The Schwarzian Theory - A Wilson Line Perspective

TL;DR

The paper develops a Wilson line perspective on Schwarzian quantum mechanics by embedding JT gravity in a constrained SL^+(2,R) BF theory and identifying Schwarzian bilocals with boundary-anchored Wilson lines.It first analyzes boundary correlators in compact BF systems and then extends to the non-compact SL(2,R) case, deriving Schwarzian correlators through Hamiltonian reduction and matching them to bilocal Schwarzian operators, including OTOs via 6j-symbols.A detailed diagrammatic framework is provided, including 3j/6j symbols and the Plancherel measure dim k = k sinh 2πk, establishing a complete bulk/boundary dictionary and laying groundwork for higher-spin and Toda generalizations.Overall, the work offers a constructive bulk derivation of Schwarzian correlators from BF/YM theory and clarifies the holographic interpretation of bilocals in the Schwarzian/JT gravity context.

Abstract

We provide a holographic perspective on correlation functions in Schwarzian quantum mechanics, as boundary-anchored Wilson line correlators in Jackiw-Teitelboim gravity. We first study compact groups and identify the diagrammatic representation of bilocal correlators of the particle-on-a-group model as Wilson line correlators in its 2d holographic BF description. We generalize to the Hamiltonian reduction of SL(2,R) and derive the Schwarzian correlation functions. Out-of-time ordered correlators are determined by crossing Wilson lines, giving a 6j-symbol, in agreement with 2d CFT results.

Figures (18)

• Figure 1: Relation through holography and dimensional reduction of 3d $\Lambda<0$ gravity, 2d JT gravity, 2d Liouville CFT and 1d Schwarzian QM.
• Figure 2: The 3d Chern-Simons / 2d WZW holographic relation, which dimensionally reduces to the 2d BF / 1d particle-on-a-group holography.
• Figure 3: Left: disk amplitude in the open channel. Middle: disk amplitude in the crossed channel. Right: Boundary circle with evolution along the circle.
• Figure 4: The disk amplitude in the open channel (left). Hilbert space slicings are denoted in red. Deforming the amplitude (either preserving the area (YM), or the boundary length (BF)) into a rectangle (right), and identifying the upper and lower boundaries, one arrives at the fundamental cylinder amplitude.
• Figure 5: Left: Hilbert space slicing in the point-defect channel, representing propagation between pointlike defects. Right: Amplitude with a Wilson line inserted.