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

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.