Monopole Constituents inside SU(n) Calorons

TL;DR

This work shows that SU($n$) calorons with unit topological charge and nontrivial holonomy ${\cal P}_\infty$ decompose into $n$ BPS monopole constituents whose masses are set by holonomy gaps $\nu_m$ and positions $\vec{y}_m$. By marrying the ADHM construction with the Nahm transformation through a Fourier approach in the algebraic gauge, the authors derive a compact, explicit action-density formula: $\frac{1}{2}\mathrm{tr} F_{\mu\nu}^2(x) = -\frac{1}{2} \partial_\mu^2 \partial_\nu^2 \log \psi(x)$, with $\psi(x)$ encoding the constituent data. The construction identifies the $m$-th constituent’s location $\vec{y}_m$ and mass $8\pi^2\nu_m/{\cal T}$ (subject to $\sum_m\nu_m=1$) and shows how the Green’s function on $S^1$ ties the ADHM/Nahm data to the caloron, reproducing known SU(2) results as a special case. The results illuminate the monopole-substructure of calorons, discuss limits and non-maximal symmetry breaking, and point to implications for QCD and potential generalizations to higher charge.

Abstract

We present a simple result for the action density of the SU(n) charge one periodic instantons - or calorons - with arbitrary non-trivial Polyakov loop P_oo at spatial infinity. It is shown explicitly that there are n lumps inside the caloron, each of which represents a BPS monopole, their masses being related to the eigenvalues of P_oo. A suitable combination of the ADHM construction and the Nahm transformation is used to obtain this result.

Figures (1)

• Figure 1: Action densities for the $SU(3)$ caloron at $x_0=0$ in the plane defined by the centers of the three constituents for $1/{\cal{T}}=1.5,3$ and 4 (increasing temperature from top to bottom). We choose mass parameters $(\nu_1,\nu_2,\nu_3)=(0.4,0.35,0.25)$, implemented by $(\mu_1,\mu_2,\mu_3)= (-17/60,-2/60,19/60)$. The constituents are located at $\vec{y}_1=(-{{{1\over 2}}},{{{1\over 2}}}, 0)$, $\vec{y}_2=(0,{{{1\over 2}}},0)$ and $\vec{y}_3=({{{1\over 2}}}, -{{{1\over 4}}},0)$, in units of ${\cal{T}}$. The profiles are given on equal logarithmic scales, cut-off at an action density below $1/e$.