Multi-Caloron solutions

Abstract

We discuss the construction of multi-caloron solutions with non-trivial holonomy, both as approximate superpositions and exact self-dual solutions. The charge k SU(n) moduli space can be described by kn constituent monopoles. Exact solutions help us to understand how these constituents can be seen as independent objects, which seems not possible with the approximate superposition. An "impurity scattering" calculation provides relatively simple expressions. Like at zero temperature an explicit parametrization requires solving a quadratic ADHM constraint, achieved here for a class of axially symmetric solutions. We will discuss the properties of these exact solutions in detail, but also demonstrate that interesting results can be obtained without explicitly solving for the constraint.

Figures (7)

• Figure 1: Approximate superposition of two $SU(2)$ charge 1 calorons with its pairs of equal mass constituents at $\vec{x}=(2,0,2),~(2,0,8)$ and $\vec{x}=(8, 0,2),~(8,0,8)$. The logarithm of the action density is plotted as a function of $x$ and $z$. The plot on the right shows one of the would-be Dirac strings, zooming in by a factor 40 on the transverse direction.
• Figure 2: Comparing the logarithm of the action density (cutoff for $\log(S)$ below -3 and above 7) as a function of $x$ and $z$ for the exact $SU(2)$ solution ($\mu_2={{{1\over 4}}}$) with charge 2 (left) with the approximate superposition of two charge one calorons (middle) and the abelian solution based on Dirac monopoles (right), all on the same scale. The two pairs of constituent monopoles are located at $\vec{x}=(0,0,6.031),~(0,0,2.031)$ and $\vec{x}=(0,0,-2.031),~(0,0,-6.031)$.
• Figure 3: Constituent locations $y_m^{(j)}$ based on Eq. (\ref{['eq:Ym']}) (i.e. $\alpha_a=0$ and $\xi_0\to0$) as a function of $\xi=\xi_1=-\xi_2$ for $y_2= -y_1=\nu_1=\nu_2={{{1\over 2}}}$ and $\rho_1=\rho_2=2$. Dashed versus full lines distinguish the magnetic charge of the constituents. The dotted lines represent the four cases shown in Figs. \ref{['fig:Wellsep']} and \ref{['fig:Overlap-xz']}.
• Figure 4: The action density (cutoff for $\log(S)$ below -3) as a function of $x$ and $z$ for the $SU(2)$ solution with charge 2 ($\mu_2={{{1\over 4}}},\, \alpha_a^j=\xi^a_0=0,\,\rho_j=2$) and increasing values of $\xi\equiv\xi_1= -\xi_2$ (see Fig \ref{['fig:Wellsep']} (left) for $\xi=3.5$), with $\xi=1.6$ (left), $\xi= 1.0$ (middle) and $\xi=0.5$ (right). Compare Fig. \ref{['fig:Ydiag']} for the corresponding constituent locations.
• Figure 5: The action density (cutoff for $\log(S)$ below -3) as a function of $z$ and $t$ (doubling the time-period) for the $SU(2)$ solution with charge 2 ($\mu_2={{{1\over 4}}},\,\xi={{{1\over 2}}},\,\alpha_a^j=0,\,\rho_j=2$) and increasing values of $\xi_0$, with $\xi_0=0$ (left), $\xi_0=0.2$ (middle) and $\xi_0=0.25$ (right), for the top row all on the same scale, zooming in on the middle region on the bottom row (not to scale). See Fig \ref{['fig:Overlap-xz']} (right) for the case of $\xi_0=0$ shown as a function of $x$ and $z$.