Quantum weights of dyons and of instantons with non-trivial holonomy

TL;DR

This work provides a complete one-loop quantum weight for KvBLL calorons with non-trivial holonomy in finite-temperature SU(2) Yang–Mills theory by deriving the exact determinant Det$(-D^2)$ in the caloron background. By partitioning space into dyon-core and far regions and employing ADHM/Nahm techniques to construct periodic Green functions, the authors obtain a closed-form expression for the determinant as a function of the holonomy $v$, temperature $T$, scale $\Lambda$, and dyon separation $r_{12}$, with controlled $1/r_{12}$ corrections. In the large separation limit, the caloron weight factorizes into the product of the two dyon measures plus a calculable interaction energy, and a two-loop running-coupling refinement is provided to improve density estimates. The paper also analyzes the competition between trivial and nontrivial holonomy, showing that at temperatures below a critical value the trivial holonomy becomes unstable and calorons tend to ionize into dyons, offering a microscopic mechanism relevant to confinement and lattice observations. Overall, the results deliver a quantitative framework connecting caloron structure, holonomy dynamics, and finite-temperature nonperturbative effects in Yang–Mills theory.

Abstract

We calculate exactly functional determinants for quantum oscillations about periodic instantons with non-trivial value of the Polyakov line at spatial infinity. Hence, we find the weight or the probability with which calorons with non-trivial holonomy occur in the Yang--Mills partition function. The weight depends on the value of the holonomy, the temperature, Lambda_QCD, and the separation between the BPS monopoles (or dyons) which constitute the periodic instanton. At large separation between constituent dyons, the quantum measure factorizes into a product of individual dyon measures, times a definite interaction energy. We present an argument that at temperatures below a critical one related to Lambda_QCD, trivial holonomy is unstable, and that calorons ``ionize'' into separate dyons.

Figures (5)

• Figure 1: The action density of the KvBLL caloron as function of $z,t$ at fixed $x=y=0$, with the asymptotic value of $A_4$ at spatial infinity ${\rm v}=0.9\pi T,\; \overline{{\rm v}}=1.1\pi T$. It is periodic in $t$ direction. At large dyon separation the density becomes static (left, $r_{12}=1.5/T$). As the separation decreases the action density becomes more like a $4d$ lump (right, $r_{12}=0.6/T$). In both plots the L,M dyons are centered at $z_{\rm L}=-{\rm v}\,r_{12}/2\pi T,\;z_{\rm M}=\overline{{\rm v}}\,r_{12}/2\pi T,\;x_{\rm L,M}=y_{\rm L,M}=0$. The axes are in units of temperature T.
• Figure 2: Potential energy as function of ${\rm v}/T$. Two minima correspond to $\frac{1}{2}{\rm Tr} L=\pm 1$, the maximum corresponds to ${\rm Tr} L=0$. The range of the holonomy where dyons experience repulsion is shown in dashing.
• Figure 3: The action density of the KvBLL caloron as function of $z,x$ at fixed $t=y=0$. At large separations $r_{12}$ the caloron is a superposition of two BPS dyon solutions (left, $r_{12}=1.5/T$). At small separations they merge (right, $r_{12}=0.6/T$). The caloron parameters are the same as in Fig. 1.
• Figure 4: Three regions of integration for well separated dyons.
• Figure 5: Free energy of the caloron gas in units of $T^3V$ at $T= 1.3\Lambda$ (dotted), $T= 1.125\Lambda$ (solid) and $T= 1.05\Lambda$ (dashed) as function of the asymptotic value of $A_4$ in units of $2\pi T$.