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AdS $N$-body problem at large spin

Petr Kravchuk, Jeremy A. Mann

TL;DR

This work establishes a semiclassical Berezin-Toeplitz quantization framework for leading-twist $N$-body states in AdS at large spin $J$, revealing a classical phase space given by $\mathrm{Gr}_{+}(2,N)$ (or ${\mathbb{C}P}^{N-2}$ after reduction) and an effective Hamiltonian encoding pairwise interactions. For $N=3$, the spectrum is controlled by Bohr-Sommerfeld quantization with acute/obtuse regions, and subleading corrections yield harmonic-oscillator-like towers for low-lying states, with permutation symmetries imposing detailed degeneracy structures. For general $N$, the problem remains semiclassical but with $N-2$ degrees of freedom, producing Weyl-law density of states and localized low-lying excitations around a unique minimum; BT quantization and pseudodifferential methods characterize the leading and subleading spectra, while numerical diagonalization confirms the analytic predictions and the Lorentzian inversion formula connections. The results provide analytical handles on multi-twist operator spectra in holographic CFTs, illuminate connections to LLL physics on hyperbolic spaces, and suggest avenues toward understanding universal multi-twist dynamics beyond holography. Overall, the paper bridges AdS/CFT, geometric quantization, and spectral analysis to reveal the structure of high-spin, multi-particle sectors in strongly coupled quantum field theories.

Abstract

Motivated by the problem of multi-twist operators in general CFTs, we study the leading-twist states of the $N$-body problem in AdS at large spin $J$. We find that for the majority of states the effective quantum-mechanical problem becomes semiclassical with $\hbar=1/J$. The classical system at $J=\infty$ has $N-2$ degrees of freedom, and the classical phase space is identified with the positive Grassmannian $\mathrm{Gr}_{+}(2,N)$. The quantum problem is recovered via a Berezin-Toeplitz quantization of a classical Hamiltonian, which we describe explicitly. For $N=3$ the classical system has one degree of freedom and a detailed structure of the spectrum can be obtained from Bohr-Sommerfeld conditions. For all $N$, we show that the lowest excited states are approximated by a harmonic oscillator and find explicit expressions for their energies.

AdS $N$-body problem at large spin

TL;DR

This work establishes a semiclassical Berezin-Toeplitz quantization framework for leading-twist -body states in AdS at large spin , revealing a classical phase space given by (or after reduction) and an effective Hamiltonian encoding pairwise interactions. For , the spectrum is controlled by Bohr-Sommerfeld quantization with acute/obtuse regions, and subleading corrections yield harmonic-oscillator-like towers for low-lying states, with permutation symmetries imposing detailed degeneracy structures. For general , the problem remains semiclassical but with degrees of freedom, producing Weyl-law density of states and localized low-lying excitations around a unique minimum; BT quantization and pseudodifferential methods characterize the leading and subleading spectra, while numerical diagonalization confirms the analytic predictions and the Lorentzian inversion formula connections. The results provide analytical handles on multi-twist operator spectra in holographic CFTs, illuminate connections to LLL physics on hyperbolic spaces, and suggest avenues toward understanding universal multi-twist dynamics beyond holography. Overall, the paper bridges AdS/CFT, geometric quantization, and spectral analysis to reveal the structure of high-spin, multi-particle sectors in strongly coupled quantum field theories.

Abstract

Motivated by the problem of multi-twist operators in general CFTs, we study the leading-twist states of the -body problem in AdS at large spin . We find that for the majority of states the effective quantum-mechanical problem becomes semiclassical with . The classical system at has degrees of freedom, and the classical phase space is identified with the positive Grassmannian . The quantum problem is recovered via a Berezin-Toeplitz quantization of a classical Hamiltonian, which we describe explicitly. For the classical system has one degree of freedom and a detailed structure of the spectrum can be obtained from Bohr-Sommerfeld conditions. For all , we show that the lowest excited states are approximated by a harmonic oscillator and find explicit expressions for their energies.

Paper Structure

This paper contains 58 sections, 277 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Some classical intuition
  3. $\mathrm{AdS}_{d+1}$ toy model
  4. Summary of the model and first-order perturbation theory
  5. Symmetries
  6. Minimal twist Hilbert space
  7. Primary states
  8. Effective pair potential
  9. Decay of two-particle and $N$-particle potentials
  10. Higher-particle interactions
  11. Toeplitz operators
  12. Two-body binding energies and Lorentzian inversion formula
  13. Numerical diagonalization at finite spin
  14. Three-body problem at large spin
  15. Line bundles on ${\mathbb{C} \mathrm{P}}^{N-2}$ and Berezin-Toeplitz quantization
  16. ...and 43 more sections

Figures (19)

  • Figure 1: A slice of $\mathrm{AdS}$ space at constant global time. Left: a two-body state in $\mathrm{AdS}$ at large $J$. Right: a hierarchical three-body state in $\mathrm{AdS}$.
  • Figure 2: A typical three-body state of the class considered in this paper.
  • Figure 3: A typical spectrum of anomalous dimensions for $N=3$ leading twist states in the model \ref{['eq:sfmodel']} as a function of spin $J$. In this figure, $\Delta_\phi = 1.234,\, \Delta_\sigma = 0.6734$, and the constant $b_0$ is defined in \ref{['eq:b0defn']}.
  • Figure 4: Same as figure \ref{['fig:sampleN3']} but for $N=4$.
  • Figure 5: Spiraling geodesics in AdS.
  • ...and 14 more figures