Re-dressing Emperor: Four Dimensional Yang-Mills Theory, Gauge Invariant Mass And Fluctuating Three Branes

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We consider the coupling between four dimensional Yang-Mills field and a three brane that fluctuates into higher dimensions. For this we interpret the Yang-Mills theory as a higher dimensional bulk gravity theory with dynamics that is governed by the Einstein action, and with a metric tensor constructed from the gauge field in a manner that displays the original gauge symmetry as an isometry. The brane moves in this higher dimensional space-time under the influence of its bulk gravity, with dynamics determined by the Nambu action. This introduces the desired interaction between the brane and the gauge field in a way that preserves the original gauge invariance as an isometry of the induced metric. After a prudent change of variables the result can be interpreted as a gauge invariant and massive vector field that propagates in the original space-time R⁴: The presence of the brane becomes entirely invisible, except for the mass.