U($N$) supersymmetric Yang-Mills theory naturally appears as the low-energy effective theory of a system of $N$ D-branes and open strings between them. Transverse spatial directions emerge from scalar fields, which are $Ntimes N$ matrices with color indices; roughly speaking, the eigenvalues are the locations of D-branes. In the past, it was argued that this simple emergent space picture cannot be used in the context of gauge/gravity duality, because the ground-state wave function delocalizes at large $N$, leading to a conflict with the locality in the bulk geometry. In this paper we show that this conventional wisdom is not correct: the ground-state wave function does not delocalize, and there is no conflict with the locality of the bulk geometry. This conclusion is obtained by clarifying the meaning of the diagonalization of a matrix in Yang-Mills theory, which is not as obvious as one might think. This observation opens up the prospect of characterizing the bulk geometry via the color degrees of freedom in Yang-Mills theory, all the way down to the center of the bulk.
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