Yang--Mills topology on four-dimensional triangulations

Abstract

We consider 4D $SU(N)$ gauge theories coupled to gravity in the Causal Dynamical Triangulations (CDT) approach, focusing on the topological classification of the gauge path integral over fixed triangulations. We discretize the topological charge and, after checking the emergence of topology and the continuum scaling on flat triangulations, we show that topology emerges on thermalized triangulations only in the so-called $C$-phase of CDT, thus enforcing the link between such phase and semiclassical spacetime. We also provide a tool to visualize the topological structures.

Figures (8)

• Figure 1: $Q_L$ and $S_{YM}$ evolution during cooling for two $SU(3)$ configurations sampled at different $\beta$ values on quasi-flat triangulations with $T = 40$.
• Figure 2: Histograms of $Q_L$ for $SU(3)$ on quasi-flat $T = 40$ triangulations after $5$ (red) and $150$ (blue) cooling steps.
• Figure 3: Topological susceptibility for $SU(3)$ gauge fields in a quasi-flat toroidal triangulation.
• Figure 4: Histograms of $Q_L$ for $SU(3)$ on triangulations thermalized in the de Sitter phase, with ${(S^1)}^4$ overall topology, and bare parameters $\kappa_0 = 4.0,\, \Delta = 0$, $T = 20$ and $N_4 = 120k$.
• Figure 5: Action descent during smoothing for a few configurations on triangulations thermalized in the de Sitter phase, but with different overall topologies.