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Abstract

In this thesis, we develop a framework to study n-extensions of Kac-Moody algebras, and use the resulting Lorentzian algebras to study Lorentzian extension of Toda field theories and their integrability. We begin our discourse by providing context and motivation for the study of these ideas, mainly through illustrating the unmatched historical successes of quantum field theory in science and the role of symmetry algebras and integrability in mathematical physics.

Continuing, we develop a new framework to extend finite, gf Kac-Moody algebras, through their simple root structure n times to what we name n-extended Lorentzian Kac- Moody algebras, g−n. Using constants from the Casimir operators of g−n, we find a novel type of decomposition of g−n. We derive conditions in which these decompositions are possible, and tabulate all possible decompositions of g−n. Applying the methods we developed in the construction of g−n, we build Lorentzian Toda field theories as extensions of Toda field theories based on gf . We find that on each subsequent addition of simple roots in the n-extension procedure results in the Lorentzian Toda field theory alternating between conformal and massive behaviour. We calculate mass ratios for the massive theories, and using the Painlev´e test, we find that some of these Lorentzian models can not be integrable.

Examining another class of Lorentzian Toda field, which we name the null root models, we show that these pass the Painlev´e test. Furthermore, we show that these models can also possess the more restrictive Painlev´e property, showing this explicitly for a spin-3 rank-2 example, meaning that this example is integrable. The procedure used is generalizable, and we therefore conclude that more models from the null root class of Lorentzian Toda field theories are very likely to also be integrable models.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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