Gauging the Schwarzian Action

TL;DR

This paper develops a gauge-invariant formulation of the Schwarzian action by promoting its global $SL(2, ext{R})$ symmetry to a local gauge symmetry, enabling new couplings and insights into topological sectors.

Contribution

It introduces a method to gauge the Schwarzian derivative, creating a gauge-invariant action with potential applications in 2D gravity and topological defect analysis.

Findings

Constructed a composite field transforming linearly under $SL(2, ext{R})$

Derived a gauge-invariant Schwarzian derivative as a bilinear invariant

Identified topological sectors related to non-trivial topologies like $S^1$

Abstract

In this work, we promote the global $SL(2,\mathbb{R})$ symmetry of the Schwarzian derivative to a local gauge symmetry. To achieve this, we develop a procedure that potentially can be generalized beyond the $SL(2,\mathbb{R})$ case: 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 $SL(2,\mathbb{R})$, 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 N\"other charges associated with the original global symmetry. The gauge-invariant Schwarzian action introduces $SL(2,\mathbb{R})$ 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., $S^1$) 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.