This paper builds a controlled way to modify two-dimensional quantum field theories while keeping a particular kind of failure of gauge symmetry, called a background gauge-field anomaly, unchanged. The deformation they study is a “double-current” change, which is driven by a product of two conserved currents. The main result is a path integral construction that produces a deformed partition function with exactly the same anomaly as the original theory.

The authors extend an earlier path integral approach by coupling the original or “seed” theory to new dynamical gauge fields and to compact Stueckelberg fields. A Stueckelberg field is a simple, periodic field that can compensate for gauge transformations. Because partition functions of anomalous theories are not ordinary functions of the background gauge field but rather sections of an “anomaly line bundle,” the construction also inserts a parallel-transport factor in that bundle. This parallel transport links the auxiliary gauge field used in the deformation to the physical background gauge field so that the deformed partition function transforms with the same finite anomaly as the seed theory.

A major simplification happens when the background gauge fields are flat, meaning they have no local field strength. In that case the Stueckelberg non-zero modes force the dynamical gauge field to be flat too. The infinite-dimensional path integral collapses to a finite-dimensional integral over holonomies, which are the gauge-field twists you get by going around cycles of the surface. On the torus (the doughnut-shaped surface) the authors work out this finite kernel explicitly. They also outline how the same idea extends to Riemann surfaces of higher genus (surfaces with more holes).

To check the formalism they apply it to the compact boson, which is equivalent to the Abelian U(1) Wess–Zumino–Witten model. There the holonomy integral becomes a Gaussian transform of the torus partition function. At zero background gauge field this changes the effective radius parameter in the spectrum (they write k→K_λ for this replacement). Importantly, certain other data — contact terms and the rules for how states shift under large gauge transformations (spectral flow) — remain determined by the original anomaly coefficient. In plain terms: the deformation changes the energy levels in the expected way, but it does not change the anomaly-controlled pieces of the theory.