Rational Conformal Field Theories and Complex Multiplication

TL;DR

This paper explores the geometric aspects of rational conformal field theories (RCFTs) on Calabi-Yau manifolds, linking complex multiplication to specific branes and proposing conditions for RCFTs in higher dimensions.

Contribution

It identifies the geometric interpretation of RCFTs on T^2, relates complex multiplication to D0-branes, and proposes a general condition for RCFTs on arbitrary Calabi-Yau n-folds.

Findings

RCFTs on T^2 correspond to specific geometric branes.

Complex multiplication characterizes certain Cardy branes as D0-branes.

Rational conformal theories are rare and possibly finite for Calabi-Yau n-folds with n>2.

Abstract

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.