[Paper Review] Arithmetic and Attractors

Gregory Moore|ArXiv.org|Jul 13, 1998

Black Holes and Theoretical Physics131 references171 citations

TL;DR

This paper establishes a deep connection between arithmetic geometry and supersymmetric black holes via the attractor mechanism in N=2 supergravity. It shows that attractor varieties in N=4 and N=8 compactifications are arithmetic, arising from elliptic curves with complex multiplication, and are linked to class numbers and ring class fields of imaginary quadratic fields, revealing a profound link between black hole entropy and number theory.

ABSTRACT

We study relations between some topics in number theory and supersymmetric black holes. These relations are based on the ``attractor mechanism'' of N=2 supergravity. In IIB string compactification this mechanism singles out certain ``attractor varieties.'' We show that these attractor varieties are constructed from products of elliptic curves with complex multiplication for N=4 and N=8 compactifications. The heterotic dual theories are related to rational conformal field theories. In the case of N=4 theories U-duality inequivalent backgrounds with the same horizon area are counted by the class number of a quadratic imaginary field. The attractor varieties are defined over fields closely related to class fields of the quadratic imaginary field. We discuss some extensions to more general Calabi-Yau compactifications and explore further connections to arithmetic including connections to Kronecker's Jugendtraum and the theory of modular heights. The paper also includes a short review of the attractor mechanism. A much shorter version of the paper summarizing the main points is the companion note entitled ``Attractors and Arithmetic'' (hep-th/9807056).

Motivation & Objective

• To explore the mathematical structure of attractor varieties in supersymmetric black holes within string compactifications.
• To investigate the role of U-duality groups and their orbits in classifying black hole solutions with fixed horizon areas.
• To connect the geometry of attractor varieties to arithmetic invariants such as class numbers and ring class fields.
• To extend the attractor mechanism to Calabi-Yau threefolds and relate them to modular forms and heights.
• To examine the implications of these structures for Kronecker’s Jugendtraum and Hilbert’s twelfth problem in number theory.

Proposed method

• Uses the attractor mechanism in N=2 supergravity to fix moduli at black hole horizons, leading to special algebraic varieties.
• Analyzes attractor varieties in IIB compactifications on K3×T2 and related FHSV models, identifying them as products of elliptic curves with complex multiplication.
• Applies the U-duality group action to classify BPS states and computes the number of U-duality inequivalent backgrounds with equal horizon area via class numbers.
• Relies on the geometry of period integrals and the Hodge structure of Calabi-Yau threefolds to define attractor points as solutions to attractor equations.
• Connects the resulting moduli spaces to class fields of imaginary quadratic fields using complex multiplication theory.
• Extends results to large complex structure limits and mirror symmetry, using the K3 mirror map and modular invariants.

Experimental results

Research questions

• RQ1 Why do attractor varieties in N=4 and N=8 compactifications exhibit arithmetic structure?
• RQ2 How are the number of U-duality inequivalent black hole backgrounds with the same horizon area related to class numbers?
• RQ3 What is the role of complex multiplication in the construction of attractor varieties?
• RQ4 How do attractor points in Calabi-Yau threefolds relate to modular forms and arithmetic heights?
• RQ5 Can the attractor mechanism provide a physical realization of Kronecker’s Jugendtraum and Hilbert’s twelfth problem?

Key findings

• Attractor varieties in N=4 and N=8 compactifications are constructed from products of elliptic curves with complex multiplication.
• The number of U-duality inequivalent backgrounds with the same horizon area is given by the class number of an imaginary quadratic field.
• The attractor varieties are defined over ring class fields of the quadratic imaginary field, linking them to class field theory.
• The attractor mechanism selects points in moduli space that are algebraic and arithmetic, with special Galois invariance.
• The construction generalizes to Calabi-Yau threefolds, where attractor points are related to modular heights and special values of L-functions.
• The results support a physical interpretation of Kronecker’s Jugendtraum, with the attractor moduli space realizing the abelian extensions of imaginary quadratic fields.