Abelian duality, walls and boundary conditions in diverse dimensions

TL;DR

The paper develops a unified framework of duality walls to track how abelian dualities act on boundary conditions and both local and nonlocal operators across dimensions 2, 3, and 4. It provides explicit mappings: in 4d, the SL(2,${\mathbb Z}$) action on boundary data and the electric-magnetic duality of Wilson/'t Hooft loops and the Chern-Simons operator; in 2d, T-duality on toroidal branes is realized as a differential-geometric Fourier-Mukai transform with D-brane charges carried by vector bundles; in 3d, a gerbe-based duality wall captures the duality between a $U(1)$ gauge theory and a scalar, clarifying how duality acts on boundary conditions and line/disorder operators. A key outcome is that dualities in the abelian case can be understood as topological/gerbe-coupled walls, with boundary degrees of freedom essential to preserving gauge invariance and topological structure, and with T-duality closely mirroring the Fourier-Mukai framework. This provides a systematic, non-supersymmetric method to map dual theories and their observables, offering new tools for analyzing boundary phenomena and nonlocal operators in abelian field theories. The results are expected to inform both mathematical formulations of dualities and practical calculations of operator maps in diverse dimensions.

Abstract

We systematically apply the formalism of duality walls to study the action of duality transformations on boundary conditions and local and nonlocal operators in two, three, and four-dimensional free field theories. In particular, we construct a large class of D-branes for two-dimensional sigma-models with toroidal targets and determine the action of the T-duality group on it. It is manifest in this formalism that T-duality transformations on D-branes are given by a differential-geometric version of the Fourier-Mukai transform.