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Introduction to conformal field theory with gauge symmetries. (English) Zbl 0873.32022
Andersen, Jørgen Ellegaard (ed.) et al., Geometry and physics. Proceedings of the conference at Aarhus University, Aarhus, Denmark, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 184, 603-745 (1997).
The author gives a survey of conformal field theory on an arbitrary Riemann surface, emphasizing the connection with moduli. The treatment is very detailed and mathematically rigorous, and could probably form the basis of an advanced course in the subject aimed at an audience familiar with the basics of algebraic geometry. The following list of contents gives some idea of the range of material covered by the article:
Introduction.
1. Compact Riemann surfaces and stable curves; 1.1. Pointed stable curves; 1.2. Deformations of Riemann surfaces and stable curves; 1.3. Versal family of pointed stable curves; 1.4. Period mappings and bidifferentials.
2. Affine Lie algebras and integrable highest weight representations; 2.1. Affine Lie algebras and integrable highest weight modules; 2.2. The Segal-Sugawara form.
3. The space of conformal blocks and correlation functions; 3.1. The space of conformal blocks; 3.2. Formal neighbourhoods; 3.3. Basic properties of the space of conformal blocks; 3.4. Correlation functions; 3.5. The Riemann sphere \(P^1\); 3.6. Elliptic curves.
4. The sheaf of conformal blocks; 4.1. The sheaf of vacua attached to a family of stable curves; 4.2. Local freeness I (the smooth case); 4.3. The family of one pointed stable curves of genus 1; 4.4. Local freeness II (the general case) and factorization.
5. Projectively flat connections and sheaf of twisted differential operators; 5.1. Projectively flat connections; 5.2. The sheaf of twisted differential operators; 5.3. The differential equation near the boundary; 5.4. The differential equation for \(g=0\).
References.
For the entire collection see [Zbl 0855.00020].

MSC:
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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