Spectral Green functions for T-duality-regularized electrostatics in finite and bounded planar geometries
We develop a finite-size formulation of T-duality-deformed electrostatics in (2+1) dimensions. The T-duality form factor regularizes the ultraviolet behaviour of the planar static Green function, while finite volume and boundary conditions control the infrared sector. We first rederive the self-energy-subtracted infinite-plane interaction, which is finite at the origin and recovers the logarithmic Coulomb law at large distances. We then define the corresponding Green function on general domains via the spectral calculus of the Laplacian and work out explicit realizations on the square torus, the grounded half-plane and the grounded strip. The torus implements charge neutrality through zero-mode subtraction, whereas Dirichlet boundaries produce finite image and self-energy effects. Finally, we introduce an additive-constant-free running interaction dimension and show that the ultraviolet flow to one dimension survives in finite volume, while the infrared behaviour is geometry-dependent.
