The Edge States of the BF System and the London Equations

Abstract

It is known that the 3d Chern-Simons interaction describes the scaling limit of a quantum Hall system and predicts edge currents in a sample with boundary, the currents generating a chiral $U(1)$ Kac-Moody algebra. It is no doubt also recognized that in a somewhat similar way, the 4d $BF$ interaction (with $B$ a two form, $dB$ the dual ${}^{\ast }j$ of the eletromagnetic current, and F the electromagnetic field form) describes the scaling limit of a superconductor. We show in this paper that there are edge excitations in this model as well for manifolds with boundaries. They are the modes of a scalar field with invariance under the group of diffeomorphisms (diffeos) of the bounding spatial two-manifold. Not all of this group seem implementable by operators in quantum theory, the implementable group being a subgroup of volume preserving diffeos. The $BF$ system in this manner can lead to the $w_{1+\infty }$ algebra and its variants. Lagrangians for fields on the bounding manifold which account for the edge observables on quantization are also presented. They are the analogues of the $1+1$ dimentional massless scalar field Lagrangian describing the edge modes of an abelian Chern-Simons theory with a disk as the spatial manifold. We argue that the addition of ``Maxwell'' terms constructed from $F{\wedge }^{\ast }F$ and $dB{\wedge }^{\ast }dB$ do not affect the edge states, and that the augmented Lagrangian has an infinite number of conserved charges- the aforementioned scalar field modes- localized at the edges. This Lagrangian is known to describe London equations and a massive vector field. A $(3+1)$ dimensional generalization of the Hall effect involving vortices coupled to $B$ is also proposed.

38 pages

Cited by in corpus (33)

• Superconductors are topologically ordered
• Topological BF field theory description of topological insulators
• Bulk-boundary correspondence in (3+1)-dimensional topological phases
• Bosonic topological phases of matter: bulk-boundary correspondence, SPT invariants and gauging
• Effective field theories for topological insulators by functional bosonization
• Canonical BF-type Topological Field Theory and Fractional Statistics of Strings
• Extended actions, dynamics of edge modes, and entanglement entropy
• Wilson operator algebras and ground states for coupled BF theories
• Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in $(3+1)$-Dimensions
• Edge Dynamics for BF Theories and Gravity
• (3+1)-dimensional topological quantum field theory from a tight-binding model of interacting spinless fermions
• Higgsless superconductivity from topological defects in compact BF terms
• Screening and Topological Order in Thin Superconducting Films
• 4-d Chern-Simons Theory: Higher Gauge Symmetry and Holographic Aspects
• String Holonomy and Extrinsic Geometry in Four-dimensional Topological Gauge Theory
• Abelian BF-Theory and Spherically Symmetric Electromagnetism
• The Spin-Statistics Connection in Quantum Gravity
• On the dual equivalence of the self-dual and topologically massive p-form models
• Quantum Topology Change in (2 + 1)d
• Effective Field Theory for a p-wave Superconductor in the Subgap Regime
• Quantum phase excitations in Ginzburg-Landau superconductors
• Gauge Invariant Factorisation and Canonical Quantisation of Topologically Massive Gauge Theories in Any Dimension
• 4D Edge Currents from 5D Chern-Simons Theory
• U_e(1)xU_g(1) Actions in 2+1-Dimensions: Full Vectorial Electric and Magnetic Fields
• Constrained field theories on spherically symmetric spacetimes with horizons
• Inequivalence of the Massive Vector Meson and Higgs Models on a Manifold with Boundary
• Quantum field theoretic representation of Wilson surfaces: I higher coadjoint orbit theory
• Constrained field theories on Kerr backgrounds
• Quantum field theoretic representation of Wilson surfaces: II higher topological coadjoint orbit model
• Remarks on a $B\wedge F$ model with topological mass from gauging spin
• Spin-Charge Soldering from Tensor Higgs mechanism
• Notes from the bulk: metric dependence of the edge states of Chern-Simons theory
• Hamiltonian BRST formalism for gauge fields on black hole spacetimes