Dipoles, Twists and Noncommutative Gauge Theory

TL;DR

The paper addresses how T-duality for gauge theories on noncommutative tori extends to sectors with twisted boundary conditions, revealing dual theories populated by constant-length dipoles with dipole-vectors $\bf L$ and twists $\boldsymbol{\alpha}$ that transform jointly under ${\rm SO}(d,d,\mathbb{Z})$ as a $2d$-dimensional representation. It establishes a concrete map between twists and dipoles via SL$(2,\mathbb{Z})$ transformations, introduces a dipole-friendly modification of the star-product, and presents a Seiberg–Witten–like bridge from nonlocal dipole theories to local variables. The work then explores several properties of dipole theories, including rational-dipole reductions to local quiver theories, the behavior under S-duality (which may render the dual theory nonlocal in time), and speculative links to the $(2,0)$ theory through higher-dimensional deformations. Overall, the study shows that dipole theories capture many qualitative features of noncommutative gauge theories in a simpler setting, offering a tractable framework to investigate nonlocal phenomena and dualities in string-inspired field theories.

Abstract

T-duality of gauge theories on a noncommutative $T^d$ can be extended to include fields with twisted boundary conditions. The resulting T-dual theories contain novel nonlocal fields. These fields represent dipoles of constant magnitude. Several unique properties of field theories on noncommutative spaces have simpler counterparts in the dipole-theories.