A stack of N D-branes can have open strings ending on them. There is a U(N) bran

Question

A stack of N D-branes can have open strings ending on them. There is a U(N) brane gauge field, and r adjoint Higgs fields, with r equal to the number of transverse spatial dimensions. The eigenvalues of the Higgs fields gives the (noncommutative) brane displacements.

Surprisingly, this is equivalent to an extremal black brane with a p-form flux of N, with only closed strings. No gauge symmetry is manifest. Why are they equivalent? What is the exact nature of this correspondence?

It's true we can have "closed" strings which stretch all the way to the horizon an infinite distance away. Their total energy is actually finite though, because of gravitational time dilation. However, no trace of any Chan-Paton factors appear anywhere.

Explanation / Answer

Your question has many layers. The most comprehensive answer would have to explain everything about the gauge/gravity or AdS/CFT duality.

Less ambitiously, there is a simple reason why a stack of D-branes behaves as a black p-brane. It carries a mass (well, the branes have a tension, the mass/energy density per unit volume), and if one has many D-branes, they collectively become heavier. Even if the coupling constant gs is small and gravity is therefore weak, a large enough number/stack of D-branes will have a sufficiently strong gravitational field proportional to gsN (essentially coming from GN?M but I don't want to discuss the subtleties by which they differ) and a large enough concentration of matter in small enough volume (measured in the transverse directions) always creates a black hole. So when gsN is larger than one in some natural units, the gravitational field is strong anyway and the branes gravitate. A better description than perturbative D-branes is to acknowledge the curved geometry around them and use the laws of general relativity (coupled to other fields) to see what's going on.

The gauge symmetry doesn't have to be manifest because gauge symmetries are redundancies, not real symmetries. All physically allowed states must be invariant under these gauge symmetries, anyway. In other words, the gauge theory is marginally confining so only the colorless objects may exist in isolation (well, conformal theories don't allow one to isolate any particles at all, but the colorless particles are closer to be able to be isolated). Only the "global" portion of the gauge symmetry may remain material, see later.

Equivalently, only gauge invariant operators such as traces of various products of the adjoint fields are truly physical and are mapped to some meaningful objects on the gravitational side. Well, you may still try to split these gauge-invariant objects into colored pieces that carry Chan-Paton indices. You will get some objects that are hard to identify on the gravitational side but they're there and their number is always effectively infinite because they correspond to microstates of some black-hole-like states.

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