A menagerie of Schwarzians: coadjoint orbits of Virasoro and near-dS$_2$ quantum gravity

Abstract

The Schwarzian theory, which governs the universal low-energy dynamics of near-extremal black holes and the SYK model, can be characterised as an integral over a particular coadjoint orbit of the Virasoro group. We describe and solve a complete classification of all possible generalised Schwarzian theories, defined by integrals over any Virasoro coadjoint orbit, including new classes of theories with qualitatively novel features. The classification of coadjoint orbits coincides with the moduli space of constant positive curvature two-dimensional Lorentzian geometries, and the associated Schwarzian theories govern associated wavefunctions in asymptotically near-dS$_2$ gravity (Jackiw-Teitelboim gravity in particular). The novel theories are inherently Lorentzian, defined by oscillatory path integrals weighted by $e^{iI}$ and force consideration of varying `coupling functions' (renormalised dilaton) which may not have definite sign. The definition of the theories involves an ambiguity, arising because the operator describing quadratic fluctuations at one loop fails to be essentially self-adjoint. This requires a choice of boundary condition, and also forces us to allow certain singularities in configurations and classical solutions. The choice is justified from the realisation in JT gravity, which naturally regulates these singularities. The path integral remains one-loop exact via fermionic localisation, but this requires additional input beyond the Duistermaat-Heckman theorem. This allows an exact computation of the path integral for all theories and all couplings, including new results for the original Schwarzian theory.

Figures (1)

• Figure 1: The moduli space of Schwarzian theories, of Virasoro coadjoint orbits, and of constant positive curvature Lorentzian 2D geometries. The dashed line in the bottom left is a family of constant curvature geometries ('big crunch' spacetimes), but does not represent a separate Schwarzian theory or coadjoint orbit since the spacetime does not contain an asymptotic future infinity $\mathscr{I}_+$. The top left branch and horizontal spine consist of $U(1)_b$ orbits parametrised by a real constant $b$, with special $SL^{(n)}(2,\mathds{R})$ points at $b=-\frac{c}{24}n^2$ for positive integers $n$ where the symmetry is enhanced. At each such vertex, a branch of 'hyperbolic' $T_{n,\Delta}$ orbits (parametrised by $\Delta>0$) emerges. These vertices in fact split into three separate points (indicated schematically by the edges of the small triangles): a pair of exceptional 'parabolic' orbits $\tilde{T}_{n,\pm}$ as well as the $SL^{(n)}(2,\mathds{R})$ orbit.