Bulk geometry in gauge/gravity duality and color degrees of freedom

$\mathrm{U}(N)$ 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 $N\times N$ 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.

Figure 1

The coherent state in ${\mathbb{R}}^{9N^2}$ defined by (15). Each gray disk and black point represents a wave packet and its center, respectively. Under the gauge transformation, the location of the center moves, but the shape of the wave packet in ${\mathbb{R}}^{9N^2}$ does not change.

Figure 2

Gauge-symmetrized version of the wave packet shown in Fig. 1 (gray ring). The wave function is localized near the gauge orbit of $\{Y_{I,\alpha }\}$ (dotted circle). The ground state (gray disk) is localized around the origin. Note that the shape and volume of the gauge orbit depend on $\{Y_{I,\alpha }\}$.