Emergent Geometry from Quantized Spacetime

TL;DR

This paper explores how a mass-deformed matrix model can produce constant curvature spacetimes like spheres and (anti-)de Sitter spaces, linking emergent geometry to Snyder algebra and its deformations.

Contribution

It demonstrates that the vacuum geometry from the mass-deformed matrix model corresponds to Snyder algebra, establishing a connection between algebraic structures and emergent curved spacetime.

Findings

Vacuum geometry is a constant curvature spacetime.

Snyder algebra is equivalent to Lorentz algebra in higher dimensions.

Emergent geometry is described by G-invariant metrics on coset manifolds.

Abstract

We examine the picture of emergent geometry arising from a mass-deformed matrix model. Because of the mass-deformation, a vacuum geometry turns out to be a constant curvature spacetime such as d-dimensional sphere and (anti-)de Sitter spaces. We show that the mass-deformed matrix model giving rise to the constant curvature spacetime can be derived from the d-dimensional Snyder algebra. The emergent geometry beautifully confirms all the rationale inferred from the algebraic point of view that the d-dimensional Snyder algebra is equivalent to the Lorentz algebra in (d+1)-dimensional {\it flat} spacetime. For example, a vacuum geometry of the mass-deformed matrix model is completely described by a G-invariant metric of coset manifolds G/H defined by the Snyder algebra. We also discuss a nonlinear deformation of the Snyder algebra.