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

TL;DR

This paper discovers a novel saddle point in the Euclidean IKKT matrix model at infinite N, revealing emergent four-dimensional geometry with SU(2) symmetry related to black hole spacetimes.

Contribution

It identifies a unique classical saddle solution in the IKKT model with Lie algebraic structure leading to emergent geometry with black hole-like features.

Findings

Derived a four-dimensional space from the Lie algebraic structure.

Found the metric has SU(2) isometry related to Taub NUT/Bolt geometry.

Connected the saddle solution to black hole physics.

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.