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Noncommutative electromagnetism as a large \(N\) gauge theory. (English) Zbl 1189.81221

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

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
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