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The mathematics of string theory. (English) Zbl 1146.58001
Summary: Over the years string theory [J. Polchinski, String theory, Cambridge: University Press, vol. I (1998; Zbl 1006.81521), vol. II (1998; Zbl 1006.81522)] has been able to enrich various fields of mathematics. Subjects like algebraic and differential geometry, topology, representation theory, infinite dimensional analysis and many others have been stimulated by new concepts such as mirror symmetry [D. A. Cox, S. Katz, Mathematical Surveys and Monographs 68. Providence; RI: AMS (1999; Zbl 0951.14026); K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, S. Vafa, R. Vakil and E. Zaslov, Clay Mathematics Monographs 1. Providence, RI: AMS (2003; Zbl 1044.14018)], quantum cohomology [Y. I. Manin, Colloquium Publications. 47. Providence, RI: AMS (1999; Zbl 0952.14032); D. McDuff and D. Salamon, University Lecture Series. 6. Providence, RI: AMS (1994; Zbl 0809.53002)] and conformal field theory [P. Di Francesco, P. Mathieu and D. Sénéchal, Graduate Texts in Contemporary Physics. New York, NY: Springer. (1997; Zbl 0869.53052)]. In fact, one can argue that this stimulating influence in mathematics will be a lasting and rewarding impact of string theory in science, whatever its final role in fundamental physics. String theory seems to be the most complex and richest mathematical object that has so far appeared in physics and the inspiring dialogue between mathematics and physics that it has triggered is blooming and spreading in wider and wider circles of mathematics.
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory