Yang--Mills topology on four-dimensional triangulations

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.

💡 Research Summary

This paper investigates four‑dimensional SU(N) Yang–Mills gauge theories coupled to gravity within the framework of Causal Dynamical Triangulations (CDT). The authors focus on the topological classification of gauge field configurations on fixed triangulations, introducing a discretized definition of the topological charge Q and studying its behavior on both quasi‑flat triangulations and thermalized CDT ensembles.

The construction begins by assigning an SU(N) parallel transporter to each 4‑simplex and placing these variables on the links of the dual graph. The elementary gauge observable is the plaquette, defined as the ordered product of link variables around a (d‑2)‑simplex (a triangle in 4D). By expanding the plaquette in the naïve continuum limit, they recover the field‑strength tensor F_{\mu\nu} and establish the relation between the bare gauge coupling g and the inverse coupling β: β = 12√5 N g². This provides a lattice‑like Yang–Mills action adapted to the irregular geometry of triangulations.

For the topological charge, the continuum expression Q = (g²/32π²)∫ ε^{\muν\alphaβ} Tr