The Polarised IKKT Matrix Model

TL;DR

<3-5 sentence high-level summary> The paper studies a supersymmetric mass deformation of the IKKT matrix model that preserves 16 supercharges and a SO(3)×SO(7) symmetry. In the large-mass limit, the dominant saddle is a fuzzy sphere whose matrix description matches a spherical Euclidean D1-brane polarised by NSNS flux in a finite cavity, establishing a matrix–gravity dictionary with the deformation parameter Ω mapping to the background flux. The authors develop a supersymmetric localisation framework that reduces the Ω-dependent matrix integral to a moduli-space integral plus one-loop determinants, yielding exact results for N=2 and a precise large-N large-Ω result compatible with perturbation theory. They also outline backreaction regimes, the corresponding dual IIB backgrounds, and the conceptual notion of timeless holography, with implications for quantum cosmology and emergent spacetime.

Abstract

We establish a correspondence between a supersymmetric mass deformation of the IKKT matrix integral at large $N$ and a background of Euclidean type IIB string theory. Both sides have sixteen supersymmetries and an $SO(3)\times SO(7)$ symmetry. In the limit of large mass the integral is dominated by a fuzzy sphere saddle point. This saddle corresponds to a Euclidean $D1$-brane in a finite, Euclidean, ellipsoidal cavity. The cavity is supported by three-form NSNS flux that polarises $N$ $D$-instantons into the $D1$-brane. We furthermore use supersymmetric localisation to show that the deformed matrix integral can be reduced to a moduli space integral, allowing exact results away from the large mass limit. At small mass the $D1$-branes can backreact on the geometry, and we discuss the possible formulation of a `timeless' holography in such regimes.

Figures (3)

• Figure 1: The Euclidean type IIB background is an ellipsoidal cavity, supported by three-form NSNS flux $H$, dilaton $e^\phi$ and axion $C_0$. The dilaton goes to zero at the boundary of the cavity while the axion diverges. A spherical $D1$-brane polarises due to the background flux and carries $N$ units of $D$-instanton charge.
• Figure 2: The expected type IIB phase diagram. The $x$ axis is (the logarithm of) the radius of the $D1$-brane in string units and the $y$ axis is the strength of gravitational backreaction. The vertical dashed line on the right corresponds to the $D1$-brane radius reaching the edge of the cavity. The dashed line on the left corresponds to the $D1$-brane radius reaching the string scale. Our analysis does not apply beyond these dashed lines. From top to bottom, the solid lines denote the conditions (\ref{['eq:dins']}), (\ref{['eq:back1']}) and (\ref{['eq:conds']}). In the top shaded region the background is given by the backreacted geometry of $N$$D$-instantons, perturbed by a three-form NSNS flux. The region below this one is described by a backreacted spherical $D1$-brane inside the cavity. The bottom shaded region correponds to a probe $D1$-brane in the cavity background. In the region immediately above this one, the $D1$-brane remains a probe but has stringy fluctuations.
• Figure 3: The phase diagram of Fig. \ref{['fig:phases']} with (the logarithm of) $\Omega$ on the $x$ axis. The shaded regions, solid lines and dashed lines are in correspondence with those in Fig. \ref{['fig:phases']}. The extra vertical solid line on the left shows the scale beyond which there is no suppression of subdominant saddles. The vertical line in the middle is the point at which Gaussian fluctuations compete with the action of the saddle.