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

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.