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Harmonic maps are generalisations of the concept of geodesics. They encompass many fundamental examples in differential geometry and have recently become of widespread use in many areas of mathematics and mathematical physics. This is an accessible introduction to some of the fundamental connections between differential geometry, Lie groups, and integrable Hamiltonian systems. The specific goal of the book is to show how the theory of loop groups can be used to study harmonic maps. By concentrating on the main ideas and examples, the author leads up to topics of current research. The book is suitable for students who are beginning to study manifolds and Lie groups, and should be of interest both to mathematicians and to theoretical physicists.

Harmonic maps. --- Loops (Group theory) --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Loop groups --- Group theory --- Maps, Harmonic --- Mappings (Mathematics)

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Wilson lines (also known as gauge links or eikonal lines) can be introduced in any gauge field theory. Although the concept of the Wilson exponentials finds an enormously wide range of applications in a variety of branches of modern quantum field theory, from condensed matter and lattice simulations to quantum chromodynamics, high-energy effective theories and gravity, there are surprisingly few books or textbooks on the market which contain comprehensive pedagogical introduction and consecutive exposition of the subject. The objective of this book is to get the potential reader acquainted with theoretical and mathematical foundations of the concept of the Wilson loops in the context of modern quantum field theory, to teach him/her to perform independently some elementary calculations with Wilson lines, and to familiarize him/her with the recent development of the subject in different important areas of research. The target audience of the book consists of graduate and postgraduate students working in various areas of quantum field theory, as well as researchers from other fields.

Loops (Group theory) --- Quantum field theory --- Gauge fields (Physics) --- Fields, Gauge (Physics) --- Gage fields (Physics) --- Gauge theories (Physics) --- Field theory (Physics) --- Group theory --- Symmetry (Physics) --- Relativistic quantum field theory --- Quantum theory --- Relativity (Physics) --- Loop groups --- Mathematics.

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The book discusses hidden symmetries in the Anti-de Sitter/conformal field theory (AdS/CFT) duality. This duality is a modern concept that asserts an exact duality between conformally invariant quantum field theories and string theories in higher dimensional Anti-de Sitter spaces, and in this way provides a completely new tool for the study of strongly coupled quantum field theories. In this setting, the book focuses on the Wilson loop, an important observable in four-dimensional maximally supersymmetric gauge theory. The dual string description using minimal surfaces enables a systematic study of the hidden symmetries of the loop. The book presents major findings, including the discovery of a master symmetry for strings in general symmetric spaces, its relation to the Yangian symmetry algebra and its action on the minimal surfaces appearing in the dual string description of the Wilson loop. Moreover, it clarifies why certain symmetries are not present on the gauge theory side for purely bosonic Wilson loops and, lastly, how the supersymmetrization of the minimal surface problem for type IIB superstrings can be undertaken. As such, it substantially increases our understanding and use of infinite dimensional symmetries occurring in the AdS/CFT correspondence.

Loops (Group theory) --- String models. --- Models, String --- String theory --- Nuclear reactions --- Loop groups --- Group theory --- Quantum Field Theories, String Theory. --- Mathematical Applications in the Physical Sciences. --- Quantum field theory. --- String theory. --- Mathematical physics. --- Physical mathematics --- Physics --- Relativistic quantum field theory --- Field theory (Physics) --- Quantum theory --- Relativity (Physics) --- Mathematics

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When close to a continuous phase transition, many physical systems can usefully be mapped to ensembles of fluctuating loops, which might represent for example polymer rings, or line defects in a lattice magnet, or worldlines of quantum particles. 'Loop models' provide a unifying geometric language for problems of this kind. This thesis aims to extend this language in two directions. The first part of the thesis tackles ensembles of loops in three dimensions, and relates them to the statistical properties of line defects in disordered media and to critical phenomena in two-dimensional quantum magnets. The second part concerns two-dimensional loop models that lie outside the standard paradigms: new types of critical point are found, and new results given for the universal properties of polymer collapse transitions in two dimensions. All of these problems are shown to be related to sigma models on complex or real projective space, CP^{n−1} or RP^{n−1} -- in some cases in a 'replica' limit -- and this thesis is also an in-depth investigation of critical behaviour in these field theories.

Physics. --- Mathematical Methods in Physics. --- Statistical Physics, Dynamical Systems and Complexity. --- Mathematical Applications in the Physical Sciences. --- Condensed Matter Physics. --- Mathematical physics. --- Physique --- Physique mathématique --- Physics --- Physical Sciences & Mathematics --- Physics - General --- Critical phenomena (Physics) --- Loops (Group theory) --- Loop groups --- Phenomena, Critical (Physics) --- Condensed matter. --- Statistical physics. --- Dynamical systems. --- Group theory --- Complex Systems. --- Statistical Physics and Dynamical Systems. --- Mathematical statistics --- Physical mathematics --- Statistical methods --- Mathematics --- Condensed materials --- Condensed media --- Condensed phase --- Materials, Condensed --- Media, Condensed --- Phase, Condensed --- Liquids --- Matter --- Solids --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Statics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics