Geometric or topological interpretation of the integer-field augmentation

Ascertain a geometric or topological interpretation for augmenting the action of the trident Lagrangian multiform with the integer-valued field Θ (and related integer parameters) that enforce equivalence of additive three-leg forms to the ABS quad equations and ensure closure properties in the presence of branch cuts.

Background

To make the corner equations strictly equivalent to the ABS quad equations, the authors introduce integer-valued fields into the action to handle branch cut choices and recover needed multiples of 2πi in the three-leg forms. This modification also underpins their closure and almost-closure results.

While effective, the conceptual meaning of these integer fields is unresolved. The authors highlight the need for a geometric or topological explanation of this addition to the discrete variational structure.