(Un)solvable Matrix Models for BPS Correlators

TL;DR

This work develops complex matrix-model representations for protected two- and three-point correlators in ${\cal N}=4$ SYM, enabling a direct link between matrix-eigenvalue densities and LLM bubbling geometries. By analyzing huge operators with $\Delta\sim N^2$ and probes of various types (light, giant, AdS), the authors map correlators to complex matrix models, derive saddle-point solutions, and identify the resulting droplet shapes with holographic backgrounds. They demonstrate exact matches with holographic vevs for several cases (notably light probes in HHL backgrounds and HHG structure constants) and reveal rich connections to isomorphic problems in random planar graphs (Potts model) and Eguchi–Kawai reductions. The framework unifies Schur-based, exponential, coherent-state, and multi-trace operator constructions, and provides analytic control over large-$N$ limits, including critical regimes and double-scaling behavior, with implications for holographic reconstruction and potential integrability structures in BPS sectors. Overall, the paper advances a concrete, analytically tractable dictionary between protected SYM correlators and dual gravitational data, offering new handles on non-perturbative AdS/CFT in bubbling geometries and related topological/string-theoretic settings.

Abstract

We propose and study a family of complex matrix models computing the protected two- and three-point correlation functions in $\mathcal{N}=4$ SYM. Our description allows us to directly relate the eigenvalue density of the matrix model for ``Huge" operators with $ Δ\sim N^2$ to the shape of droplets in the dual Lin-Lunin-Maldacena (LLM) geometry. We demonstrate how to determine the eigenvalue distribution for various choices of operators such as those of exponential, character, or coherent state type, which then allows us to efficiently compute one-point functions of light chiral primaries in generic LLM backgrounds. In particular, we successfully match the results for light probes with the supergravity calculations of Skenderis and Taylor. We provide a large $N$ formalism for one-point functions of ``Giant" probes, such as operators dual to giant graviton branes in LLM backgrounds, and explicitly apply it for particular backgrounds. We also explicitly compute the correlator of three huge half-BPS operators of exponential type and stacks of determinant operators by reducing them to the known matrix model problems such as the Potts or $O(n)$ model on random planar graphs. Finally, we point out a curious relation between the correlators of $\frac{1}{4}$-BPS and $\frac{1}{8}$-BPS coherent state operators and the Eguchi-Kawai reduction of the Principal Chiral Model in $2D$ and $3D$ correspondingly.

Figures (16)

• Figure 1: Left: Young tableau consisting of $m$ large rectangular blocks. We parametrize it by the heights of the blocks $L_i$ and by their lenghts $K_i$. Right: An LLM droplet corresponding to this Young tableau (we take the case of 2 rectangular blocks for simplicity). We parametrize the annuli by their inner radii $R_i$ and their outer radii $r_i$. See the paragraph below on the general YT for the expression for $r_i$ and $R_i$ in terms of $K_i$ and $L_i$. See Appendix \ref{['App-densities-LLM']} for explicit formulas for $r_i$ and $R_i$ in terms of $L_i,K_i$.
• Figure 2: Two-dimensional density of eigenvalues $\rho_{\mathcal{D}}(z,\bar{z})$, corresponding to the Schur polynomial operator with rectangular Young tableau with $N$ rows and $N$ columns, and it's projection onto the $x$-axis.
• Figure 3: Shapes of domains $\mathcal{D}$ (droplets) for different regimes in the complex matrix model: a) Blue: subcritical $r= (r_{c}-0.1),\,\,\tau=0.4$; b) Orange: critical $r= r_{c},\,\,\tau=0.4$; c) Red: supercritical (forming intersections) $r= (r_{c}+0.118),\,\,\tau=0.4$; Green: "astroid" $r= r_{c},\,\,\tau=0$.
• Figure 4: Shapes of domain $\mathcal{D}$ (droplet) for the double-critical regime $m=3$ given by the curve \ref{['doublec']}. Notice that the beaks (on the top and bottom) are less pronounced than in the $m=2$ case.
• Figure 5: The support of eigenvalue distribution for the coherent state \ref{['twoEvCS']} with $\frac{p}{N} = \frac{2}{3}$ for various centers. When $\alpha_2=\frac{1+\sqrt 2}{\sqrt 3}\approx 1.394$, the droplet degenerates into two discs.