发表机构
Department of Physics and Center for Theoretical Physics, National Taiwan University（台湾大学物理系与理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究IKKT矩阵模型中李代数鞍点，揭示时间涌现需可解根，空间可半单，为时空维度与符号涌现提供判据。
AI 中文摘要
在IKKT矩阵模型中，时空的维度、几何和度量符号必须从矩阵动力学中涌现。我们对模型的李代数鞍点进行分类：经典解中，十个厄米矩阵在酉表示中实现一个有限维实李代数。对于此类，运动方程归结为关于代数上对称双线性形式$h$的单一代数方程，涌现时空的符号可从$h$读出。两个事实控制结果。首先，当复化代数是单李代数时，Casimir阻碍仅允许$h=0$；这解释了为何模糊球和膨胀宇宙解需要质量项或红外正则化。其次，半单代数仅通过不同单理想之间的耦合允许鞍点；此类鞍点首先在六维出现，并且仅从十二维起达到所有符号。这些事实给出时间涌现的判据。在至多十一维中每个非退化洛伦兹鞍点——因此十维模型的每个洛伦兹背景——具有非零可解根。半单部分的唯一非平凡可能性是洛伦兹代数$\mathfrak{so}(1,3)$，它纯类空，并且每个类时方向在根中有分量。当半单部分非平凡时，它仅从十维起作用于根；在十维，根必须是阿贝尔的并且必须携带Weyl旋量表示，这解出洛伦兹运动方程；非阿贝尔根首次出现在十一维。结果，空间可以是半单的；时间必须是可解的。
英文摘要
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.
Comments17 pages, 0 figures
