Dynamical Covariant Quantum Spacetime with Fuzzy Extra Dimensions in the IKKT model

TL;DR

This work constructs a class of $SO(1,3)$-covariant, $k=-1$ FLRW covariant quantum spacetimes with dynamical fuzzy extra dimensions within the undeformed IKKT matrix model. By reformulating the dynamics in terms of covariant conservation laws for the $R$-symmetry current and the matrix energy-momentum tensor, it demonstrates a classical stabilization mechanism for the internal space via nonzero internal angular momentum, yielding a large UV-IR hierarchy. Incorporating quantum effects, the authors argue that the late-time evolution selects a critical scaling $\alpha(\tau)\sim e^{-3\tau/4}$, which keeps the dilaton constant and produces a linear cosmic expansion $a(t)\sim t$, with constant gauge couplings and KK masses. On this background, fluctuations realize a higher-spin gauge theory including gravity, and the effective geometry reduces to a standard 4D spacetime in local normal coordinates, forming a higher-spin manifold with tractable local physics. The work lays out a detailed roadmap for including loop corrections and matter, aiming to connect this emergent framework to broader cosmological and gravitational phenomena within the IKKT model.

Abstract

We consider general $k=-1$ FLRW covariant quantum spacetimes $\mathcal{M}^{3,1} \times \mathcal{K}$ with fuzzy extra dimensions $\mathcal{K}$ as classical solutions of the IKKT matrix model. The coupled equations of motion are recast in terms of conservation laws, which allow to determine the evolution of spacetime in a transparent way. We show that $\mathcal{K}$ is stabilized as a classical solution in the presence of a large $R$ charge, corresponding to internal angular momentum. This provides a mechanism to maintain a large hierarchy between UV and IR scales. We also argue that the evolution of spacetime is determined by a balance between classical and quantum effects, leading to a cosmic scale factor $a(t) \sim t$ and constant dilaton at late times. On such a background, the undeformed IKKT model leads to a higher-spin gauge theory including gravity.

Figures (2)

• Figure 1: The behavior of $A(\tau)=\alpha^2(\tau)$, $\chi(\tau)$, $\omega(\tau)$, and their ratio with the asymptotic exponential scaling are shown.
• Figure 2: Numerical solutions for non-rotating $f(\tau)$ on the background with $\alpha\sim x_4^{-1}$.