$\mathrm{SL}(2,\mathbb{R})$ Gauge Theory, Hyperbolic Geometry and Virasoro Coadjoint Orbits

TL;DR

This paper establishes a comprehensive correspondence between all Virasoro coadjoint orbits and moduli spaces of hyperbolic metrics via $ ext{SL}(2, ext{R})$ gauge theory, revealing new geometric and physical insights in 2D gravity.

Contribution

It proves that every Virasoro orbit corresponds to a hyperbolic moduli space, using gauge theory, and explores the geometric and physical implications of this correspondence.

Findings

All Virasoro orbits are realized as hyperbolic moduli spaces.

Large gauge transformations generate geometries with no constant representative.

New topological sectors of 2D gravity are characterized by twisted boundary conditions.

Abstract

It has long been known that the moduli space of hyperbolic metrics on the disc can be identified with the Virasoro coadjoint orbit $\mathrm{Diff}^+(S^1) / \mathrm{SL}(2,\mathbb{R})$. The interest in this relationship has recently been revived in the study of two-dimensional JT gravity and it raises the natural question if all Virasoro orbits $\mathcal{O}$ arise as moduli spaces of hyperbolic metrics. In this article, we give an affirmative answer to this question using $\mathrm{SL}(2,\mathbb{R})$ gauge theory on a cylinder $S$: to any $L\in\mathcal{O}$ we assign a flat $\mathrm{SL}(2,\mathbb{R})$ gauge field $A_L = (g_L)^{-1} dg_L$, and we explain how the global properties and singularities of the hyperbolic geometry are encoded in the monodromies and winding numbers of $g_L$, and how they depend on the Virasoro orbit. In particular, we show that the somewhat mysterious geometries associated with Virasoro orbits with no constant representative $L$ arise from large gauge transformations acting on standard (constant $L$ ) funnel or cuspidal geometries, shedding some light on their potential physical significance: e.g. they describe new topological sectors of two-dimensional gravity, characterised by twisted boundary conditions. Using a gauge theoretic gluing construction, we also obtain a complete dictionary between Virasoro coadjoint orbits and moduli spaces of hyperbolic metrics with specified boundary projective structure.