Topological phase characterization independent of global winding in 4D compact U(1) lattice gauge theory

Determine whether a topological characterisation of the confining and Coulomb phases in four-dimensional pure compact U(1) lattice gauge theory can be formulated without relying on non-trivial global winding of monopole current loops around the spacetime torus, so that the characterisation remains valid on lattices with trivial homotopy such as lattice discretizations of the 4-sphere where all loops are contractible.

Background

Kerler et al. characterized the phase structure on toroidal lattices via percolating monopole current networks and Dirac sheets with non-trivial winding, linking confinement to global topological features on T^4.

Subsequent studies showed that the phase transition also occurs on lattice representations of the 4-sphere (S^4), where all loops are contractible and global winding cannot occur, raising the issue of whether phases can be distinguished through topology without invoking winding around the torus.

This observation motivates a precise question about formulating a topological phase characterization that is applicable both on toroidal lattices and on spherical lattices, thus independent of global winding.