Chapter 4 deals with the extension of dual symmetry to superspace. We explain this for super <I>O</I>(<I>l</I>) sigma models and derive an infinite series of nonlocal conserved charges in terms of both superfields and component fields. We elaborate on the BIZZ procedure in the SPCM and derive Lie algebra valued nonlocal charges in this model. We also show that the nonlocal charges are in involution with the local charges in the SPCM. At the end of the dissertation we give a summary of the main results and briefly discuss some open problems for further research. There follow appendices which collect some background material on Lie algebras, invariant tensors, Yangian algebras, and some details of counting about quantum conservation of higher spin local currents in the PCMs and the SPCMs.