Non-constant Non-commutativity in 2d Field Theories and a New Look at Fuzzy Monopoles

TL;DR

This paper explores scalar and gauge theories on two-dimensional noncommutative spaces with curvature and variable non-commutativity, developing a new approach that avoids singular maps and analyzing monopole solutions and gauge transformations.

Contribution

It introduces a novel method to formulate theories on curved noncommutative spaces without singular algebraic maps, and studies monopoles and gauge transformations in this framework.

Findings

Magnetic monopole solutions on fuzzy sphere are found.

Classical magnetic charge quantization is not required.

Solutions to Maxwell equations are stable under noncommutative corrections.

Abstract

We write down scalar field theory and gauge theory on two-dimensional noncommutative spaces ${\cal M}$ with nonvanishing curvature and non-constant non-commutativity. Usual dynamics results upon taking the limit of ${\cal M}$ going to i) a commutative manifold ${\cal M}_0$ having nonvanishing curvature and ii) the noncommutative plane. Our procedure does not require introducing singular algebraic maps or frame fields. Rather, we exploit the K\"ahler structure in the limit i) and identify the symplectic two-form with the volume two-form. As an example, we take ${\cal M}$ to be the stereographically projected fuzzy sphere, and find magnetic monopole solutions to the noncommutative Maxwell equations. Although the magnetic charges are conserved, the classical theory does not require that they be quantized. The noncommutative gauge field strength transforms in the usual manner, but the same is not, in general, true for the associated potentials. We develop a perturbation scheme to obtain the expression for gauge transformations about limits i) and ii). We also obtain the lowest order Seiberg-Witten map to write down corrections to the commutative field equations and show that solutions to Maxwell theory on ${\cal M}_0$ are stable under inclusion of lowest order noncommutative corrections. The results are applied to the example of noncommutative AdS${}^2$.