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Conformal field theory and geometry of strings. (English) Zbl 0826.58007

Feldman, J. (ed.) et al., Mathematical quantum theory I: Field theory and many body theory. Proceedings of the Canadian Mathematical Society annual seminar, held in Vancouver, Canada, August 4-14, 1993. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 7, 57-97 (1994).
The subject of this pedagogical and highly readable article is a discussion of conformal field theory models (which may be considered as tree-level solutions of string theory) from the point of view of noncommutative geometry as introduced by A. Connes [Publ. Math., Inst. Hautes Étud. Sci. 62, 257-360 (1985; Zbl 0592.46056)]. Already the simplest sigma models (with target spaces \(S^1\) and \(T^2\), respectively) exhibit the “stringy” phenomena of target space duality and mirror symmetry. Further topics treated in detail are: Wess-Zumino- Witten and “coset” theories, supersymmetric conformal field theories in general with an introduction to noncommutative de Rham calculus, supersymmetric WZW and coset models, and the relation between \(N = 2\) supersymmetric conformal field theory models and Calabi-Yau geometry (which lies at the core of mirror symmetry).
For the entire collection see [Zbl 0807.00020].
Reviewer: H.Rumpf (Wien)

MSC:

46L85 Noncommutative topology
46L87 Noncommutative differential geometry
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
46L55 Noncommutative dynamical systems
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory

Citations:

Zbl 0592.46056
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