Edge dynamics from the path integral — Maxwell and Yang-Mills

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 edge 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. Recently in [26], using canonical quantization of the Maxwell action, we presented an alternative procedure that gives rise to a canonical structure on the boundary of Maxwell: instead of saving large gauge-invariance by introducing a new variable, we abandon it completely and promote the large gauge degrees of freedom φ themselves to physical variables, conjugate to the boundary charges E(x). This promotion of large gauge parameters to physical boundary variables is a key ingredient in topological theories such as 3d Chern-Simons, where they make up the boundary theory dual to the entire bulk theory.The question arises whether there is a boundary action that gives rise to this canonical structure. 1 Given such an action, it would be possible to study boundary correlation functions and real-time dynamics. In this work, we derive this boundary edge action for Maxwell and Yang-Mills theory. The result is a Lorentzian action that depends on the boundary gauge field g and the boundary current density J :where A[J ] denotes the on-shell evaluation of the gauge field sourced by the boundary current J .1 A similar goal is being pursued by Geiller and Jai-akson (private communication). 5)2 Note that J α is a contravariant tensor density of weight +1, as with our definition it already includes the metric pullback √ −h factor on ∂M. We use this definition to match more naturally with the canonical formalism in curved space, where conjugate momenta are tensor densities. 3 We will show further on that ∂nφ does not figure in the action, and hence only the variables φ| ∂M and ∂αφ| ∂M appear.

Kinematical gravitational charge algebra

“…We focus on the kinematics of the theory, analyzing the SU(2) gauge transformations and space diffeomorphisms while postponing studying the fate of time diffeomorphisms and the time evolution of the geometry to future work. In the first part of this work, we show that the constraints generating the SU(2) gauge transformations and space diffeomorphisms can be recast in terms of conservation of boundary charges satisfying a Poincaré algebra ISU (2). While the SU(2) sector of Poincaré obviously corresponds to the local gauge transformations, the space diffeomorphisms are now written as field dependent translations.…”

“…As it is well-known, any first class constraint also plays the role of canonical generators for an associated gauge transformation. This dual role of the constraints, often phrased by pointing out that each first class constraint kills two degrees of freedom, reflects the fact that initial data differing by an infinitesimal gauge transformation again solve 2 The conservation law in the bulk d A Σ = 0 implies by an integration by parts that the boundary charge is given by the bulk integral of the covariant variation of the electric gauge parameter:…”

“…There is growing understanding that edge modes must play a key role in quantum gravity. They are central in the quantization of 2d Yang-Mills and 2d gravity [2][3][4], They play a key role in 3d quantum gravity [5][6][7][8]. They are essential to the understanding of boundary dynamics and to the construction of defects operators in Chern-Simons theory [9][10][11][12].…”

An exact quantization of Jackiw-Teitelboim gravity

“…63 62 There's no obstruction for such extensions since G B is simply connected. 63 Note that in the world-line action (E.11), the fields (x ν , k µ (x)) take values in the co-tangent bundle T * Σ. The path integration measure is the natural one induced by the symplectic structure of T * Σ.…”

Strolling along gravitational vacua

“…The next question is to find the theory governing the dynamics of edge modes. This question has been tackled in [21,23]. Our adiabatic analysis induces a dynamics for edge modes which at first glance is a truncation of that of [23] 17 , while the connection to the boundary action of [21] is less clear.…”