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Aspects of conformal field theory from Calabi-Yau arithmetic. (English) Zbl 1061.11032
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 233-253 (2003).
Summary: This paper describes a framework in which techniques from arithmetic algebraic geometry are used to formulate a direct and intrinsic link between the geometry of Calabi-Yau manifolds and aspects of the underlying conformal field theory. As an application the algebraic number field determined by the fusion rules of the conformal field theory is derived from the number theoretic structure of the cohomological Hasse-Weil $$L$$-function determined by Artin’s congruent zeta function of the algebraic variety. In this context a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.
See also the author’s paper (jointly with S. Underwood) in J. Geom. Phys. 48, No. 2–3, 169–189 (2003; Zbl 1033.11030).
For the entire collection see [Zbl 1022.00014].

MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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