Determine the dominant IKKT vacuum

Determine whether a single saddle point dominates the Lorentzian IKKT matrix integral or whether many saddle points, potentially with different dimensions, contribute significantly, and establish whether the dominant vacua are four-dimensional.

Background

The Lorentzian IKKT model admits many candidate saddle points, including noncommutative quantum planes of different dimensions. The paper adopts a perturbative approach around selected backgrounds rather than determining the vacuum selected by the full matrix integral. The dominance question is therefore central to establishing whether the model dynamically selects physically relevant 3+1-dimensional spacetime.

The authors note numerical evidence favoring 3+1-dimensional configurations, but state that the issue has not been analytically settled and may require new techniques.

References

On the other hand, it is not clear whether one saddle point will dominate the matrix integral, or many different saddle points (possibly with different dimensions) contribute significantly, with complex weights. Although this is not essential for the physics on some background of interest, it would be desirable to show that the ''dominant'' vacua are 3+1 dimensional.

— Quantum spacetime and gravity from the IKKT matrix model: an invitation  (2609.08955 - Steinacker, 8 Sep 2026) in Section 4.1, “Vacua, frame and geometry”