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Rational Conformal Field Theories and Complex Multiplication

Gukov, Sergei and Vafa, Cumrun (2004) Rational Conformal Field Theories and Complex Multiplication. Communications in Mathematical Physics, 246 (1). pp. 181-210. ISSN 0010-3616. doi:10.1007/s00220-003-1032-0.

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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 the 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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Gukov, Sergei0000-0002-9486-1762
Additional Information:© 2004 Springer-Verlag. Received: 18 July 2003. Accepted: 26 August 2003. Published online: 23 January 2004. We would like to thank D. Kazhdan and B. Mazur for many illuminating discussions on complex multiplication. We are also grateful to J. de Jong, J. Maldacena, K. Oguiso, H. Ooguri, F. Oort, A. Recknagel, S. Shenker, F. Rodriguez-Villegas, and E. Witten for valuable discussions. This research was partially conducted during the period S.G. served as a Clay Mathematics Institute Long-Term Prize Fellow. The work of S.G. is also supported in part by grant RFBR No. 01-02-17488, and the Russian President’s grant No. 00-15-99296. The work of C.V. is supported in part by NSF grants PHY-9802709 and DMS 0074329. Communicated by Y. Kawahigashi.
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Russian Foundation for Basic Research01-02-17488
Russian Foundation for Basic Research00-15-99296
Clay Mathematics InstituteUNSPECIFIED
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ID Code:66710
Deposited By: Ruth Sustaita
Deposited On:06 May 2016 19:59
Last Modified:11 Nov 2021 00:01

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