Gauging the Schwarzian action

In this work, we promote the global SL(2,R) symmetry of the Schwarzian derivative to a local gauge symmetry. To achieve this, we develop a procedure that potentially can be generalized beyond the SL(2,R) case to any symmetry where the fundamental fields transform nonlinearly: We first construct a composite field from the fundamental field and its derivative such that it transforms linearly under the group action. Then we write down its gauge-covariant extension and apply standard gauging techniques. Applying this to the fractional linear representation of SL(2,R), we obtain the gauge-invariant analogue of the Schwarzian derivative as a bilinear invariant of covariant derivatives of the composite field. The framework enables a simple construction of Noether charges associated with the original global symmetry. The gauge-invariant Schwarzian action introduces SL(2,R) gauge potentials, allowing for locally invariant couplings to additional fields, such as fermions. While these potentials can be gauged away on topologically trivial domains, non-trivial topologies (e.g., S1) lead to distinct topological sectors. We mention that in the context of two-dimensional gravity, these sectors could correspond to previously discussed defects in the bulk theory.