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 int- ductory text to conformal ?eld theory, but also to get in contact with some recent developments. In this way we take a number of remarks and contributions by re- ers of the ?rst edition into consideration who appreciated the rather detailed and self-contained exposition in the ?rst part of the notes but asked for more details for the second part. The enlarged edition also re?ects experiences made in seminars on the subject. The interest in conformal ?eld theory has grown during the last 10 years and several texts and monographs re?ecting different aspects of the ?eld have been p- lished as, e. g. , the detailed physics-oriented introduction of Di Francesco, Mathieu, 1 and Sen ´ echal ´ [DMS96*], the treatment of conformal ?eld theories as vertex - gebras by Kac [Kac98*], the development of conformal ?eld theory in the context of algebraic geometry as in Frenkel and Ben-Zvi [BF01*] and more general by Beilinson and Drinfeld [BD04*]. There is also the comprehensive collection of ar- clesbyDeligne,Freed,Witten,andothersin[Del99*]aimingtogiveanintroduction to strings and quantum ?eld theory for mathematicians where conformal ?eld 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.
The first part of this book gives a detailed, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in two dimensions. In particular, the conformal groups are determined and the appearance 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. The second part surveys some 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.
The substantially revised and enlarged second edition makes in particular the second part of the book more self-contained and tutorial, with many more examples given. Furthermore, two new chapters on Wightman's axioms for quantum field theory and vertex algebras broaden the survey of advanced topics. An outlook making the connection with most recent developments has also been added.
