The second edition of these notes has been completely rewritten and substantially expanded, with the intention not only to improve the use of the book as an introductory text to conformal field theory, but also to get in contact with some recent developments.

In this way, we take a number of remarks and contributions by readers of the first edition into consideration. They appreciated the rather detailed and self-contained exposition in the first part of the notes, but asked for more details for the second part. The enlarged edition also reflects experiences made in seminars on the subject.

The interest in conformal field theory has grown during the last 10 years, and several texts and monographs reflecting different aspects of the field have been published, such as the detailed, physics-oriented introduction of Di Francesco, Mathieu, and Sénéchal, the treatment of conformal field theories as vertex algebras by Kac, the development of conformal field theory in the context of algebraic geometry as in Frenkel and Ben-Zvi and, more generally, by Beilinson and Drinfeld.

There is also the comprehensive collection of articles by Deligne, Freed, Witten, and others in [Del99*], aiming to give an introduction to strings and quantum field theory for mathematicians, where conformal field theory is one of the main parts of the text.

The present expanded notes complement these publications by giving an elementary and comparatively short, mathematics-oriented introduction, focusing on some main principles.

Table of Contents

Part I Mathematical Preliminaries

1 Conformal Transformations and Conformal Killing Fields

2 The Conformal Group

3 Central Extensions of Groups

4 Central Extensions of Lie Algebras and Bargmann’s Theorem

5 The Virasoro Algebra

Part II First Steps Toward Conformal Field Theory

6 Representation Theory of the Virasoro Algebra

7 String Theory as a Conformal Field Theory

8 Axioms of Relativistic Quantum Field Theory

9 Foundations of Two-Dimensional Conformal Quantum Field Theory

10 Vertex Algebras

11 Mathematical Aspects of the Verlinde Formula

A Some Further Developments

Review

From the reviews of the second edition:

"The book under review is the second … edition of the original text, thereby reflecting various recent developments in the field during the past ten years, on the one hand, and making especially the physics part of the notes more detailed, self-contained and tutorial on the other. … this highly appreciated mediator between contemporary mathematics and physical quantum field theory has grown into an even more valuable source for students, teachers, and active researchers in this fascinating area of modern science." (Werner Kleinert, Zentralblatt MATH, Vol. 1161, 2009)

“The second edition of this classical introduction to conformal field theory for a mathematically oriented audience has been substantially revised and expanded … . many new examples and statements have been included throughout the whole text. … Completeness, the extremely clear exposition and well-chosen references at the end of each chapter make this introduction an unmissable reference for the mathematician aiming to learn the basics of conformal field theory.” (Domenico Fiorenza, Mathematical Reviews, Issue 2011 a)