Topology of SU(N) gauge theories at T=0 and T=Tc

TL;DR

This study analyzes the topological structure of SU($N$) lattice gauge theories up to $N=8$, focusing on zero temperature and near the deconfinement temperature $T_c$. Using cooling-probed lattice methods, the authors measure the topological susceptibility $χ_t$ and the instanton-size density $D(ρ)$ across temperatures and colors, expressing results at fixed $a=1/5T_c$ to compare $N$-dependence. They find that $χ_t$ approaches a finite, nonzero limit as $N\to\infty$ in the confined phase, while in the deconfined phase topological fluctuations are exponentially suppressed with increasing $N$, effectively vanishing at large $N$ even at $T_c$. The instanton-size distributions corroborate this: confinement yields a $D(ρ)$ that sharpens with $N$, approaching a delta function around $ρ_c \sim 1/T_c$, whereas deconfinement suppresses all instanton sizes, with rare large-$ρ$ fluctuations persists only as lattice artifacts. Together, these results support a simple large-$N$ picture in which topology survives in confinement but is erased in deconfinement, providing insight into $U_A(1)$ restoration and the nonperturbative structure of QCD in the $N\to\infty$ limit.

Abstract

We calculate the topological charge density of SU(N) lattice gauge fields for values of N up to N=8. Our T=0 topological susceptibility appears to approach a finite non-zero limit at N=infinity that is consistent with earlier extrapolations from smaller values of N. Near the deconfining temperature Tc we are able to investigate separately the confined and deconfined phases, since the transition is quite strongly first order. We find that the topological susceptibility of the confined phase is always very similar to that at T=0. By contrast, in the deconfined phase at larger N there are no topological fluctuations except for rare, isolated and small instantons. This shows that as N->infinity the large-T suppression of large instantons and the large-N suppression of small instantons overlap, even at T=Tc, so as to suppress all topological fluctuations in the deconfined phase. In the confined phase by contrast, the size distribution is much the same at all T, becoming more peaked as N grows, suggesting that D(rho) is proportional to a delta function at N=infinity, centered on rho close to 1/Tc.

Figures (13)

• Figure 1: The topological susceptibility, $\chi_t$, in units of the string tension, $\sigma$, plotted versus $1/N^2$. The calculations are performed at a fixed lattice spacing $a \simeq 1/5 T_c$. The line is a large-$N$ extrapolation that includes $O(1/N^2)$ and $O(1/N^4)$ corrections.
• Figure 2: The ratio of the topological susceptibility, $\chi_t$, in the deconfined and confined phases at $T\simeq T_c$.
• Figure 3: The topological susceptibility, $\chi_t$, in units of the string tension, $\sigma$, plotted versus $T/T_c$ for $SU(6)$, plotted separately for the confined, $\circ$, and deconfined, $\bullet$, phases. The calculations are performed at a lattice spacing $a \simeq 1/5 T_c$.
• Figure 4: The instanton size density, $D(\rho)$, for $N=2(\star),3(+),4(\circ),6(\times),8(\bullet)$ on (mostly) $10^4$ lattices with $a= 1/5T_c$.
• Figure 5: The average number of instantons versus $N$ at $a=1/5T_c$ on a $10^4$ volume, from the calculations listed in Table \ref{['table_chi5a']}