Noncommutative electromagnetism as a large N gauge theory

We map noncommutative (NC) U(1) gauge theory on ℝCd×ℝNC2n to U(N→∞) Yang–Mills theory on ℝCd, where ℝCd is a d-dimensional commutative spacetime while ℝNC2n is a 2n-dimensional NC space. The resulting U(N) Yang–Mills theory on ℝCd is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang–Mills theory onto ℝCd. We show that the gauge-Higgs system (Aμ,Φa) in the U(N→∞) Yang–Mills theory on ℝCd 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{N}}=4$\end{document} vector multiplet in the AdS/CFT duality. We further elucidate the emergent gravity by showing that the gauge-Higgs system (Aμ,Φa) in half-BPS configurations describes self-dual Einstein gravity.