Rational Conformal Field Theories and Complex Multiplication

Details Rational Conformal Field Theories and Complex Multiplication Rational Conformal Field Theories and Complex Multiplication

2002-03-22

arxiv.org/abs/hep-th/0203213v2

Commun.Math.Phys. 246 (2004) 181-210

Physics

High Energy Physics

High Energy Physics - Theory

39 pages, 6 figures, harvmac; references added

Scientific paper

10.1007/s00220-003-1032-0

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT's corresponding to T^2 target and identify the Cardy branes with geometric branes. The T^2's leading to RCFT's admit ``complex multiplication'' which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT's on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.

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