Conformal Field Theory and Solvable Lattice Models
Conformal Field Theory and Solvable Lattice Models

Author: M Jimbo

Publisher: Elsevier

Published: 2012-12-02

Total Pages: 439

ISBN-13: 0323150357

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Advanced Studies in Pure Mathematics, 16: Conformal Field Theory and Solvable Lattice Models contains nine papers based on the symposium "Conformal field theory and solvable lattice models" held at RIMS, Kyoto, May 1986. These papers cover the following active areas in mathematical physics: conformal field theory, solvable lattice models, affine and Virasoro algebra, and KP equations. The volume begins with an analysis of 1 and 2 point correlation functions of the Gibbs measure of random matrices. This is followed by separate chapters on solvable solid-on-solid (SOS) models; lectures on conformal field theory; the construction of Fermion variables for the 3D Ising Model; and vertex operator construction of null fields (singular vertex operators) based on the oscillator representation of conformal and superconformal algebras with central charge extention. Subsequent chapters deal with Hecke algebra representations of braid groups and classical Yang-Baxter equations; the relationship between the conformal field theories and the soliton equations (KdV, MKdV and Sine-Gordon, etc.) at both quantum and classical levels; and a supersymmetric extension of the Kadomtsev-Petviashvili hierarchy.

Lattice Models and Conformal Field Theory
Lattice Models and Conformal Field Theory

Author: Franck Gabriel

Publisher: American Mathematical Society, Courant Institute of Mathematical Sciences at New York University

Published: 2024-08-23

Total Pages: 219

ISBN-13: 1470456184

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This book introduces the mathematical ideas connecting Statistical Mechanics and Conformal Field Theory (CFT). Building advanced structures on top of more elementary ones, the authors map out a well-posed road from simple lattice models to CFTs. Structured in two parts, the book begins by exploring several two-dimensional lattice models, their phase transitions, and their conjectural connection with CFT. Through these lattice models and their local fields, the fundamental ideas and results of two-dimensional CFTs emerge, with a special emphasis on the Unitary Minimal Models of CFT. Delving into the delicate ideas that lead to the classification of these CFTs, the authors discuss the assumptions on the lattice models whose scaling limits are described by CFTs. This produces a probabilistic rather than an axiomatic or algebraic definition of CFTs. Suitable for graduate students and researchers in mathematics and physics, Lattice Models and Conformal Field Theory introduces the ideas at the core of Statistical Field Theory. Assuming only undergraduate probability and complex analysis, the authors carefully motivate every argument and assumption made. Concrete examples and exercises allow readers to check their progress throughout.

A Mathematical Introduction to Conformal Field Theory
A Mathematical Introduction to Conformal Field Theory

Author: Martin Schottenloher

Publisher: Springer Science & Business Media

Published: 2008-09-15

Total Pages: 153

ISBN-13: 3540706909

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Part I gives a detailed, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in two dimensions. The conformal groups are determined and the appearence of the Virasoro algebra in the context of the quantization of two-dimensional conformal symmetry is explained via the classification of central extensions of Lie algebras and groups. Part II surveys more advanced topics of conformal field theory such as the representation theory of the Virasoro algebra, conformal symmetry within string theory, an axiomatic approach to Euclidean conformally covariant quantum field theory and a mathematical interpretation of the Verlinde formula in the context of moduli spaces of holomorphic vector bundles on a Riemann surface.

Algebraic Analysis of Solvable Lattice Models
Algebraic Analysis of Solvable Lattice Models

Author: Michio Jimbo

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 180

ISBN-13: 0821803204

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Based on the NSF-CBMS Regional Conference lectures presented by Miwa in June 1993, this book surveys recent developments in the interplay between solvable lattice models in statistical mechanics and representation theory of quantum affine algebras. Because results in this subject were scattered in the literature, this book fills the need for a systematic account, focusing attention on fundamentals without assuming prior knowledge about lattice models or representation theory. After a brief account of basic principles in statistical mechanics, the authors discuss the standard subjects concerning solvable lattice models in statistical mechanics, the main examples being the spin 1/2 XXZ chain and the six-vertex model. The book goes on to introduce the main objects of study, the corner transfer matrices and the vertex operators, and discusses some of their aspects from the viewpoint of physics. Once the physical motivations are in place, the authors return to the mathematics, covering the Frenkel-Jing bosonization of a certain module, formulas for the vertex operators using bosons, the role of representation theory, and correlation functions and form factors. The limit of the XXX model is briefly discussed, and the book closes with a discussion of other types of models and related works.

Strings, Conformal Fields, and Topology
Strings, Conformal Fields, and Topology

Author: Michio Kaku

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 544

ISBN-13: 1468403974

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Following on the foundations laid in his earlier book "Introduction to Superstrings", Professor Kaku discusses such topics as the classification of conformal string theories, the non-polynomial closed string field theory, matrix models, and topological field theory. The presentation of the material is self-contained, and several chapters review material expounded in the earlier book. This book provides students with an understanding of the main areas of current progress in string theory, placing the reader at the forefront of current research.

Quantum and Non-Commutative Analysis
Quantum and Non-Commutative Analysis

Author: Huzihiro Araki

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 452

ISBN-13: 9401728232

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In the past decade, there has been a sudden and vigorous development in a number of research areas in mathematics and mathematical physics, such as theory of operator algebras, knot theory, theory of manifolds, infinite dimensional Lie algebras and quantum groups (as a new topics), etc. on the side of mathematics, quantum field theory and statistical mechanics on the side of mathematical physics. The new development is characterized by very strong relations and interactions between different research areas which were hitherto considered as remotely related. Focussing on these new developments in mathematical physics and theory of operator algebras, the International Oji Seminar on Quantum Analysis was held at the Kansai Seminar House, Kyoto, JAPAN during June 25-29, 1992 by a generous sponsorship of the Japan Society for the Promotion of Science and the Fujihara Foundation of Science, as a workshop of relatively small number of (about 50) invited participants. This was followed by an open Symposium at RIMS, described below by its organizer, A. Kishimoto. The Oji Seminar began with two key-note addresses, one by V.F.R. Jones on Spin Models in Knot Theory and von Neumann Algebras and by A. Jaffe on Where Quantum Field Theory Has Led. Subsequently topics such as Subfactors and Sector Theory, Solvable Models of Statistical Mechanics, Quantum Field Theory, Quantum Groups, and Renormalization Group Ap proach, are discussed. Towards the end, a panel discussion on Where Should Quantum Analysis Go? was held.

Integrable Systems
Integrable Systems

Author: John P. Harnad

Publisher: American Mathematical Soc.

Published:

Total Pages: 284

ISBN-13: 9780821870228

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This volume presents the papers based upon lectures given at the 1999 Séminaire de Mathémathiques Supérieurs held in Montreal. It includes contributions from many of the most active researchers in the field. This subject has been in a remarkably active state of development throughout the past three decades, resulting in new motivation for study in r s3risingly different directions. Beyond the intrinsic interest in the study of integrable models of many-particle systems, spin chains, lattice and field theory models at both the classical and the quantum level, and completely solvable models in statistical mechanics, there have been new applications in relation to a number of other fields of current interest. These fields include theoretical physics and pure mathematics, for example the Seiberg-Witten approach to supersymmetric Yang-Mills theory, the spectral theory of random matrices, topological models of quantum gravity, conformal field theory, mirror symmetry, quantum cohomology, etc. This collection gives a nice cross-section of the current state of the work in the area of integrable systems which is presented by some of the leading active researchers in this field. The scope and quality of the articles in this volume make this a valuable resource for those interested in an up-to-date introduction and an overview of many of the main areas of study in the theory of integral systems.

Quantum Group Symmetry And Q-tensor Algebras
Quantum Group Symmetry And Q-tensor Algebras

Author: Lawrence C Biedenharn

Publisher: World Scientific

Published: 1995-08-31

Total Pages: 305

ISBN-13: 9814500135

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Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations of the q-deformed angular momentum group are discussed, including the case where q is a root of unity, and general results are obtained for all unitary quantum groups using the method of algebraic induction. Tensor operators are defined and discussed with examples, and a systematic treatment of the important (3j) series of operators is developed in detail. This book is a good reference for graduate students in physics and mathematics.