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

A bstract 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 3 . 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 𝒫 𝒪 ⊂ 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 𝒫 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.