Boundary Theory and Rigidity in Relative Deligne Cohomology

This project develops a boundary theory for abelian gauge fields using the framework of relative Deligne cohomology. We introduce a precise formulation of boundary gauge data, identify the associated obstruction classes, and establish a rigidity phenomenon showing that admissible boundary configurations form a diagonal subset with no nontrivial deformations. A central result is a cylinder matching theorem for Deligne 1-cocycles, which characterizes when boundary data on a cylindrical pair (X, D) = (Y 0, 1, Y₀ Y₁) admit a bulk extension. The obstruction is shown to lie in a relative cohomology group and its vanishing forces the two boundary gauge sectors to coincide. This work provides a mathematically rigorous reformulation of Maxwell theory with boundary and suggests potential extensions to non-abelian gauge theories and dynamically changing topology. The project hosts the manuscript, supplementary notes, and diagrams associated with the development of this boundary formalism.