Bubbling Geometries for Half BPS Wilson lines

TL;DR

This work constructs bubbling geometries in IIB and M-theory that encode the holographic duals of half-BPS Wilson lines and surface operators. By enforcing the exact symmetry of straight Wilson lines, it derives a base 2D manifold framework in which the 10D/11D supergravity equations reduce to a set of harmonic and flux relations, culminating in a continuous maya-diagram description that mirrors Gaussian matrix-model eigenvalue distributions. The approach connects geometric boundary data to matrix-model saddles and provides a mechanism to label Wilson-line operators via a monomial–Young diagram correspondence, while outlining M-theory analogs for higher-dimensional defects. The results suggest a deep link between flux-quantized bubbling patterns and operator representations, with potential extensions to holes in eigenvalue distributions and anti-symmetric representations via AdS_2×S^4 D5-branes. Overall, the paper advances a unified bubbling-geometry framework for half-BPS non-local operators in AdS/CFT and lays groundwork for further exploration of defects in diverse dimensions.

Abstract

We consider the supergravity backgrounds that correspond to supersymmetric Wilson line operators in the context of AdS/CFT correspondence. We study the gravitino and dilatino conditions of the IIB supergravity under the appropriate ansatz, and obtain some necessary conditions for a supergravity background that preserves the same symmetry as the supersymmetric Wilson lines. The supergravity solutions are characterized by continuous version of maya diagrams. This diagram is related to the eigenvalue distribution of the Gaussian matrix model. We also consider the similar backgrounds of the 11-dimensional supergravity.

Figures (7)

• Figure 1: The black and white pattern that appears on the boundary of the base 2-dimensional space. At a black point, $S^2$ shrinks, and at the white point $S^4$ shrinks.
• Figure 2: The pattern that characterizes the $AdS_5\times S^5$ solution. The black segment expresses $S^5$.
• Figure 3: Figure (A) expresses the $x$-axis of the $AdS_5\times S^5$. The $AdS_2\times S^2$ D3-brane looks as a point like object on the $x$-axis. It sits on the point $x=\sqrt{4\pi g_s N+\frac{k^2(2\pi g_s)^2}{4}}$. On the other hand, figure (B) represents the eigenvalue distribution of the matrix model. The black bar is the $\lambda_2,\dots,\lambda_N$. Only $\lambda_1$ separates from the other eigenvalues. Its position is $\lambda_1=\sqrt{2\hbar N+\frac{k^2\hbar^2}{4}}$.
• Figure 4: An example of maya diagram.
• Figure 5: Correspondence between maya diagrams and Young diagrams. Here we show the Young diagram which corresponds to the example of figure \ref{['mayaex']}. The figure of (A) is the line made by replacing a white dot with horizontal segment and a black dot with vertical segment of unit length. The "height" of this figure is equal to the number of black dots $N$. The figure (B) shows how to make the Young diagram from the figure (A).