Geometry in transition in four dimensions: A model of emergent geometry in the early universe

We study a six matrix model with global $SO(3)\times SO(3)$ symmetry containing at most quartic powers of the matrices. This theory exhibits a phase transition from a geometrical phase at low temperature to a Yang-Mills matrix phase with no background geometrical structure at high temperature. This is an exotic phase transition in the same universality class as the three matrix model but with important differences. The geometrical phase is determined dynamically, as the system cools, and is given by a fuzzy sphere background ${\mathbf{S}}_N^2\times {\mathbf{S}}_N^2$, with an Abelian gauge field which is very weakly coupled to two normal scalar fields.

Figure 1

The action for the 6D Yang-Mills matrix model.

Figure 2

The specific heat for the 6D Yang-Mills matrix model.

Figure 3

The phase diagram of the 6D Yang-Mills matrix model.

Figure 4

The eigenvalue distribution for $X_3$ across the transition line.

Figure 5

The eigenvalue distributions for $X_3$ for small values of $M$ and $\tilde{\alpha }$.

Figure 6

The eigenvalue distributions for $X_3$ for large values of $M$ and small values $\tilde{\alpha }$.