Topological aspects of ${\mathbf{G}}_2$ Yang-Mills theory

Yang-Mills theory and QCD are well defined for any Lie group as gauge group. The choice ${\mathrm{G}}_2$ is of great interest, as it is the smallest group with trivial center and being at the same time accessible to simulations. This theory has been found to have many properties in common with SU(3) Yang-Mills theory and QCD, permitting us to study the role of the center. Herein, these investigations are extended to topological properties of ${\mathrm{G}}_2$ Yang-Mills theory. After giving the instanton construction for ${\mathrm{G}}_2$, topological lumps with instanton topological charge are identified in cooled lattice configurations. The corresponding topological susceptibility is determined in the vacuum and at low and high temperatures, showing a significant response to the phase structure of the theory.

Figure 1

Left panel: Cooling histories of the topological charge $Q$ for a ${12}^4$ lattice at $\beta =9.515$. Right panel: The absolute value of $Q$ for the same cooling histories (black lines) compared to the cooling curves of the action divided by the naive action $S_0=14{\pi }^2\beta$ of a $Q=1$ object (gray lines).

Figure 2

Hyper surfaces with action density (top four rows) and topological charge density (bottom four rows) of a configuration obtained after 1500 cooling sweeps from a ${16}^4$ lattice Monte Carlo configuration generated at $\beta =9.515$.

Figure 3

Histogram of the topological charges per configuration after 1500 sweeps on a ${12}^4$ lattice at $\beta =9.515$.

Figure 4

The average of the topological charge, of the absolute value of the topological charge, and topological susceptibility as a function of the cooling sweeps for various volumes and discretizations. Only the measurements performed for every 20th cooling sweep are shown.

Figure 5

Volume and discretization dependence of the average absolute value of the topological charge density $|Q|/V$ and of the fourth root of the topological susceptibility after 1500 cooling sweeps.

Figure 6

The topological susceptibility as a function of temperature on both the $6\times {12}^3$ and the $6\times {16}^3$ lattices. For comparison the results for the Polyakov loop and the chiral condensate are also shown, both from Ref. 8.