Non-constant non-commutativity in 2d field theories and a new look at fuzzy monopoles

We write down scalar field theory and gauge theory on two-dimensional non-commutative spaces M with non-vanishing curvature and non-constant non-commutativity. Usual dynamics results upon taking the limit of M going to (i) a commutative manifold M0 having non-vanishing curvature and (ii) the non-commutative plane. Our procedure does not require introducing singular algebraic maps or frame fields. Rather, we exploit the Kähler structure in the limit (i) and identify the symplectic two-form with the volume two-form. As an example, we take M to be the stereographically projected fuzzy sphere, and find magnetic monopole solutions to the non-commutative Maxwell equations. Although the magnetic charges are conserved, the classical theory does not require that they be quantized. The non-commutative 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 M0 are stable under inclusion of lowest order non-commutative corrections. The results are applied to the example of non-commutative AdS2.