KdV conformal symmetry breaking in nearly AdS$_{2}$

We study the gauge theory formulation of Jackiw-Teitelboim gravity and propose Korteweg-de Vries asymptotic conditions that generalize the asymptotic dynamics of the theory. They permit to construct an enlarged set of boundary actions formed by higher order generalizations of the Schwarzian derivative that contain the Schwarzian as lower term in a tower of SL$(2,\mathbb{R})$ invariants. They are extracted from the KdV Hamiltonians and can be obtained recursively. As a result, the conformal symmetry breaking observed in nearly AdS$_{2}$ is characterized by a much larger set of dynamical modes associated to the stationary KdV hierarchy. We study quantum perturbation theory for the generalized Schwarzian action including the symplectic measure and compute the one-loop correction to the partition function. We find that despite the non-linear nature of the higher-Schwarzian contribution, it acquires a manageable expression that renders a curious leading temperature dependence on the entropy $S=\#T^{a}$ for $a$ an odd integer.