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All-fermion electrodynamics and fermion number anomaly inflow

S. M. Kravec, John McGreevy, Brian Swingle

TL;DR

The paper shows that 3+1D all-fermion electrodynamics can arise at the boundary of a 4+1D bosonic bulk (BdC theory), establishing bulk nontriviality through edge-regulator obstructions and fermion-number anomaly inflow. It provides a constructive coupled-layer framework in which dyon-string condensation yields the BdC bulk and reproduces the edge all-fermion spectrum, while connecting to BF theory in 3+1D. A key result is that all-fermion electrodynamics cannot be bosonically regulated in strict 3+1D, but can be realized with fermionic regulators, highlighting a deep link between higher-dimensional SPT physics, anomaly inflow, and surface topological orders such as the all-fermion toric code. The findings illuminate how fermion-number anomalies and duality symmetries constrain regulators and surface terminations in bosonic SPTs, with implications for the realization of anomalous edge theories in condensed matter and lattice gauge theory contexts.

Abstract

We demonstrate that 3+1-dimensional quantum electrodynamics with fermionic charges, fermionic monopoles, and fermionic dyons arises at the edge of a 4+1-dimensional gapped state with short-range entanglement. This state cannot be adiabatically connected to a product state, even in the absence of any symmetry. This provides independent evidence for the obstruction found by arXiv:1306.3238 to a 3+1-dimensional short-distance completion of all-fermion electrodynamics. The non-triviality of the bulk is demonstrated by a novel fermion number anomaly.

All-fermion electrodynamics and fermion number anomaly inflow

TL;DR

The paper shows that 3+1D all-fermion electrodynamics can arise at the boundary of a 4+1D bosonic bulk (BdC theory), establishing bulk nontriviality through edge-regulator obstructions and fermion-number anomaly inflow. It provides a constructive coupled-layer framework in which dyon-string condensation yields the BdC bulk and reproduces the edge all-fermion spectrum, while connecting to BF theory in 3+1D. A key result is that all-fermion electrodynamics cannot be bosonically regulated in strict 3+1D, but can be realized with fermionic regulators, highlighting a deep link between higher-dimensional SPT physics, anomaly inflow, and surface topological orders such as the all-fermion toric code. The findings illuminate how fermion-number anomalies and duality symmetries constrain regulators and surface terminations in bosonic SPTs, with implications for the realization of anomalous edge theories in condensed matter and lattice gauge theory contexts.

Abstract

We demonstrate that 3+1-dimensional quantum electrodynamics with fermionic charges, fermionic monopoles, and fermionic dyons arises at the edge of a 4+1-dimensional gapped state with short-range entanglement. This state cannot be adiabatically connected to a product state, even in the absence of any symmetry. This provides independent evidence for the obstruction found by arXiv:1306.3238 to a 3+1-dimensional short-distance completion of all-fermion electrodynamics. The non-triviality of the bulk is demonstrated by a novel fermion number anomaly.

Paper Structure

This paper contains 15 sections, 46 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The $BdC$ model coupled to matter
  3. $BdC$ summary
  4. Coupling to strings (matter)
  5. Edge physics
  6. The Bad Thing that Happens on ${\mathbb C}{\mathbb P}^2$
  7. Coupled Layer Construction
  8. Warmup: deconstruction of lattice electrodynamics
  9. Dyon string condensation in more detail
  10. Alternative description of layer construction
  11. Extension to $D=3+1$ and derivation of $BF$ theory
  12. Fermion number anomaly inflow
  13. Consequences for all-fermion toric code
  14. Lattice bosons for duality-symmetric surface QED
  15. More details on monopole strings and vortex sheets in 5d abelian gauge theory

Figures (4)

  • Figure 1: A depiction of the calculation of dyon statistics. The spikes represent the flux produced by the dyon at the center.
  • Figure 2: A representation of the coupled layer construction, following PhysRevB.87.235122. The layers are coupled by condensing the objects circled in red.
  • Figure 3: Two representations of the (warmup) coupled-layer construction for $D=4+1$ Maxwell theory with gauge group $\textsf{U}(1)_o \times \textsf{U}(1)_e$. The top figure is the direct analog of the previous figure; the bottom is a 'quiver' or 'moose' diagram familiar from the high energy physics literature.
  • Figure :