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<p>The near-singularity BKL dynamics of five dimensional gravity and supergravity (and also an extended four-dimensional supergravity) is known to be given by the billiard problem of a particle within a fundamental domain of the Bianchi groups PSL(2, 𝒪) ⊂ PSL(2, ℂ), acting on ℍ<sub>3</sub>. Here 𝒪 are the Gaussian or Eisenstein integers, which define a square or triangular lattice in ℂ. The Wheeler-DeWitt wavefunctions near the singularity are, correspondingly, automorphic Maaß forms of PSL(2, 𝒪). We show how these wavefunctions are associated to certain <i>L</i>-functions evaluated along their critical axis. Each of these <i>L</i>-functions admits an Euler product representation over the complex primes 𝒫<sub>𝒪</sub> ⊂ 𝒪. From this fact we write the <i>L</i>-function as the trace over an auxiliary Hilbert space of charged harmonic oscillators, labeled by the complex primes 𝒫<sub>𝒪</sub>. In this way we have constructed a ‘dual’ primon gas partition function for the wavefunction of the universe close to a five dimensional cosmological singularity.</p>

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Wheeler-DeWitt wavefunctions for 5d BKL dynamics, automorphic L-functions and complex primon gases

  • 期刊论文

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  • 发表日期：

    2025年11月25日

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  • Marine De Clerck,
  • Sean A. Hartnoll,
  • Ming Yang

摘要

The near-singularity BKL dynamics of five dimensional gravity and supergravity (and also an extended four-dimensional supergravity) is known to be given by the billiard problem of a particle within a fundamental domain of the Bianchi groups PSL(2, 𝒪) ⊂ PSL(2, ℂ), acting on ℍ3. Here 𝒪 are the Gaussian or Eisenstein integers, which define a square or triangular lattice in ℂ. The Wheeler-DeWitt wavefunctions near the singularity are, correspondingly, automorphic Maaß forms of PSL(2, 𝒪). We show how these wavefunctions are associated to certain L-functions evaluated along their critical axis. Each of these L-functions admits an Euler product representation over the complex primes 𝒫𝒪 ⊂ 𝒪. From this fact we write the L-function as the trace over an auxiliary Hilbert space of charged harmonic oscillators, labeled by the complex primes 𝒫𝒪. In this way we have constructed a ‘dual’ primon gas partition function for the wavefunction of the universe close to a five dimensional cosmological singularity.