The Schwarzian theory describes the universal low-energy dynamics of near-extremal black holes and the SYK model. It can be characterized as an integral over a particular coadjoint orbit of the Virasoro group — one specific orbit among many. What about the others?
This paper classifies and solves them all. Every Virasoro coadjoint orbit defines a generalized Schwarzian theory, and the complete classification reveals new classes with features qualitatively different from the original. The new theories are inherently Lorentzian: their path integrals are oscillatory, weighted by e^{iI} rather than e^{-I}, and cannot be Wick-rotated to Euclidean signature.
The classification has a geometric interpretation. The coadjoint orbits of the Virasoro group coincide with the moduli space of constant positive curvature two-dimensional Lorentzian geometries. Each orbit is a family of spacetimes, and the associated Schwarzian theory governs wavefunctions in asymptotically near-de Sitter gravity — the cosmological setting, not the black hole setting.
The path integrals for all theories are computed exactly via fermionic localization, but the localization requires careful treatment. The quadratic fluctuation operator at one loop fails to be essentially self-adjoint, forcing a choice of boundary condition. Certain field configurations that would normally be excluded — singularities in the classical solutions — must be admitted. The JT gravity embedding regulates these singularities naturally, justifying the choices.
The original Schwarzian theory was one species in a menagerie. The classification reveals the full population, and the new members live in de Sitter space rather than anti-de Sitter. The mathematics is the same group; the physics is a different universe.