Lie Algebra Saddles in the IKKT Matrix Model and Criteria for the Emergence of Time

Abstract

In the IKKT matrix model, the dimension, geometry, and metric signature of spacetime must emerge from the matrix dynamics. We classify the Lie-algebraic saddles of the model: classical solutions in which the ten Hermitian matrices realize a finite-dimensional real Lie algebra in a unitary representation. For this class the equations of motion collapse to a single algebraic equation for a symmetric bilinear form $h$ on the algebra, and the signature of the emergent spacetime can be read off from $h$. Two facts control the outcome. First, when the complexified algebra is simple, a Casimir obstruction allows only $h=0$; this explains why fuzzy spheres and the expanding-universe solutions need mass terms or infrared regulators. Second, semisimple algebras admit saddles only through couplings between distinct simple ideals; such saddles first appear in six dimensions and reach every signature only from twelve. These facts yield criteria for the emergence of time. Every nondegenerate Lorentzian saddle in at most eleven dimensions---hence every Lorentzian background of the ten-dimensional model---has a nonzero solvable radical. The only nontrivial possibility of the semisimple part is the Lorentz algebra $\mathfrak{so}(1,3)$, which is purely spacelike, and every timelike direction has a component in the radical. When the semisimple part is nontrivial, it acts on the radical only from ten dimensions on; at ten dimensions, the radical must be abelian and must carry the Weyl-spinor representation, which solves the Lorentzian equations of motion; a nonabelian radical first appears in eleven dimensions. As a result, space can be semisimple; time must be solvable.