From the Schwinger Model on a Torus to a Bosonic Vacuum Effective Field Theory with a Compact Holonomy Mode

This work studies the massive Schwinger model—two-dimensional QED with a single Dirac fermion—formulated on a torus. Building on the classic analyses of Schwinger and Coleman, and on detailed studies of the model on the circle and torus by Sachs, Wipf, Joos, and Azakov, the focus is on the compact vacuum degree of freedom provided by the spatial holonomy of the gauge field: a global Wilson-line mode that cannot be gauged away and carries a vacuum energy that depends on the geometry of the torus. Starting from the exact spectrum of the Dirac operator, the holonomy-dependent vacuum energy is computed and an effective field theory for the vacuum mode is constructed, taking the form of a generalized sine-Gordon model. In an extended low-energy description—where the holonomy zero mode is promoted to a slowly varying field—this effective theory supports holonomy domain walls (kinks). Their tension and thickness are evaluated, and the setup naturally accommodates Jackiw–Rebbi–type fermionic zero modes localized on these walls. By introducing a small CP-violating tilt of the potential, modelled phenomenologically as an energy splitting between neighbouring holonomy vacua, false-vacuum decay is analysed in close analogy with the Schwinger pair-production mechanism. Conceptually, the paper fits into a broader “bosonic vacuum” viewpoint in which the physical vacuum is treated as a medium with its own collective degrees of freedom. The compact holonomy mode in the Schwinger model provides a concrete, fully calculable example of such a bosonic vacuum excitation. In this sense, the results complement existing pictures in which effective scalar “vacuum fields” emerge from fermion loops and anomalies (e.g. the anomaly-induced scalar modes studied by Blaschke, Carballo-Rubio, and Mottola in 2D), by showing—within a simple textbook model—how an effective field theory for a global vacuum degree of freedom arises from a local gauge theory on a compact space.