[Paper Review] Attractors and Arithmetic

Gregory Moore|arXiv (Cornell University)|Jul 8, 1998

Mathematical and Theoretical Analysis21 references57 citations

TL;DR

This paper establishes a deep connection between attractor black holes in Calabi-Yau compactifications of IIB string theory and arithmetic geometry, showing that attractor varieties for N=4,8 supersymmetry arise from products of elliptic curves with complex multiplication. The key result is that the moduli space of such black holes is governed by class field theory, linking string theory to number theory through Galois actions and height functions.

ABSTRACT

We consider attractor varieties arising in the construction of dyonic black holes in Calabi-Yau compactifications of IIB string theory. We show that the attractor varieties are constructed from products of elliptic curves with complex multiplication for $\mathcal{N}=4,8$ compactifications. The heterotic dual theories are related to rational conformal field theories. The emergence of curves with complex multiplication suggests many interesting connections between arithmetic and string theory. This paper is a brief overview of a longer companion paper entitled ``Arithmetic and Attractors,'' hep-th/9807087.

Motivation & Objective

• To explore the emergence of arithmetic structures in attractor black hole solutions within Calabi-Yau compactifications of IIB string theory.
• To investigate the role of complex multiplication in the moduli space of attractor varieties for N=4 and N=8 supersymmetric compactifications.
• To establish a correspondence between attractor points and algebraic number fields, particularly class fields of imaginary quadratic fields.
• To examine whether attractor mechanisms in general N=2 compactifications exhibit arithmetic properties analogous to those in N=4,8 cases.
• To probe the physical significance of Galois symmetry in unifying U-duality-inequivalent black hole backgrounds.

Proposed method

• Analyzes the attractor equations for dyonic black holes in d=4, N=2 supergravity coupled to abelian vectormultiplets, derived from type IIB compactification on Calabi-Yau threefolds.
• Identifies the attractor condition as a Hodge decomposition requirement: γ = γ^{3,0} + γ^{0,3}, which fixes the complex structure moduli to isolated points in the moduli space.
• Applies the central charge minimization principle |Z(z;γ)|² to show that attractor points correspond to local minima of the BPS mass, linking dynamics to stability.
• Uses the F-map (K3 mirror map) to relate special values of modular forms (like j(τ)) to arithmetic data in elliptically fibered K3 surfaces.
• Applies class field theory to show that attractor moduli for N=8 compactifications lie in class fields of imaginary quadratic fields K_D, with Galois group Gal(ĤK_D / K_D) acting on U-duality inequivalent black hole solutions.
• Employs Faltings height functions and logarithmic estimates to compare black hole entropy S = π√I₄(γ) with arithmetic invariants of abelian varieties derived from attractor points.

Experimental results

Research questions

• RQ1 Do attractor varieties in Calabi-Yau compactifications of IIB string theory exhibit arithmetic structure, particularly through complex multiplication?
• RQ2 Can the attractor mechanism in N=4 and N=8 compactifications be fully described by class field theory of imaginary quadratic fields?
• RQ3 Is there a generalization of the j-invariant map to higher-dimensional attractor moduli spaces, analogous to the mirror map for K3 surfaces?
• RQ4 Do multiple attractor points exist for a single charge in the moduli space, and what does this imply for black hole entropy and duality?
• RQ5 Can the growth of BPS state degeneracies (entropy) be quantitatively linked to arithmetic invariants like Faltings height?

Key findings

• Attractor varieties for N=4 and N=8 compactifications arise from products of elliptic curves with complex multiplication, linking string theory to algebraic number theory.
• The attractor points correspond to special values in class fields of imaginary quadratic fields, with the Galois group Gal(ĤK_D / K_D) permuting U-duality-inequivalent black hole solutions.
• For the two-parameter family of degree-8 Calabi-Yau threefolds in P^{1,1,2,2,2}, the attractor conjectures are verified along a special divisor, confirming arithmetic structure in moduli.
• The paper provides a preliminary estimate suggesting that the logarithmic Faltings height of the abelian variety X_γ satisfies h(X_γ / ĤK) ∼ κ log(S/π), with κ expected to be a small rational number.
• Multiple attractor points for a single charge are found in a specific example, implying multiple basins of attraction in the dynamical system, challenging the uniqueness of entropy assignment.
• The F-map for K3 surfaces maps quadratic imaginary y-values to arithmetic (algebraic) coefficients (α, β), generalizing the behavior of the j-invariant to higher-dimensional moduli.