Periodic Instantons with non-trivial Holonomy

TL;DR

This work constructs charge-one caloron solutions with non-trivial holonomy in SU(2) by merging the Nahm transformation with the ADHM formalism and by deriving an explicit Green's function to express the gauge potential and action density. The caloron exhibits a two-monopole constituent structure whose masses and separation are controlled by the holonomy parameter, and its moduli space is a hyperkähler manifold given by ${\rm I R}^3 \times S^1 \times {\rm Taub-NUT}/Z_2$, with the Taub-NUT parameter encoding the holonomy. The authors provide a detailed two-step construction, including the periodic chain interpretation and the explicit form of the gauge field, and discuss the implications for abelian projection and finite-temperature QCD. The work highlights the deep link between calorons and monopole dynamics, offering a framework for exploring the monopole content of topological gauge configurations and potential lattice applications.

Abstract

We present the detailed derivation of the charge one periodic instantons - or calorons - with non-trivial holonomy for SU(2). We use a suitable combination of the Nahm transformation and ADHM techniques. Our results rely on our ability to compute explicitly the relevant Green's function in terms of which the solution can be conveniently expressed. We also discuss the properties of the moduli space, R^3 X S^1 X Taub-NUT/Z_2 and its metric, relating the holonomy to the Taub-NUT mass parameter. We comment on the monopole constituent description of these calorons, how to retrieve topological charge in the context of abelian projection and possible applications to QCD.

Figures (6)

• Figure 1: The function $\Theta_\omega(z)$.
• Figure 2: Time evolution of the caloron solution. During one period (${\cal{T}}=1$), we plot the "energy" as a function of time, $E(t)\equiv-\frac{1}{16\pi^2} \int_{{\rm I R}^3}{\rm tr} F^2$, for $\rho=0.1,0.2,0.3,0.5,1.0,2.0$. For small values of $\rho$, the caloron is short-lived and instanton-like, whereas for large values, $\rho > 1$, the profile flattens and the caloron becomes static and monopole-like.
• Figure 3: Shown are caloron profiles for $\omega=0.125$ (${\cal{T}}=1$), with $\rho=0.8,1.2,1.6$ (from bottom to top). This illustrates the growing separation of the two lumps with $\rho$. Once the constituents are separated, the lumps are spherically symmetric and do not change their shape upon further separation. Vertically is plotted the action density at $x_0=0$, on equal logarithmic scales for all profiles. They were cut off at an action density below $1/e^2$.
• Figure 4: Profiles for calorons at $\omega\!=\!0$, 0.125, 0.25 (from left to right) with $\rho={\cal{T}}=1$. The axis connecting the lumps, separated by a distance $\pi$ (for $\omega\neq0$), corresponds to the direction of $\hat{a}$. The other direction indicates the distance to this axis, making use of the axial symmetry of the solutions. The mass ratio of the two lumps is approximately $\omega/\bar{\omega}$, i.e. zero (no second lump), a third and one (equal masses), for the respective values of $\omega$. Vertically is plotted the action density at $x_0=0$, on equal logarithmic scales for all profiles. They were cut off at an action density below $1/e$.
• Figure 5: The function $\int^z_0\Theta_\omega(z')dz'$.