Generalized Schwarzian Dynamics from a Bulk-First BF Perspective

Abstract

We investigate the emergence of generalized Schwarzian dynamics from a bulk-first BF perspective. Starting from two-dimensional BF gravity, we analyze the associated boundary phase space and its Drinfeld-Sokolov reductions. For the sl(2,R) theory, we recover the ordinary Schwarzian action as the reduced boundary dynamics arising from a particular sector of the BF asymptotic phase space. We then extend this construction to sl(3,R), where the reduced dynamics is governed by the second and third Wilczynski invariants, providing a natural higher-rank generalization of the Schwarzian derivative. In this framework, generalized Schwarzian dynamics emerges directly from flat BF connections and their companion forms rather than being introduced as an independent boundary theory. We further relate the resulting projective invariants to Casimir charges, monodromy data, and generalized Schwarzian thermodynamics, including monodromy spectra and semiclassical thermodynamics. In particular, constant projective invariants determine the corresponding Casimir sectors and monodromy data, which in turn organize the thermodynamic structure of the theory. Our results provide a unified bulk-first description of Schwarzian and generalized Schwarzian dynamics and reveal a direct link between BF gravity, asymptotic symmetry reductions, projective geometry, and boundary thermodynamics.