Complex Multiplication Symmetry of Black Hole Attractors

We show how Moore's observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne's period conjecture.Comment: 28 page

Twenty-five years of two-dimensional rational conformal field theory

In this article we try to give a condensed panoramic view of the development of two-dimensional rational conformal field theory in the last twenty-five years.Comment: A review for the 50th anniversary of the Journal of Mathematical Physics. Some references added, typos correcte

On Fields of rationality for automorphic representations

This paper proves two results on the field of rationality \Q(\pi) for an automorphic representation $\pi$, which is the subfield of \C fixed under the subgroup of \Aut(\C) stabilizing the isomorphism class of the finite part of $\pi$. For general linear groups and classical groups, our first main result is the finiteness of the set of discrete automorphic representations $\pi$ such that $\pi$ is unramified away from a fixed finite set of places, $\pi_\infty$ has a fixed infinitesimal character, and [\Q(\pi):\Q] is bounded. The second main result is that for classical groups, [\Q(\pi):\Q] grows to infinity in a family of automorphic representations in level aspect whose infinite components are discrete series in a fixed $L$-packet under mild conditions

Symmetric boundary conditions in boundary critical phenomena

Conformally invariant boundary conditions for minimal models on a cylinder are classified by pairs of Lie algebras $(A,G)$ of ADE type. For each model, we consider the action of its (discrete) symmetry group on the boundary conditions. We find that the invariant ones correspond to the nodes in the product graph $A \otimes G$ that are fixed by some automorphism. We proceed to determine the charges of the fields in the various Hilbert spaces, but, in a general minimal model, many consistent solutions occur. In the unitary models $(A,A)$, we show that there is a unique solution with the property that the ground state in each sector of boundary conditions is invariant under the symmetry group. In contrast, a solution with this property does not exist in the unitary models of the series $(A,D)$ and $(A,E_6)$. A possible interpretation of this fact is that a certain (large) number of invariant boundary conditions have unphysical (negative) classical boundary Boltzmann weights. We give a tentative characterization of the problematic boundary conditions.Comment: 13 pages, REVTeX; reorganized and expanded version; includes a new section on unitary minimal models; conjectures reformulated, pointing to the generic existence of negative boundary Boltzmann weights in unitary model