Chern-Simons gravity and $\mathrm{PT}$ symmetry

This paper considers the possibility that, starting from a relativistic Hermitian quantum field theory in the ultraviolet (UV) regime, and applying a nonperturbative renormalization-group (RG) flow, we arrive at a situation where there are infrared (IR) singularities in the RG flow of couplings. The latter can be resolved by assuming that the theory can have a phase described by a related non-Hermitian $\mathrm{PT}$-symmetric modification in the IR. The UV-to-IR (Hermitian-to-$\mathrm{PT}$-symmetric) transition can occur in a single renormalization-group flow of the pertinent couplings, as demonstrated in concrete examples. When embedded in a gravitational setting such a transition can lead to a repulsive gravity phase. If there is a RG flow to a repulsive $\mathrm{PT}$ -symmetric gravity, then this would be an alternative to dark energy. The discussion here is presented in the context of a string-inspired Chern-Simons gravitational effective action, which involves a pseudoscalar (axionlike) field coupled to Abelian gauge fields and gravity; it may also hold more generally in gravity with torsion. The validity of such a scenario in realistic theories might alleviate the need for de Sitter phases in the current epoch of cosmological evolution, thus avoiding their associated conceptual and technical complications.