Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-02-06
Nucl.Phys.B745:236-259,2006
Physics
High Energy Physics
High Energy Physics - Theory
27 pages
Scientific paper
10.1016/j.nuclphysb.2006.04.001
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${}^2$.
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