Holography from number theory: Emergent holographic AdS$_2$ space-time from exact BPS black hole microstate counting

Technical Summary: Holography from Number Theory

Problem Statement
The paper addresses a fundamental question in quantum gravity: how to derive the holographic bulk space-time and the statistical origin of Bekenstein-Hawking entropy directly from the exact microscopic degeneracy formulae of BPS black holes. While exact degeneracy formulae for 4D BPS black holes (specifically 1/2 BPS in $N=4$ and 1/8 BPS in $N=8$ theories) are known to be expressed as Rademacher expansions of modular and Jacobi forms, the holographic interpretation of the individual terms in these expansions has remained incomplete. Previous work [6] identified the modified Bessel function component of the Rademacher series with the heat kernel of the de Alfaro-Fubini-Furlan (DFF) conformal quantum mechanics (CQM) model, but a full derivation of the emergent bulk geometry and the holographic encoding of the entire Rademacher sum (including Kloosterman sums) from a CQM dual was lacking.

Methodology
The authors employ a "bottom-up" approach, starting from the number-theoretic degeneracy formulae and reconstructing the holographic dual. The methodology proceeds through the following steps:

1. Microscopic Origin (Calogero to DFF): The paper revisits the $N$-particle rational Calogero model in a harmonic trap, which models BPS black hole microstates. In the large $N$ limit, the authors demonstrate that the system reorganizes into a collective field theory where the radial sector reduces to a single DFF model with a harmonic trapping potential ($DFF_\omega$), evaluated on a specific vacuum state $|0\rangle$ with zero radial momentum.
2. Heat Kernel Matching: The authors compute the Euclidean heat kernel of the $DFF_\omega$ model. By evaluating this kernel on complexified paths (specifically from $x_0$ to $ix_0$) and amputating the ground state wavefunction factors, they show that the resulting expression exactly reproduces the modified Bessel function term ($\hat{I}_a(z)$) found in the Rademacher expansion of the black hole degeneracy.
3. Kloosterman Sum Derivation: To account for the remaining terms in the Rademacher series, the authors derive the classical and generalized Kloosterman sums from a quantum mechanical perspective. They introduce Wilson lines (electric and magnetic charges) in the Euclidean action to generate the classical sums. For the generalized sums appearing in the $N=8$ case (associated with the multiplier system of weak Jacobi forms), they construct an auxiliary Hilbert space and a time-dependent Hamiltonian involving quadratic potentials and delta-function boundary terms. The evolution operator of this system yields the required multiplier system phases.
4. Emergent Geometry via Krylov Basis: To derive the bulk space-time, the authors analyze the time evolution of the $DFF_\omega$ vacuum state $|0\rangle$ in the Krylov basis. They express the evolution as a hopping equation on a discrete lattice indexed by energy levels. Taking the continuum limit of this discrete Schrödinger equation yields a wave equation invariant under an $SL(2, \mathbb{R})$ symmetry group.
5. Geometrisation and Entanglement: By demanding that this $SL(2, \mathbb{R})$ symmetry corresponds to the isometry group of the boundary of an $AdS_2$ space-time, the authors identify the emergent metric as the Poincaré patch of $AdS_2$. The radial coordinate $r$ is identified with the energy scale of the boundary CQM. Finally, they compute the vacuum entanglement entropy of a segment in the associated chiral 2D CFT (derived from the 2D gravity theory of the $AdS_2$ attractor) and show that, via conformal transformation to a thermal cylinder, it reproduces the leading Bekenstein-Hawking entropy.

Key Contributions and Results

• Exact Encoding of Rademacher Terms: The paper explicitly demonstrates that every term in the Rademacher expansion for BPS black hole degeneracies (excluding the overall phase) is encoded in the $DFF_\omega$ CQM. The modified Bessel function arises from the amputated heat kernel on complexified paths, while the Kloosterman sums arise from Wilson line holonomies and specific Hamiltonian evolution in an extended Hilbert space.
• Emergence of $AdS_2$: The authors derive the near-horizon $AdS_2$ attractor geometry directly from the dynamics of the boundary CQM. The radial coordinate of the emergent $AdS_2$ space-time is shown to be the continuum limit of the discrete energy spectrum index (Krylov complexity) of the CQM, establishing the $AdS_2$/CQM correspondence from first principles of the microscopic counting.
• Entropy as Entanglement: The Bekenstein-Hawking entropy is recovered as the thermal entropy of the chiral 2D CFT dual to the $AdS_2$ bulk. This thermal entropy is derived from the vacuum entanglement entropy of a finite segment in the CFT, computed using the central charge determined by the ground state energy of the $DFF_\omega$ model.
• Unification of Number Theory and Holography: The work provides a concrete dictionary between number-theoretic data (modular forms, Rademacher sums, Kloosterman sums) and physical quantities in a holographic quantum mechanical model (heat kernels, Wilson lines, Krylov evolution).

Significance
The paper claims to provide the first known holographic derivation of exact BPS black hole entropy in string theory starting from a candidate holographic CQM dual. By inferring the holographic CQM and the emergent $AdS_2$ bulk directly from the automorphic structures governing BPS state degeneracies, the authors demonstrate that the near-horizon geometry and the statistical entropy are intrinsic consequences of the underlying conformal quantum mechanics. This work bridges the gap between the exact mathematical formulae for black hole microstates and the geometric realization of the holographic principle, showing that the $AdS_2$ attractor geometry emerges naturally from the temporal diffusion of the CQM vacuum in the Krylov basis.