Edge Dynamics from the Path Integral: Maxwell and Yang-Mills

TL;DR

The paper builds a first-principles boundary action for edge dynamics in Maxwell and Yang–Mills theories by path-integrating bulk degrees of freedom across a boundary, promoting large gauge transformations to dynamical boundary fields. It demonstrates that the resulting action yields the correct canonical structure and enables explicit computations of edge partition functions and correlators, including Maxwell edge states in flat space and in Rindler, and the 2d YM boundary as a particle-on-G model with exact disk/annulus correspondences. It then connects boundary bilocals to boundary-anchored Wilson lines in 2d YM, providing diagrammatic and group-theoretic tools to compute edge correlators and to understand which bulk data are captured by the boundary theory. The work also explores horizon physics, showing localization to static charges and deriving a horizon entropy with UV divergences, and discusses the quasi-topological character of 2d YM and its BF-limit, suggesting avenues for holographic and quantum-gravity connections. Overall, it exposes a rich boundary dynamics framework that encodes substantial, though not all-encompassing, bulk information and offers a controlled setting to study edge modes and their entropic content in gauge theories.

Abstract

We derive an action describing edge dynamics on interfaces for gauge theories (Maxwell and Yang-Mills) using the path integral. The canonical structure of the edge theory is deduced and the thermal partition function calculated. We test the edge action in several applications. For Maxwell in Rindler space, we recover earlier results, now embedded in a dynamical canonical framework. A second application is 2d Yang-Mills theory where the boundary action becomes just the particle-on-a-group action. Correlators of boundary-anchored Wilson lines in 2d Yang-Mills are matched with, and identified as correlators of bilocal operators in the particle-on-a-group edge model.

Figures (6)

• Figure 1: Cartoon of the $A$ functional space. The blue lines are the physical fields. The bulk field is gauge-fixed to a single copy per gauge orbit. At the boundary the full would-be gauge orbit is rendered physical, parametrized by the gauge parameter $\phi$.
• Figure 2: Plane dividing space in two halves. Surface charges and currents source bulk electric and magnetic fields.
• Figure 3: Rindler space embedded in Minkowski space. The dashed blue line represents the regularized horizon (as a brick wall) that is taken in the limit to the actual null horizon.
• Figure 4: Left: cutting a sphere along an identity Wilson loop causes a decomposition into two disks. Middle: cutting a sphere through an additional Wilson loop produces two disks with single boundary-anchored Wilson lines. Right: two additional intersecting Wilson loops leads to two disks that each contain two crossing boundary-anchored Wilson lines.
• Figure 5: Three examples of a Wilson line network in 2d YM. The red regions cannot be reproduced from the edge theory perspective.