Current Algebras On Riemann Surfaces
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| Author | : Oleg K. Sheinman |
| Publisher | : Walter de Gruyter |
| Total Pages | : 164 |
| Release | : 2012-10-01 |
| Genre | : Mathematics |
| ISBN | : 3110264528 |
This monograph is an introduction into a new and fast developing field on the crossroads of infinite-dimensional Lie algebra theory and contemporary mathematical physics. It contains a self-consistent presentation of the theory of Krichever-Novikov algebras, Lax operator algebras, their interaction, representation theory, relations to moduli spaces of Riemann surfaces and holomorphic vector bundles on them, to Lax integrable systems, and conformal field theory. For beginners, the book provides a short way to join in the investigations in these fields. For experts, it sums up the recent advances in the theory of almost graded infinite-dimensional Lie algebras and their applications. The book may serve as a base for semester lecture courses on finite-dimensional integrable systems, conformal field theory, almost graded Lie algebras. Majority of results are presented for the first time in the form of monograph.
| Author | : Jouko Mickelsson |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 325 |
| Release | : 2013-03-09 |
| Genre | : Science |
| ISBN | : 1475702957 |
Let M be a smooth manifold and G a Lie group. In this book we shall study infinite-dimensional Lie algebras associated both to the group Map(M, G) of smooth mappings from M to G and to the group of dif feomorphisms of M. In the former case the Lie algebra of the group is the algebra Mg of smooth mappings from M to the Lie algebra gof G. In the latter case the Lie algebra is the algebra Vect M of smooth vector fields on M. However, it turns out that in many applications to field theory and statistical physics one must deal with certain extensions of the above mentioned Lie algebras. In the simplest case M is the unit circle SI, G is a simple finite dimensional Lie group and the central extension of Map( SI, g) is an affine Kac-Moody algebra. The highest weight theory of finite dimensional Lie algebras can be extended to the case of an affine Lie algebra. The important point is that Map(Sl, g) can be split to positive and negative Fourier modes and the finite-dimensional piece g corre sponding to the zero mode.
| Author | : Benson Farb |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 371 |
| Release | : 2013-08-16 |
| Genre | : Mathematics |
| ISBN | : 0821898876 |
Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.
| Author | : Rick Miranda |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 414 |
| Release | : 1995 |
| Genre | : Mathematics |
| ISBN | : 0821802682 |
In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking centre stage. But the main examples come fromprojective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Dualtiy Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves andcohomology are introduced as a unifying device in the later chapters, so that their utility and naturalness are immediately obvious. Requiring a background of one term of complex variable theory and a year of abstract algebra, this is an excellent graduate textbook for a second-term course in complex variables or a year-long course in algebraic geometry.
| Author | : Martin Schlichenmaier |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 228 |
| Release | : 2010-02-11 |
| Genre | : Science |
| ISBN | : 3540711759 |
This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.
| Author | : Martin Schlichenmaier |
| Publisher | : Walter de Gruyter GmbH & Co KG |
| Total Pages | : 453 |
| Release | : 2014-08-19 |
| Genre | : Mathematics |
| ISBN | : 3110381478 |
Krichever and Novikov introduced certain classes of infinite dimensional Lie algebras to extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them to a more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are still manageable. This book gives an introduction for the newcomer to this exciting field of ongoing research in mathematics and will be a valuable source of reference for the experienced researcher. Beside the basic constructions and results also applications are presented.
| Author | : Philippe Francesco |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 908 |
| Release | : 2012-12-06 |
| Genre | : Science |
| ISBN | : 1461222567 |
Filling an important gap in the literature, this comprehensive text develops conformal field theory from first principles. The treatment is self-contained, pedagogical, and exhaustive, and includes a great deal of background material on quantum field theory, statistical mechanics, Lie algebras and affine Lie algebras. The many exercises, with a wide spectrum of difficulty and subjects, complement and in many cases extend the text. The text is thus not only an excellent tool for classroom teaching but also for individual study. Intended primarily for graduate students and researchers in theoretical high-energy physics, mathematical physics, condensed matter theory, statistical physics, the book will also be of interest in other areas of theoretical physics and mathematics. It will prepare the reader for original research in this very active field of theoretical and mathematical physics.
| Author | : Sergei D. Silvestrov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 308 |
| Release | : 2008-11-18 |
| Genre | : Mathematics |
| ISBN | : 3540853324 |
This book explores the cutting edge of the fundamental role of generalizations of Lie theory and related non-commutative and non-associative structures in mathematics and physics.
| Author | : Masaki Kashiwara |
| Publisher | : Academic Press |
| Total Pages | : 501 |
| Release | : 2014-05-10 |
| Genre | : Mathematics |
| ISBN | : 1483267946 |
Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of his 60th Birthday, Volume II is a collection of research papers on algebraic analysis and related topics in honor to Professor Mikio Sato’s 60th birthday. This volume is divided into 29 chapters and starts with research works concerning the fundamentals of KP equations, strings, Schottky problem, and the applications of transformation theory for nonlinear integrable systems to linear prediction problems and isospectral deformations,. The subsequent chapters contain papers on the approach to nonlinear integrable systems, the Hodge numbers, the stochastic different equation for the multi-dimensional weakly stationary process, and a method of harmonic analysis on semisimple symmetric spaces. These topics are followed by studies on the quantization of extended vortices, moduli space for Fuchsian groups, microfunctions for boundary value problems, and the issues of multi-dimensional integrable systems. The remaining chapters explore the practical aspects of pseudodifferential operators in hyperfunction theory, the elliptic solitons, and Carlson’s theorem for holomorphic functions. This book will prove useful to mathematicians and advance mathematics students.
| Author | : Mario Rasetti |
| Publisher | : World Scientific |
| Total Pages | : 234 |
| Release | : 1990 |
| Genre | : Science |
| ISBN | : 9789810202255 |
http://www.worldscientific.com/worldscibooks/10.1142/1095
