Noncommutative electromagnetism as a large N gauge theory

Abstract

We map noncommutative (NC) U(1) gauge theory on ℝ ×ℝ 2 to U(N→∞) Yang–Mills theory on ℝ , where ℝ is a d-dimensional commutative spacetime while ℝ 2 is a 2n-dimensional NC space. The resulting U(N) Yang–Mills theory on ℝ is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang–Mills theory onto ℝ . We show that the gauge-Higgs system (A μ ,Φ a ) in the U(N→∞) Yang–Mills theory on ℝ 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 ${\mathcal{N}}=4$ vector multiplet in the AdS/CFT duality. We further elucidate the emergent gravity by showing that the gauge-Higgs system (A μ ,Φ a ) in half-BPS configurations describes self-dual Einstein gravity.