[Paper Review] BPS black holes, quantum attractor flows and automorphic forms

Murat Günaydin, Andrew Neitzke+2 more|arXiv (Cornell University)|Dec 22, 2005

Black Holes and Theoretical Physics56 references82 citations

TL;DR

This paper proposes that exact degeneracies of four-dimensional BPS black holes in N≥2 supergravities with symmetric scalar manifolds are counted by Fourier coefficients of automorphic forms associated with the three-dimensional U-duality group G3. By radial quantization of the attractor flow, the BPS Hilbert space is identified as a unipotent representation of G3, and the black hole degeneracies are conjectured to arise from generalized Fourier coefficients of modular forms for G3, offering a non-perturbative realization of the OSV conjecture via automorphic forms.

ABSTRACT

We propose a program for counting microstates of four-dimensional BPS black holes in N >= 2 supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degeneracies are counted by Fourier coefficients of modular forms for the three-dimensional U-duality group, associated to special "unipotent" representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.

Motivation & Objective

• To understand the microscopic origin of BPS black hole entropy beyond the large charge limit.
• To establish a connection between black hole degeneracies and automorphic forms for the three-dimensional U-duality group G3.
• To provide a quantum mechanical framework for BPS black holes using radial quantization of the attractor flow.
• To generalize the Ooguri-Strominger-Vafa conjecture by identifying black hole degeneracies as Fourier coefficients of modular forms.

Proposed method

• Performing a timelike reduction of four-dimensional N≥2 supergravity to three-dimensional Euclidean gravity coupled to a non-linear sigma model with G3 symmetry.
• Mapping stationary, spherically symmetric BPS solutions to geodesic flows on the scalar manifold M∗3 = G3/H3.
• Radial quantization of the attractor flow by replacing geodesics with square-integrable wave functions in L2(G3/H3).
• Identifying the BPS Hilbert space as a unipotent representation of G3, with a distinguished spherical vector generalizing the topological string amplitude.
• Conjecturing that exact black hole degeneracies are generalized Fourier coefficients of automorphic forms constructed from these unipotent representations.
• Using the quasi-conformal realization of G3 on electric and magnetic charges to relate black hole states across different charge orbits.

Experimental results

Research questions

• RQ1 How can the attractor flow in four-dimensional N≥2 supergravity be recast as geodesic motion in three dimensions?
• RQ2 What is the role of the three-dimensional U-duality group G3 as a spectrum-generating symmetry for BPS black hole states?
• RQ3 Can the BPS Hilbert space in radial quantization be identified as a unipotent representation of G3?
• RQ4 How do the Fourier coefficients of automorphic forms for G3 relate to exact black hole degeneracies?
• RQ5 In what way does this framework generalize or realize the Ooguri-Strominger-Vafa conjecture non-perturbatively?

Key findings

• The BPS Hilbert space in radial quantization of the attractor flow realizes a unipotent representation of the three-dimensional U-duality group G3.
• The unipotent representation is of unusually small functional dimension and contains a distinguished spherical vector that generalizes the topological string amplitude Ψtop.
• The exact degeneracies of BPS black holes are conjectured to be generalized Fourier coefficients of automorphic forms associated with the unipotent representation of G3.
• The proposal provides a non-perturbative realization of the OSV conjecture, where |Ψtop|2 is replaced by the square of a wave function in a unitary representation of G3.
• The spectrum-generating symmetry G3(Z) is conjectured to control exact black hole degeneracies, linking number theory and quantum black hole physics.
• The framework explains mysterious modular properties observed in BPS degeneracies and suggests a deeper role for automorphic forms in quantum gravity.