Chaos and moduli space volumes in unorientable JT gravity

Abstract

We show the late time, or $\tau-$scaled, limit of the canonical spectral form factor (SFF) in unorientable JT gravity agrees with universal random matrix theory (RMT) up to genus one in the topological expansion, establishing a key signature of quantum chaos for the time-reversal symmetric case. The loop equations for an orthogonal matrix model with spectral curve $y(z) \propto \sin(2\pi z)$ are used to compute the moduli space volumes of unorientable surfaces. The divergences of the unorientable volumes are regularized by first regularizing the resolvents of the orthogonal matrix model. To this end, we make use of the large $p$ limit of the $(2,2p+1)$ minimal string model. Using properties of the volumes and the loop equations, we derive streamlined formulas to compute the volumes for one and two boundaries, giving explicit results up to genus one. We find the general structure of the unorientable volumes to be written in terms of multiple polylogarithms and zeta values, with weight determined by the genus, number of boundaries, and number of crosscaps. In the $\tau-$scaled limit, contributions to the SFF from the divergent parts of the volume cancel, and the SFF becomes finite and independent of regularization. The SFF from universal RMT is a distinct computation, that depends on the leading order energy density of JT gravity, which we also derive up to genus one.