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 Nöther 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.
| Comments: | 22 pages |
| Subjects: | Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th) |
| Cite as: | arXiv:2507.04091 [math-ph] |
| (or arXiv:2507.04091v3 [math-ph] for this version) | |
| https://doi.org/10.48550/arXiv.2507.04091 arXiv-issued DOI via DataCite |
|
| Journal reference: | J. Math. Phys. 67, 053504 (2026) |
| Related DOI: | https://doi.org/10.1063/5.0302560
DOI(s) linking to related resources |
From: Allen Stern [view email]
[v1]
Sat, 5 Jul 2025 16:31:49 UTC (18 KB)
[v2]
Tue, 15 Jul 2025 00:26:36 UTC (19 KB)
[v3]
Tue, 26 May 2026 03:15:04 UTC (22 KB)
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