Universal Dynamics in Non-Orientable CFT${}_2$

Two-dimensional conformal field theories (CFTs) defined on non-orientable Riemann surfaces obey consistency Cardy conditions analogous to those in the orientable case. We revisit those conditions for irrational theories with central charge $c>1$ in the context of two-point functions of primaries on the Real Projective plane ${\mathbb{R}\mathbb{P}}^2$ and the partition function on the Klein bottle ${\mathbb{K}}^2$. Using the irrational versions of the Virasoro fusion and modular kernels we derive universal expressions for the non-orientable CFT data at large conformal dimension, assuming a gap in the spectrum of scalar primaries. In particular, we derive asymptotic formulas at finite central charge for the averaged Light-Light-Heavy product $C_{LLH}\times {\mathrm{\Gamma }}_H$ of OPE coefficients with the ${\mathbb{R}\mathbb{P}}^2$ one-point function normalizations, as well as for the parity-weighted density of heavy scalar primaries (or equivalently the density of heavy ${\mathrm{\Gamma }}_H^2$). We discuss the gravitational interpretation of the results.