Yu Lu

Giuseppe Morandi

Stig Lundqvist

This volume contains a set of pedagogical reviews covering the most recent applications of low-dimensional quantum field theory in condensed matter physics, written by experts who have made major contributions to this rapidly developing field of research. The main purpose is to introduce active young researchers to new ideas and new techniques which are not covered by the standard textbooks. Contents: Some Geometry and Topology (G Marmo & G Morandi)Gauge Symmetries, Topology and Quantisation (A P Balachandran)The Chern-Simons-Landau-Ginzburg Theory of the Fractional Quantum Hall Effect (S C Zhang)Universitality in the Fractional Quantum Hall (E Fradkin & A Lopez)Anyons and Anyon Superconductivity (A L Fetter)Bosonization: How to Make It Work for You in Condensed Matter (R Shankar)Methods of Conformal Field Theory in Condensed Matter Physics (A W W Ludwig)Integrable Models in Condensed Matter Physics (N Andrei)Quantum Antiferromagnets in Two Dimensions (S Sachdev) Readership: Researchers and graduate students. keywords:

Low-Dimensional Quantum Field Theories for Condensed Matter Physicists

Giuseppe Marmo

José F. Cariñena

Alberto Ibort

This book describes, by using elementary techniques, how some geometrical structures widely used today in many areas of physics, like symplectic, Poisson, Lagrangian, Hermitian, etc., emerge from dynamics. It is assumed that what can be accessed in actual experiences when studying a given system is just its dynamical behavior that is described by using a family of variables ("observables" of the system). The book departs from the principle that ''dynamics is first'' and then tries to answer in what sense the sole dynamics determines the geometrical structures that have proved so useful to describe the dynamics in so many important instances. In this vein it is shown that most of the geometrical structures that are used in the standard presentations of classical dynamics (Jacobi, Poisson, symplectic, Hamiltonian, Lagrangian) are determined, though in general not uniquely, by the dynamics alone. The same program is accomplished for the geometrical structures relevant to describe quantum dynamics. Finally, it is shown that further properties that allow the explicit description of the dynamics of certain dynamical systems, like integrability and super integrability, are deeply related to the previous development and will be covered in the last part of the book. The mathematical framework used to present the previous program is kept to an elementary level throughout the text, indicating where more advanced notions will be needed to proceed further. A family of relevant examples is discussed at length and the necessary ideas from geometry are elaborated along the text. However no effort is made to present an ''all-inclusive'' introduction to differential geometry as many other books already exist on the market doing exactly that. However, the development of the previous program, considered as the posing and solution of a generalized inverse problem for geometry, leads to new ways of thinking and relating some of the most conspicuous geometrical structures appearing in Mathematical and Theoretical Physics.

Giuseppe Morandi

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