Einstein gravity from a matrix integral -- Part I

TL;DR

We construct regular, horizonless Euclidean IIB geometries dual to the mass-deformed IKKT matrix model, described by a single axially symmetric potential $V(\rho,z)$ and preserving $\,SO(7)\times SO(3)\,$ with 16 supersymmetries. Flux quantization ties disk data in a 4D electrostatic setup to the dimensions and degeneracies of $SU(2)$ irreps, yielding a one-to-one map between backreacted geometries and fuzzy-sphere vacua. The on-shell action and a polarized D1-brane probe reproduce the expected matrix-model saddles, with a scaling form $S \sim N^2/\lambda^{2/3}$ controlled by a dimensionless parameter $\xi$, and a 12D uplift framing the holographic picture. The results provide a concrete holographic realization of the polarized IKKT model and lay groundwork for a localization-based dictionary and future extensions with branes and monodromies.

Abstract

We construct backreacted geometries dual to the supersymmetric mass deformation of the IKKT matrix model. They are Euclidean type IIB supergravity solutions given in terms of an electrostatic potential, having $SO(7)\times SO(3)$ isometry and 16 supersymmetries. Quantizing the fluxes, we find that the supergravity solutions are in one-to-one correspondence with fuzzy sphere vacua of the matrix model.

Figures (4)

• Figure 1: Picture of the geometry in the $(\rho,z)$ plane. The 10d geometry is obtained by fibering an $S^2$ and an $S^6$. The blue lines are defined by $\dot{V}\equiv \partial_\rho V=0$ and are the regions where $S^2$ shrinks. The red line is the $z$-axis $\rho=0$ where the $S^6$ shrinks. We can construct 7-cycles $\Sigma_7$ as the product of the 6-sphere times a segment on the $(\rho,z)$ plane whose endpoints have $\rho=0$ (where $S^6$ shrinks). Similarly, there are 3-cycles $\Sigma_3$ given by the 2-sphere times a segment connecting points where $S^2$ shrinks.
• Figure 2: Correspondence between the fuzzy sphere vacua of the matrix model and the dual geometries. For each spin $j_s$$SU(2)$ irreducible representation of dimension $N_s=2j_s+1$ we put a disk at position $z_s \sim N_s$. The number $n_s$ of copies of this representation appearing determines the charge of that disk $Q_s \sim n_s$. Those integers are related to the $D1$ brane charge and $NS5$ brane charge by $N_{D1,s}=n_s$ and $N_{NS5,s} = N_s-N_{s-1}$.
• Figure 3: Summary of our analysis of the supergravity regime. We study the scaling of the string coupling $g_s e^\phi$ and the curvature probed by the Ricci scalar in regions ① to ⑦. The scaling behaviors are given in table \ref{['tab:summary of validity analysis']}. We find that the strongest conditions come from infinity and the tip of the disk (region ① and ②).
• Figure 4: Integration contour for the on-shell action. The interior is a patch covering the entire spacetime minus regions of measure zero. All cycles in that patch are contractible.