A prediction for bubbling geometries

TL;DR

This work computes the vacuum expectation values of supersymmetric circular Wilson loops in N=4 Yang-Mills theory within the parameter regime that admits smooth bubbling geometry duals. By mapping these loops to Gaussian matrix models, the author derives explicit strong-coupling predictions for the loop vevs in terms of the Young diagram data, linking them to the on-shell supergravity action on bubbling geometries. The results reveal a universal structure: for a general Young diagram with edge lengths n_I and k_I, the leading vev factorizes into a sum of terms proportional to λ/(8N) times K_I^2, where K_I are cumulative sums of k_I, and the D5 and D3 brane pictures provide complementary interpretations as geometric transitions and bound-state branes. The analysis is carried out via two-matrix models for rectangular diagrams and a general g-level construction, with complementary bound-state pictures showing the same leading behavior, thereby strengthening the AdS/CFT correspondence for non-local operators and providing concrete predictions for dual bubbling geometries.

Abstract

We study the supersymmetric circular Wilson loops in N=4 Yang-Mills theory. Their vacuum expectation values are computed in the parameter region that admits smooth bubbling geometry duals. The results are a prediction for the supergravity action evaluated on the bubbling geometries for Wilson loops.

Figures (12)

• Figure 1: Gravity duals of Wilson loops in ${\mathcal{N}}=4$ Yang-Mills. A string corresponds to the fundamental representation, a D3-brane to a symmetric representation, a D5-brane to an anti-symmetric representation, and a bubbling geometry to the representation specified by a rectangular Young diagram. (D3-brane ..., D5-brane ...) in the figure should be replaced by (D3-brane wrapping $S^3\subset AdS_5$, D3-brane wrapping $S^3\subset S^5$) for local operators in ${\mathcal{N}}=4$ Yang-Mills, and by (D-brane, anti-D-brane) for Wilson loops in Chern-Simons theory.
• Figure 2: The Young diagram $R$, shown rotated and inverted, is specified by the lengths $n_I$ and $k_I$ of the edges. Equivalently, $n_I$ and $k_I$ denote the lengths of the black and white regions in the Maya diagram. $n_{g+1}$ is defined by $\sum_{I=1}^{g+1} n_I=N$.
• Figure 3: A rectangular Young diagram with $n$ rows and $k$ columns.
• Figure 4: The eigenvalue distributions in the rectangular case. The $m$-eigenvalues shown as black lines split into two groups, $\{m_i^{(1)}\}$ on the right and $\{m_i^{(2)}\}$ on the left. The $u$-eigenvalues are distributed uniformly along the red line in the imaginary direction.
• Figure 5: The eigenvalue bound states ($m$-$v$)${}_a$. The distance between the two eigenvalues in the $a$-th bound state from the left is $1/(k+a)$. The distance between neighboring bound states is much larger and is of the order $\sqrt\lambda/N$.