Edge States in Gauge Theories: Theory, Interpretations and Predictions

TL;DR

The paper develops a unifying framework for edge states in gauge theories on manifolds with boundaries, showing that boundary-localized observables can exist across dimensions depending on carefully chosen boundary conditions. It analyzes 1+1, 2+1, and 3+1D models—Maxwell, Chern-Simons, Maxwell-Chern-Simons, Higgs, and BF theories—framing edge phenomena through boundary-condition parameters λ and μ that link to physical quantities like penetration depth in superconductors. Key contributions include identifying when edge observables are well-defined, deriving their spectra and Coulomb energies, and connecting these edge states to topological field theories and monopole solitons. The work suggests experimental routes to test λ-dependent edge-state behavior in thin devices and highlights the role of boundary physics in topological solitons and higher-dimensional gauge theories.

Abstract

Gauge theories on manifolds with spatial boundaries are studied. It is shown that observables localized at the boundaries (edge observables) can occur in such models irrespective of the dimensionality of spacetime. The intimate connection of these observables to charge fractionation, vertex operators and topological field theories is described. The edge observables, however, may or may not exist as well-defined operators in a fully quantized theory depending on the boundary conditions imposed on the fields and their momenta. The latter are obtained by requiring the Hamiltonian of the theory to be self-adjoint and positive definite. We show that these boundary conditions can also have nice physical interpretations in terms of certain experimental parameters such as the penetration depth of the electromagnetic field in a surrounding superconducting medium. The dependence of the spectrum on one such parameter is explicitly exhibited for the Higgs model on a spatial disc in its London limit. It should be possible to test such dependences experimentally, the above Higgs model for example being a model for a superconductor. Boundary conditions for the 3+1 dimensional $BF$ system confined to a spatial ball are studied. Their physical meaning is clarified and their influence on the edge states of this system (known to exist under certain conditions) is discussed. It is pointed out that edge states occur for topological solitons of gauge theories such as the 't Hooft-Polyakov monopoles.

Figures (4)

• Figure 1: This figure gives the plot of $G_{0}(\omega R)$ vs $\omega R$.
• Figure 2: This figure gives the plot of $G_{1}(\omega R)$ vs $\omega R$.
• Figure 3: This figure gives the plot of $G_{10}(\omega R)$ vs $\omega R$.
• Figure 4: Figure showing a strip coming out from the boundary of ball