The Schwarzian Zoo

The Schwarzian theory governs the low-energy dynamics of near-extremal black holes and the SYK model. It can be characterized as a path integral over a particular coadjoint orbit of the Virasoro group — the orbit corresponding to a constant negative curvature geometry. But the Virasoro group has many coadjoint orbits, and each defines a different theory.

This paper classifies all of them.

Each Virasoro coadjoint orbit corresponds to a constant-curvature two-dimensional Lorentzian geometry. The familiar Schwarzian is the orbit of negative curvature. Positive curvature orbits give theories relevant to near-dS₂ gravity — Jackiw-Teitelboim gravity with positive cosmological constant. These are Lorentzian theories with oscillatory path integrals, qualitatively different from the Euclidean Schwarzian.

The coupling functions vary across orbits. Some orbits have smooth couplings; others have singularities requiring boundary condition choices. These choices are not ambiguities to be resolved but physical parameters: in the JT gravity realization, they correspond to choices about which geometries contribute to the path integral and which singularities are allowed.

Despite the oscillatory nature of the Lorentzian path integrals, they remain one-loop exact via fermionic localization. This is a powerful computational tool — it means exact results are available across all orbits, all coupling regimes, even in theories that haven’t been studied before.

The familiar Schwarzian is one animal in a zoo. The classification reveals the rest: novel theories with distinct physical content, governed by the same algebraic structure (Virasoro coadjoint orbits) but with different dynamics. Most of the zoo has been unexplored.