Extended actions, dynamics of edge modes, and entanglement entropy
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Springer
Received: December Revised: August Accepted: August Published: September
17, 18, 26, 21,
2019 2020 2020 2020
Extended actions, dynamics of edge modes, and entanglement entropy
a
Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, UMR 5672, F-69342 Lyon, France b Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, ON, N2L 2Y5, Canada c Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada
E-mail: [email protected], [email protected] Abstract: In this work we propose a simple and systematic framework for including edge modes in gauge theories on manifolds with boundaries. We argue that this is necessary in order to achieve the factorizability of the path integral, the Hilbert space and the phase space, and that it explains how edge modes acquire a boundary dynamics and can contribute to observables such as the entanglement entropy. Our construction starts with a boundary action containing edge modes. In the case of Maxwell theory for example this is equivalent to coupling the gauge field to boundary sources in order to be able to factorize the theory between subregions. We then introduce a new variational principle which produces a systematic boundary contribution to the symplectic structure, and thereby provides a covariant realization of the extended phase space constructions which have appeared previously in the literature. When considering the path integral for the extended bulk + boundary action, integrating out the bulk degrees of freedom with chosen boundary conditions produces a residual boundary dynamics for the edge modes, in agreement with recent observations concerning the contribution of edge modes to the entanglement entropy. We put our proposal to the test with the familiar examples of Chern-Simons and BF theory, and show that it leads to consistent results. This therefore leads us to conjecture that this mechanism is generically true for any gauge theory, which can therefore all be expected to posses a boundary dynamics. We expect to be able to eventually apply this formalism to gravitational theories. Keywords: Gauge Symmetry, Chern-Simons Theories ArXiv ePrint: 1912.06025
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)134
JHEP09(2020)134
Marc Geillera and Puttarak Jai-aksonb,c
Contents 1 Motivations
1
2 Extended phase spaces
5
3 Extended actions
6 9 10 10 11 14 15 17 20 20 21 24 25
5 Perspectives
29
A Covariant phase space
32
B Maxwell theory in radial gauge
33
C Maxwell-Chern-Simons theory in temporal gauge
36
D Boundary conditions as boundary sources
40
E Hamiltonian quantization of the chiral bosons
41
F Extended action and phase space for non-Abelian theories F.1 Chern-Simons theory F.2 Yang-Mills theory F.3 BF theory
46 46 47 48
–i–
JHEP09(2020)134
4 Examples 4.1 Chern-Simons theory 4.1.1 Extended phase space 4.1.2 Boundary dynamics 4.1.3 Gluing of subregions 4.1.4
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