Exact Caloron solutions with non-trivial holonomy

Here we make available the C-programmes that can be used for making plots of the energy density ((SU(2) and SU(3)) and the trace of the Polyakov loop and the zero-mode density (SU(2) only). The output (Profile[b]) is in a format that easily allows one to plot it using the Mathematica command ListPlot3D[Profile[b]].

A number of these programmes were used to compare lattice and analytic results as discussed in the paper:

Calorons on the Lattice - a new Perspective, by M. García Pérez, A. González-Arroyo, A. Montero and P. van Baal, JHEP 06 (1999) 001.

The analytic results are based on the papers by T.C. Kraan and P. van Baal,

Exact T-duality between Calorons and Taub-NUT spaces, Phys. Lett. B428 (1998) 268-276;
Periodic Instantons with non-trivial Holonomy, Nucl. Phys. B533 (1998) 627-659;
Monopole constituents inside SU(n) Calorons, Phys. Lett. B435 (1998) 389-395.

A short summary of the analytic results can be found in

Constituent monopoles without gauge fixing, presented at Lattice'98, Boulder, Nucl. Phys. B(Proc.Suppl.) 73 (1999) 554-556.

Warning: avoid being exactly at the location of a constituent monopole or for t=0 at "the center of mass" (gauge singularity) to avoid numerical instabilities.

Available are the following modules:

calor-su2.c which calculates the double d'Alembertian of log(psi) for SU(2). The action density is obtained by dividing this result by -2.
su2-A-F.c which calculates the gauge fields A and F for SU(2) with constituents along the z-axis, with the z coordinate of the first constituent larger than that of the second. If this is not the case one first needs to perform a rotation to achieve this!
calor-su3.c which calculates the double d'Alembertian of log(psi) for SU(3). The action density is obtained by dividing this result by -2.

These modules are called in the C-programmes:

su2.c to compute gauge fields, action density and Polyakov loop at a given point for SU(2).
su2-t-z.c to provide an SU(2) action density profile as a function of t and z.
su2-r-z.c to provide an SU(2) action density profile as a function of r and z.
su2-lat.c to provide SU(2) profiles at fixed y (and t) for the action density or Polyakov loop, for arbitrary locations of the two constituents, in a form suitable for comparison with lattice data.
su3.c to compute the action density at a given point for SU(3).
su3-x-y.c to provide an SU(3) action density profile in the plane of the three constituents.

To run the C-programmes on a UNIX platform use:

makefile to compile the above mentioned programmes
(make su2 su3 su2-t-z su2-r-z su3-x-y su2-lat).
The programmes require input to specify the mass parameters of the constituent monopoles (nu1=2omega, nu2=1-nu1 for SU(2) and nu1, nu2, nu3=1-nu1-nu2 for SU(3)) and their positions, labelled below by (x1,y1,z1), (x2,y2,z2) (and for SU(3) (x3,y3,z3)). Where not to be entered, the time-period beta is put to 1. For SU(2), unless noted differently, we place the constituents along the z-axis at z1=nu2*rho²*pi and z2=-nu1*rho²*pi, entered by specifying omega and rho.

From the command line each of these programmes can be run as
(if not correctly entered the text below will be echoed)
- usage: su2 omega rho x y z t nt
  (nt=steps for computing the Polyakov loop)
  Samples.
- usage: su2-t-z omega rho x y z s nz nt nn
  (Profile[0]={t[mt]=-0.5+mt*nn/nt, z[mz]=z+mz*s} with mt=0,....,nt-1 and mz=0,...,nz-1 and s the z-step size)
  Samples discussed below.
- usage: su2-r-z omega rho r z t s nr nz
  (Profile[0]={x[mr]=r+mr*s, z[mz]=z+mz*s} with mr=0,...,nr-1 and mz=0,...,nz-1 and s the x- and z-step size)
  Sample corresponding to figure 2 (middle) shown in Nucl. Phys. B(Proc.Suppl.) 73 (1999) 554-556.
- usage: su2-lat omega x1 y1 z1 x2 y2 z2 -t0 ns nt n b
  (ns³ X nt lattice points; For b=0: action density including n "mirrors", otherwise: trace of the Polyakov loop using b steps. Listed at y=0, t=0 (can be adjusted by shifting y1, y2 and t0)).
  Samples discussed below.
- usage: su3 nu1 nu2 x1 x2 y1 y2 x y z t
  (z1=z2=z3=0, x3=-x1-x2, y3=-y1-y2)
  Samples.
- usage: su3-x-y nu1 nu2 x1 y1 x2 y2 x3 y3 x y z t s nx ny beta
  (time-period=beta; z1=z2=z3=0, Profile[0]={x[mr]=x+mx*s, y[my]=y+my*s} with mx=0,...,nx-1 and my=0,...,ny-1 and s the x- and y-step size)
  Sample corresponding to figure 1 (top) shown in the Nucl. Phys. B(Proc.Suppl.) 73 (1999) 554-556.
See also the source codes for explanations.

All these programmes can also be run through UNIX shell programmes. One can use figs.run as samples for su2-lat and su2-t-z. Output is written to files ppp.*, collected here. The shell programme figs.run generates all the analytic data for the figures in the paper JHEP 06 (1999) 001. Figure numbers refer to that paper.

The analytic result for the SU(2) fermion zero-mode density is based on the paper:

Weyl-Dirac zero-mode for calorons, by M. García Pérez, A. González-Arroyo, C. Pena and P. van Baal, Phys. Rev. D60 (1999) 031901 (Rapid Comm.), hep-th/9905016.
See also: Nahm dualities on the torus -- a synthesis, by M. García Pérez, A. González-Arroyo, C. Pena and P. van Baal, Nucl. Phys. B564 (2000) 159-181, hep-th/9905138.

The following module was added to the existing ones:

zm-su2.c which calculates the d'Alembertian of f_x(.5,.5) for SU(2). The zero-mode (anti-periodic boundary conditions in time) density is obtained by dividing this result by -4pi².

This module is called in the SU(2) C-programmes (beware when your file system is case insensitive):

SU2.c to compute gauge fields, action density, zero-mode density and Polyakov loop at a given point.
SU2-t-z.c to provide action and zero-mode density profiles as a function of t and z.
SU2-r-z.c to provide an action and zero-mode density profiles as a function of r and z.
SU2-lat.c to provide profiles at fixed y (and t) for the action density, zero-mode density or Polyakov loop, for arbitrary locations of the two constituents, in a form suitable for comparison with lattice data.

These are to be used exactly as (and make obsolete) the earlier SU(2) programmes, replacing su2 by SU2 everywhere (use the same makefile to compile these programmes: make SU2 SU2-t-z SU2-r-z SU2-lat). The only change is that now also the fermion zero-mode density is given (Profile[0]=action density, Nzmsq(X) or Profile[-1]=anti-periodic zero-mode density, Profile[-2]=periodic zero-mode density).

usage: SU2 omega rho x y z t nt
(nt=steps for computing the Polyakov loop)
Samples.
usage: SU2-t-z omega rho x y z s nz nt nn
(Profile[0/-1/-2]={t[mt]=-0.5+mt*nn/nt, z[mz]=z+mz*s} with mt=0,....,nt-1 and mz=0,...,nz-1 and s the z-step size)
Sample based in part on figure 5 in Nucl. Phys. B(Proc.Suppl.) 73 (1999) 554-556.
usage: SU2-r-z omega rho r z t s nr nz
(Profile[0/-1/-2]={x[mr]=r+mr*s, z[mz]=z+mz*s} with mr=0,...,nr-1 and mz=0,...,nz-1 and s the x- and z-step size)
Sample corresponding to figure 1 shown in Phys. Rev. D60 (1999) 031901 (Rapid Comm.).
usage: SU2-lat omega x1 y1 z1 x2 y2 z2 -t0 ns nt n b
(ns³ X nt lattice points; For b=0: action density including n "mirrors", b>0: trace of the Polyakov loop using b steps, b=-1 the anti-periodic zero-mode density, b=-2 the periodic zero-mode density (b=-3 adds both zero-mode densities). Listed at y=0, t=0 (can be adjusted by shifting y1, y2 and t0)).
Sample corresponding to figure 3 in Nucl. Phys. B564 (2000) 159-181.

For convenience one can download all C-programmes through the uufiles c.uu.

Disclaimer: Users themselves are responsible for the validity of the results obtained with these programmes.

Results for SU(N>2) zero-modes (see a movie with changing boundary conditions for SU(3)) can be found in :

Exact fermion zero-mode for the new calorons, by M.N. Chernodub, T.C. Kraan and P. van Baal, Nucl.Phys. B(Proc.Suppl.) 83-84 (2000) 556-558, hep-lat/9907001.
Instantons versus Monopoles, by P. van Baal, in: Lattice fermions and structure of the vacuum, eds. V. Mitrjushkin and G. Schierholz (Kluwer, Dordrecht, 2000), pp. 269-279, hep-th/9912035.
Abelian projected monopoles - to be or not to be, by P. van Baal, Nucl. Phys. B(Proc.Suppl.)106 (2002) 586-588, hep-lat/0108027.

Results for higher topological charge can be found in (see some movies for exact SU(2) solutions with charge 2) :

Multi-Caloron solutions, by F. Bruckmann and P. van Baal, Nucl. Phys. B645 (2002) 105-133, hep-th/0209010.
Constituent monopoles through the eyes of fermion zero-modes, by F. Bruckmann, D. Nógrádi and P. van Baal, Nucl. Phys. B666 (2003) 195-227, hep-th/0305063.
Higher charge calorons with non-trivial holonomy, by F. Bruckmann, D. Nógrádi and P. van Baal, Nucl. Phys. B698 (2004) 233-254, hep-th/0404210.

Pierre van Baal, last updated June 2004. Page first created March 1999.