Rational Conformal Field Theories and Complex Multiplication

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.

39 pages, 6 figures, harvmac; references added

References in corpus (3)

• Moduli Stabilization from Fluxes in a Simple IIB Orientifold
• Conformal boundary states for free bosons and fermions
• Boundary States of c=1 and 3/2 Rational Conformal Field Theories

Cited by in corpus (35)

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• Topological Transition on the Conformal Manifold
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• Fixing All Moduli for M-Theory on K3xK3
• (2,2) Superconformal Bootstrap in Two Dimensions
• Narain to Narnia
• The Entropic Principle and Asymptotic Freedom
• Enhanced Symmetries in Multiparameter Flux Vacua
• Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions
• On the Classification of Elliptic Fibrations modulo Isomorphism on K3 Surfaces with large Picard Number
• Arithmetic and Hyperbolic Structures in String Theory
• Gauge Threshold Corrections for N = 2 Heterotic Local Models with Flux, and Mock Modular Forms
• Revisiting arithmetic solutions to the $W=0$ condition
• Examples of Calabi-Yau threefolds parametrised by Shimura varieties
• Complex Multiplication of Exactly Solvable Calabi-Yau Varieties
• T-Duality in Gauged Linear Sigma-Models with Torsion
• String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
• Local models of heterotic flux vacua: spacetime and worldsheet aspects
• Threshold corrections in heterotic flux compactifications
• New supersymmetric index of heterotic compactifications with torsion
• Maximal automorphisms of Calabi-Yau manifolds versus maximally unipotent monodromy
• Level Crossings, Attractor Points and Complex Multiplication
• Complex multiplication, rationality and mirror symmetry for abelian varieties
• On Rational Points in CFT Moduli Spaces
• W=0 Complex Structure Moduli Stabilization on CM-type K3 x K3 Orbifolds:---Arithmetic, Geometry and Particle Physics---
• Compactified Strings as Quantum Statistical Partition Function on the Jacobian Torus
• CY Manifolds with Locally Symmetric Moduli Spaces
• Black Holes and Hurwitz Class Numbers
• Abelian Varieties, RCFTs, Attractors, and Hitchin Functional in Two Dimensions
• Attractor Flows from Defect Lines
• On the classification of duality defects in $c=2$ compact boson CFTs with a discrete group orbifold
• Rational conformal field theory with matrix level and strings on a torus
• A note on varieties of weak CM-type
• Examples of Calabi-Yau 3-manifolds with complex multiplication
• Counting Unimodular Lattices in ${\mathbb{R}}^{r,s}$