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Quantization, Classical and Quantum Field Theory and Theta Functions
Andrei Tyurin, Steklov Institute of Mathematical Sciences, Moscow, Russia
A co-publication of the AMS and Centre de Recherches Mathématiques.

CRM Monograph Series
2003; 136 pp; hardcover
Volume: 21
ISBN-10: 0-8218-3240-9
ISBN-13: 978-0-8218-3240-0
List Price: US$50
Member Price: US$40
Order Code: CRMM/21
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This book is written by a well-known expert in classical algebraic geometry. Tyurin's research was specifically in explicit computations to vector bundles on algebraic varieties. This is the only available monograph written from his unique viewpoint.

Ordinary (abelian) theta functions describe properties of moduli spaces of one-dimensional vector bundles on algebraic curves. Non-abelian theta functions, which are the main topic of this book, play a similar role in the study of higher-dimensional vector bundles. The book presents various aspects of the theory of non-abelian theta functions and the moduli spaces of vector bundles, including their applications to problems of quantization and to classical and quantum conformal field theories.

The book is an important source of information for specialists in algebraic geometry and its applications to mathematical aspects of quantum field theory.

Titles in this series are co-published with the Centre de Recherches Mathématiques.


Graduate students and research mathematicians interested in algebraic geometry and its applications to mathematical physics.


"The book opens a beautiful and grandiose view to a fascinating part of geometry, notably symplectic and algebraic geometry, and, as usual in Andrei Tyurin's work, it has a very geometric flavor. ...The ideal reader should preferably dispose of some basic knowledge of classical algebraic geometry or theory of Riemann surfaces and of symplectic geometry, then he will benefit quite a lot from reading this book."

-- Zentralblatt MATH

Table of Contents

  • Quantization procedure
  • Algebraic curves = Riemann surfaces
  • Non-abelian theta functions
  • Symplectic geometry of moduli spaces of vector bundles
  • Two versions of CQFT
  • Three-valent graphs
  • Analytical aspects of the theory of non-abelian theta functions
  • BPU-map
  • The main weapon
  • Bibliography
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