# Noncommutative Electromagnetism As A Large N Gauge Theory

## TL;DR

This paper demonstrates how noncommutative U(1) gauge theories can be mapped to large N Yang-Mills theories, revealing emergent geometries and gravity, including connections to AdS/CFT and self-dual Einstein gravity.

## Contribution

It establishes a detailed correspondence between noncommutative gauge theories and large N Yang-Mills theories, uncovering emergent gravitational geometries from gauge-Higgs systems.

## Key findings

- Mapping of NC U(1) gauge theory to large N Yang-Mills theory.
- Emergent geometry in higher-dimensional spacetime from gauge-Higgs systems.
- Connection between BPS configurations and self-dual Einstein gravity.

## Abstract

We map noncommutative (NC) U(1) gauge theory on R^d_C X R^{2n}_{NC} to U(N -> \infty) Yang-Mills theory on R^d_C, where R^d_C is a d-dimensional commutative spacetime while R^{2n}_{NC} is a 2n-dimensional NC space. The resulting U(N) Yang-Mills theory on R^d_C is equivalent to that obtained by the dimensional reduction of (d+2n)-dimensional U(N) Yang-Mills theory onto R^d_C. We show that the gauge-Higgs system (A_\mu,\Phi^a) in the U(N -> \infty) Yang-Mills theory on R^d_C 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 (A_\mu,\Phi^a) in half-BPS configurations describes self-dual Einstein gravity.