Excitations in the Higher Lattice Gauge Theory Model for Topological Phases III: the 3+1d Case

Abstract

In this, the third paper in our series describing the excitations of the higher lattice gauge theory model for topological phases, we will examine the 3+1d case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge.

A toy model for categorical charges

“…where the blue-shaded plaquettes are those for which the 1-flatness/plaquette stabiliser conditions are violated. Note however that interpreting this excitation strictly as a 1-flux requires some care as the cube operators are defined to vanish whenever the 1-flatness is not everywhere satisfied around the cube (see [76] for a discussion on the interplay between 1-and 2-fluxes). The domain wall n is distinct from the trivial one 1 E1 since we cannot annihilate the resulting magnetic excitation with a local or genuine line-like operator without creating additional excitations or altering the condensation defect itself.…”