Noncommutative Electromagnetism As A Large N Gauge Theory

We map noncommutative (NC) U(1) gauge theory on IRdC × IR NC to U(N → ∞) Yang-Mills theory on IRdC , where IR d C is a d-dimensional commutative spacetime while IR 2n NC is a 2n-dimensional NC space. The resulting U(N) Yang-Mills theory on IRdC is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang-Mills theory onto IRdC . We show that the gauge-Higgs system (Aμ,Φ) in the U(N → ∞) Yang-Mills theory on IRdC leads to an emergent geometry in the (d+2n)-dimensional spacetime whose metric was determined by Ward a long time ago. In particular, the 10-dimensional gravity for d = 4 and n = 3 corresponds to the emergent geometry arising from the 4-dimensional N = 4 vector multiplet in the AdS/CFT duality. We further elucidate the emergent gravity by showing that the gauge-Higgs system (Aμ,Φ) in half-BPS configurations describes selfdual Einstein gravity. PACS numbers: 11.10.Nx, 02.40.Gh, 04.50.+h