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Lectures on conformal field theory. (English) Zbl 1170.81430
Deligne, Pierre (ed.) et al., Quantum fields and strings: a course for mathematicians. Vol. 1, 2. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. Providence, RI: AMS, American Mathematical Society (ISBN 0-8218-1987-9/vol.1; 0-8218-1988-7/vol.2; 0-8218-1198-3/set). 727-805 (1999).
As the author states: “Over the last decade and a half, conformal field theory (CFT) has been one of the main domains of interaction between theoretical physics and mathematics. The present review is designed as an introduction to the subject aimed at mathematicians$$\ldots.$$” It contains a wealth of material. In these lectures the concept of action functional is stressed. The presentation is divided into four lectures. (1) Conformal free fields—simple functional integrals with the subtopics: What is a quantum field theory? Euclidean free field and Gaussian functional integrals. Feynman-Kac formula. Massless free fields with values in $$S^1$$. Toroidal compactifications: the partition and correlation functions. (2) Axiomatic approaches to CFT, with the subtopics: Conformal field theory data. Conformal Ward identities. Physical positivity and Hilbert space picture. Virasoro algebra and its primary fields. Highest weight representations. Segal’s axioms and vertex operator algebras. (3) Perturbative analysis of two-dimensional sigma models with the subtopics: 1PI effective action and large deviations. Geometric sigma models. Regularization and renormalization. Renormalization group effective actions. Background field effective action. Dimensional regularization. Renormalization of the sigma models to 1 loop. Renormalization group analysis of sigma models. (4) Exact solutions of the Wess-Zumino-Witten model and coset theories with the subtopics: WZW model. Gauge symmetry Ward identities. Scalar product of nonabelian theta functions. KZB connection. Coset theories. WZW factory. Each of the lectures closes with a very helpful hint to the literature.
For the entire collection see [Zbl 0984.00503].

##### MSC:
 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations 81T17 Renormalization group methods applied to problems in quantum field theory 81T45 Topological field theories in quantum mechanics