A New Type of Saddle in the Euclidean IKKT Matrix Model and Its Emergent Geometry

TL;DR

This work identifies a novel nontrivial saddle of the Euclidean IKKT matrix model by restricting $A_\mu$ to generators of $\mathfrak{so}(n,m)$ and exploiting an infinite-dimensional representation, yielding a unique $\mathfrak{so}(1,3)$ solution. Using a coadjoint-orbit construction with fixed Casimirs, the authors obtain a four-dimensional manifold $\mathcal{M}_4$ endowed with a nondegenerate Kirillov-Kostant-Souriau Poisson structure, enabling a semi-classical readout of geometry. A test scalar analysis then yields an effective metric $G_{ab}$ on $\mathcal{M}_4$ with an $SU(2)$ isometry and asymptotic cylinder-like behavior, connecting to Taub-NUT/Bolt and potentially black hole physics. The results provide a concrete route to gravity-like emergent geometry within the IKKT framework and motivate further numerical, stability, and holographic investigations.

Abstract

We study the equation of motion of the Euclidean IKKT matrix model, and realize a new type of classical saddle that only exists in $N\rightarrow\infty$ limit. Under the assumption that the matrices are the generators of $\mathfrak{so}(n,m)$, we identify a unique solution, that is, $\mathfrak{so}(1,3)$. Even though it has $6$ generators and thus $6$ non-zero matrices, they are not independent due to the $2$ Casimir constraints in $\mathfrak{so}(1,3)$. Exploiting the Lie-algebraic structure and the Casimir constraints, we derive a four-dimensional space that a test scalar propagates on. The associated metric possesses $\mathrm{SU}(2)$ isometry, which is closely related to the Taub NUT/Bolt geometry and, more broadly, to black hole physics.

Figures (1)

• Figure 1: Geometry of effective metric \ref{['eq:Gab']}. Here, we emphasize the relation between the radius of $\chi$ direction and $r$, so each point in the plot is an $S_{\mathrm{squash}}^2$ parametrized by $\theta, \varphi$. Note that one can easily check that the curvature is finite everywhere.