Wheeler-DeWitt wavefunctions for 5d BKL dynamics, automorphic L-functions and complex primon gases
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| Format: | Preprint |
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2025
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| author | De Clerck, Marine Hartnoll, Sean A. Yang, Ming |
| author_facet | De Clerck, Marine Hartnoll, Sean A. Yang, Ming |
| contents | 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 $PSL(2,{\mathcal O}) \subset PSL(2,\mathbb{C})$, acting on $\mathbb{H}_3$. Here ${\mathcal O}$ are the Gaussian or Eisenstein integers, which define a square or triangular lattice in $\mathbb{C}$. The Wheeler-DeWitt wavefunctions near the singularity are, correspondingly, automorphic Maass forms of $PSL(2,{\mathcal O})$. 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 ${\mathcal{P}}_{\mathcal O} \subset {\mathcal O}$. 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 ${\mathcal{P}}_{\mathcal O}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_08788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Wheeler-DeWitt wavefunctions for 5d BKL dynamics, automorphic L-functions and complex primon gases De Clerck, Marine Hartnoll, Sean A. Yang, Ming High Energy Physics - Theory General Relativity and Quantum Cosmology 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 $PSL(2,{\mathcal O}) \subset PSL(2,\mathbb{C})$, acting on $\mathbb{H}_3$. Here ${\mathcal O}$ are the Gaussian or Eisenstein integers, which define a square or triangular lattice in $\mathbb{C}$. The Wheeler-DeWitt wavefunctions near the singularity are, correspondingly, automorphic Maass forms of $PSL(2,{\mathcal O})$. 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 ${\mathcal{P}}_{\mathcal O} \subset {\mathcal O}$. 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 ${\mathcal{P}}_{\mathcal O}$. 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. |
| title | Wheeler-DeWitt wavefunctions for 5d BKL dynamics, automorphic L-functions and complex primon gases |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2507.08788 |