(Un)solvable Matrix Models for BPS Correlators

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 $ \Delta \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.