††institutetext: Department of Physics, University of Toronto, Toronto, ON M5S
1A7, Canada

# Numerical fractional instantons in $SU(2)$: center vortices, monopoles, and
a sharp transition between them

F. David Wandler<EMAIL_ADDRESS>
###### Abstract

We use a numerical cooling algorithm to study fractional instantons in $SU(2)$
pure Yang-Mills on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$,
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, and
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$. We confirm that the
fractional instantons are center vortices on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ and monopoles on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, and we calculate several properties
relevant to using these solutions for semiclassical calculations. On
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$, we interpolate between
the large ${\mathbb{T}}^{2}_{*}$ limit and the large $S^{1}$ limit to study
how the solutions interpolate between center vortices and monopoles. We find
that they are separated by a sharp transition, with ’t Hooft’s constant field
strength solutions living at the transition point. These results contrast but
do not contradict recent results suggesting continuity between vortices and
monopoles.

##  1 Introduction

In the past two decades, considerable progress has been made in understanding
the confinement of 4D non-abelian gauge theories through center-stabilized
compactifications of Yang-Mills (YM) theory111The literature is now far too
large to include every possible reference. Instead, we focus on the initial
papers related to each model. We also recommend Poppitz:2021cxe  as a review
of center-stabilization and confinement on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$.. These theories involve starting
with 4D Yang-Mills theory with one or more compact (but finite) directions and
the addition of a stabilizing feature to ensure that the center symmetry does
not break. The preservation of center symmetry ensures the theory remains in
the confining regime. Interestingly, these theories allow for semiclassical
confinement mechanisms that rely on fractional instantons, instantons with
fractional topological charge.

For example, supersymmetric YM (SYM), deformed YM (dYM), and adjoint QCD
(QCD(adj)) can all be shown to confine on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ (with periodic fermions) through
mechanisms involving monopole instantons Unsal2008adj ; Unsal2009bions ;
Shifman:2008ja ; Unsal:2008ch . These theories all abelianize in the IR, and
crucially, the monopole-instanton configurations have magnetic charges in the
abelian EFT. The condensation of these charges (or pairs of charges called
bions) causes confinement.

Another example is pure YM on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$.
Here the ∗ denotes the inclusion of an ’t Hooft flux (i.e. twisted boundary
conditions) on the ${\mathbb{T}}^{2}$. This theory was shown to abelianize to
a ${\mathbb{Z}}_{2}$ gauge theory and confine due to center vortices
Tanizaki:2022ngt .

While these theories are all shown to be in the confining phase, the
confinement mechanisms are very different. This raises the natural question:
to what extent are the different confinement mechanisms related and are the
theories continuously connected?

In search of an answer, the author with Erich Poppitz wrote Poppitz2023 . In
this paper, we presented a pair of classical vacua that should allow for a
semiclassical understanding of continuity of dYM in the semi-infinite volume
limit, ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times
S^{1}\rightarrow{\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$. We worked out that
these vacua, which had been considered previously in Perez:2013aa  and
Unsal:2020yeh , had many desirable properties, such as abelianization of the
gauge group and exchange under center symmetry. We then confirmed that they
were stable to quantum fluctuations through calculating the one-loop potential
on the holonomy. However, we did not explore the full semiclassical
calculations leading to confinement on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$.

The main difficulty in extending Poppitz2023  to a full semiclassical analysis
and opening up the entire landscape of
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ theories was a lack of
understanding of the fractional instantons. The present article makes progress
in understanding these fractional instanton solutions through numerics. In
particular, we find lattice approximations of the fractional instanton
solutions by minimizing the twisted $SU(2)$ Wilson action Wilson:1974sk  using
the cooling algorithm from GarciaPerez:1989gt ; GarciaPerez:1992fj ;
GONZALEZARROYO1998273 , which was adapted from the algorithms used in
Berg:1981nw  and Teper:1985rb .

We first confirm the existence of center vortices on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$, strengthening the initial
evidence from GONZALEZARROYO1998273  and supporting the confinement mechanism
proposed in Tanizaki:2022ngt . We then confirm the existence of monopole
instantons on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, agreeing with
analytical work in GPY1981 ; Unsal2008adj ; Unsal2009bions ; Shifman:2008ja ;
Unsal:2008ch .

Having confirmed the algorithm through its results in the two known regimes,
we consider the fractional instantons on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ as it interpolates
between the large torus (i.e. ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$) and
large circle (i.e. ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$) limits. This
is done by running the cooling algorithm on lattices with different sizes. We
use $N_{i}$ to denote the number of lattice sites in the $i$-direction. The
$N_{i}$ are directly analogous to the lengths of the dimensions of the torus
in the continuum limit.

While holding $N_{0}$ fixed at a large value and varying the remaining
$N_{i}$, we find that both monopoles and center vortices can exist, but that
there is a sharp transition between lattices that permit center vortices and
lattices that permit monopoles. Namely, given the dimensionless parameter

$\Delta=\frac{N_{1}N_{2}-N_{0}N_{3}}{\sqrt{N_{0}N_{1}N_{2}N_{3}}}\,,$ (1)

the type of fractional instanton present is determined by the sign of
$\Delta$. Center vortices are found whenever $\Delta$ is negative and
monopoles whenever $\Delta$ is positive. This suggests a phase diagram-like
structure for the parameter space, which is shown in Figure 19.

These solutions can be directly applied to the confinement problem with
varying degrees of success. The center vortices are known to cause confinement
on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ through direct calculation of
the area law of Wilson loops as presented in Tanizaki:2022ngt . Here we find
that, on ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ with small
${\mathbb{T}}^{2}_{*}$, they also cause confinement through instanton effects
that preserve the center symmetry. This leads to a nice continuous
understanding of the confinement mechanism through an entire landscape of
theories along the ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times
S^{1}\rightarrow{\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ limit.

In contrast, it is known that the monopole instanton configurations are only
useful for explaining confinement on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$
in the presence of center stabilization (i.e. either adjoint periodic fermions
or a non-local deformation potential). One limitation of the work presented
here is that the basic cooling algorithm we employ is only suitable for
studying pure YM, hence our findings do not directly provide a confinement
mechanism. However, they are useful for understanding the properties of
monopoles, in general. While there would be quantitative differences (such as
differing monopole diameters) between our solutions and the solutions in dYM,
we expect that the same qualitative features between solutions, such as the
shape of the surrounding magnetic field and the total magnetic flux.

Another interesting limitation of the monopole solutions is that the
$\Delta>0$ region of parameter space, which permits monopoles, becomes
vanishingly small in the zero temperature limit. This suggests that monopoles
do not appear in pure YM on ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times
S^{1}$ at zero temperature, though they do appear at finite temperatures with
large ${\mathbb{T}}^{2}_{*}$.

We also look at the "critical" case, $\Delta=0$. Here the lattice is self-dual
and permits the constant field strength solutions of ’t Hooft tHooft:1981nnx .
We find that our solutions on the lattice agree with these analytical
solutions.

While preparing this article, the author became aware of hayashi2024unifying
and guvendik2024metamorphosis . In contrast to the sharp transition apparent
in our numerical results, these works provide evidence of a continuity between
center vortices and monopole instantons. These results are not contradictory
to ours since they consider different geometrical setup and/or theories
differing from pure YM. The cooling algorithm presented here would be useful
in investigating the claims in hayashi2024unifying . A similar approach, but
with an upgraded algorithm capable of handling the deformation potential,
would be useful for interrogating the claims in guvendik2024metamorphosis  and
for investigating confinement mechanisms on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ with large
${\mathbb{T}}^{2}_{*}$. We discuss these results and the possibility of
numerical investigations in greater detail in Sections 6 and 7.

The data and code used in this study will be available on request starting
September 2024.

###  1.1 Layout

This paper is organized as follows: Section 2 outlines basics of the classical
twisted Wilson action, the cooling algorithm we use to minimize it, and the
notation that we use throughout the rest of the article. Section 3 discusses
features that are true of all lattice sizes, such as the vacuum states without
fractional instantons, the saturation of the BPS bound for all our final
configurations, and the moduli space of the fraction instantons. Section 4
discusses the properties of fractional instantons in the
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times{\mathbb{R}}$ limit and confirms
that they are center vortices. Similarly, Section 5 discusses the properties
of fractional instantons in the ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ limit
and confirms that they are monopoles. Section 6 discusses the fractional
instantons on ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ that
interpolate between the
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times{\mathbb{R}}$ and
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ limits. Its main result is the
"phase" diagram in Figure 19, which illustrates the sharp transition we find
between the center vortex solutions and the monopole solutions. The rest of
Section 6 is devoted to demonstrating the important features of solutions in
each regime and at the transition point. Finally, Section 7 wraps up the
discussion and presents some interesting directions for future study.

##  2 Lattice basics, notation, and conventions

###  2.1 Notation

In this work, we focus on 4-dimensional square lattices with periodic boundary
conditions. We label the directions 0, 1, 2, and 3, and use $N_{i}$ to denote
the number of sites in each direction. Throughout this article, we investigate
a variety of lattice sizes, which we present in the format: $N_{0}\times
N_{1}\times N_{2}\times N_{3}$.

While the Euclidean lattice action treats all directions the same, we find it
useful to consider the $0$ direction as the Wick rotation of the time
direction. It is purposely always taken to be the largest dimension, since we
are interested in approximating the zero temperature limit:
${\mathbb{R}}\times T^{2}_{*}\times S^{1}$.

Our lattice sites are labelled by
$x\in{\mathbb{Z}}_{N_{0}}\times{\mathbb{Z}}_{N_{1}}\times{\mathbb{Z}}_{N_{2}}\times{\mathbb{Z}}_{N_{3}}$,
hence all arithmetic involving lattice sites is done modulo the number of
sites in each direction. For the purposes of this arithmetic, we introduce
$e_{\mu}\in{\mathbb{Z}}_{N_{0}}\times{\mathbb{Z}}_{N_{1}}\times{\mathbb{Z}}_{N_{2}}\times{\mathbb{Z}}_{N_{3}}$
as the unit vector in the $\mu^{\text{th}}$ direction. Individual components
of $x$ are denoted by $x^{\mu}$ for $\mu=0,1,2,3$.

The link variable on the link extending in the positive $\mu$ direction from
site $x$ is denoted by $U_{\mu}(x)\in SU(2)$. Plaquette variables are defined
as

$\square_{\mu\nu}(x)=U_{\mu}(x)U_{\nu}(x+e_{\mu})U^{\dagger}_{\mu}(x+e_{\nu})U^{\dagger}_{\nu}(x)\,.$
(2)

The action we consider is a twisted version of the Wilson action Wilson:1974sk
:

$S=\frac{1}{2g^{2}}\sum_{x}\sum_{0\leq\mu<\nu\leq
3}\Tr\left(1-(-1)^{t_{\mu\nu}(x)}\square_{\mu\nu}(x)\right)\,.$ (3)

For the purposes of minimizing this classical action, the value of $g$ is
irrelevant, so we set it to 1 for convenience.

Here $t_{\mu\nu}(x)$ is a binary value that indicates the presence of a twist
in the $\mu\nu$ plaquette at $x$. If $t_{\mu\nu}$ is 0 everywhere, then there
is no twist in the system and the action is the usual Wilson action. Since we
are interested in understanding the system with an ’t Hooft flux through the
$12$-plane, we introduce a single twist in every $12$-plane in the system. In
other words, $t_{\mu\nu}(x)=1$ if $\mu=1$, $\nu=2$ and $x^{1}=x^{2}=0$. This
is equivalent to introducing ’t Hooft’s twisted boundary conditions with
$n_{12}=1$.

When inspecting configurations that approximate continuum configurations with
integer topological charge, such as the vacuum states for
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$, we do not introduce any
additional twists (i.e. all other $t_{\mu\nu}$ are set to 0). However, the
fractional instanton configurations that we wish to investigate have
fractional topological charge. Topological charge in the presence of twists
take values in the set
$\varepsilon^{\mu\nu\rho\lambda}\frac{n_{\mu\nu}n_{\rho\lambda}}{2}+{\mathbb{Z}}$.
Thus, to ensure we approximate fractional topological charges, we must also
include a twist in the $03$-plane. To accomplish this we keep the twists in
plaquettes in the $12$-plane the same as above and set $t_{\mu\nu}(x)=1$ if
$\mu=0$, $\nu=3$, and $x^{0}=x^{3}=0$). All other $t_{\mu\nu}(x)$ are set to
0. This is equivalent to boundary conditions with $n_{12}=n_{03}=1$ and all
other components vanishing, leading to topological charges in
$\frac{1}{2}+{\mathbb{Z}}$.

Given a topological charge, $Q_{top}$, in the continuum, the Bogomol’nyi-
Prasad-Sommerfield (BPS) bound Bogomolny:1975de ; Prasad:1975kr  is given by

$S_{cont}\geq\frac{8\pi^{2}}{g^{2}}\left|Q_{top}\right|\,.$ (4)

We find that our lattice approximations of minimal continuum configurations
with given topological charge222Here I am being very careful about the fact
that the lattice configurations do not have nice cleanly defined topological
charges like the continuum configurations they are approximating. In the rest
of this article, I will be more relaxed with this language, and leave this
footnote to avoid confusion. also approximately saturate the BPS bound. This
statement will be made more precise in Section 3.2, but it is a good test that
our approach is making meaningful approximations of the continuum theory.

For analyzing our classical configurations we must use gauge invariant
quantities333It is possible to gauge fix everything and compare configurations
directly, but this is computationally complicated and does not allow for an
easy comparison between differently sized lattices or with the continuum
limit.. On the lattice, all gauge invariant quantities are traces of Wilson
loops. Certain Wilson loops will be very important in the following sections,
so we introduce special notation for them. The first examples are the
plaquette variables $\square_{\mu\nu}(x)$ defined above. These are useful
because $\square_{\mu\nu}(x)\sim e^{ia^{2}F_{\mu\nu}(x)}$, where $a$ is the
lattice spacing and $F^{\mu\nu}$ is the standard YM field strength tensor in
the continuum limit. Hence the plaquette variables give us access to local
gauge invariant information. However, local information is not sufficient to
tell us everything about the system; in particularly, it is useful to consider
the global information contained in loops that wind around our torus. These
holonomies are defined as

$W_{\mu}(x)=U_{\mu}(x)U_{\mu}(x+e_{\mu})U_{\mu}(x+2e_{\mu})\cdots
U_{\mu}(x+(N_{\mu}-1)e_{\mu})\,.$ (5)

Recall that addition of the lattice site vectors is always done modulo the
lattice size in each direction.

###  2.2 The cooling algorithm

The cooling algorithm employed in this study is based on the algorithm used by
Gonzalez-Arroyo, Garcia Perez, Soderberg, and Montero in GarciaPerez:1989gt ;
GarciaPerez:1992fj ; GONZALEZARROYO1998273 . The algorithm works by
considering a single link variable at a time and giving it the value that
minimizes the action while keeping all other links fixed. This procedure is
then repeated for every link in the lattice. Minimizing each link variable
once is known as a sweep. Sweeps are repeated until a suitable convergence
criteria is met.

In our case, the convergence criteria used is that the action changes by less
than 1 ppm for 100 sweeps. We also require that the final value of the action
is lower than 40. This was a practical measure to ensure the algorithm would
not finish prematurely, since there is a tendency for the algorithm to sit in
local minima for many, many sweeps before finding the true minima. This is
demonstrated in the step-like pattern in Fig. 1 which plots the action as a
function of the number of sweeps performed for a particular run. The height of
the plateaus seems in line with the local minima being integer multiples of
the instanton action above the BPS bound. This suggests that, as the algorithm
minimizes, it "gets stuck" at states that are the desired ground state plus
some configuration of instantons/antinstantons. In all cases, however, the
action eventually settled to near the BPS bound.

Figure 1: The action of the configuration as a function of the number of
sweeps of the cooling algorithm. This particular example is for the lattice
configuration $48\times 48\times 48\times 12$.

The key feature of the algorithm that allows it to leave the local minima
seems to be the fact that the link variables are replaced by their global
minima (given the local neighbourhood). This can be done efficiently because
of two features unique to $SU(2)$. To see how this works consider a specific
link variable, $U\in SU(2)$. This link variable enters the action through the
trace of six plaquettes. Hence the contribution of $U$ to the action that we
want to minimize can be written as

$-\sum_{j=1}^{3}\Tr\left(UV_{j}^{\dagger}\right)-\sum_{j=4}^{6}\Tr\left(U^{{\dagger}}V_{j}\right)\,,$
(6)

where we have ignored an overall positive factor because it does not affect
the where the minimum occurs. The $V_{j}$ are the combined products of the
rest of the links making up the plaquettes. In the case of $SU(2)$, the traces
are always a real number, hence for all $A\in SU(N)$,

$\Tr\left(A^{\dagger}\right)=\Tr\left(A\right)^{*}=\Tr\left(A\right)\,$ (7)

Thus we can rearrange 6 to

$-\sum_{j=1}^{6}\Tr\left(UV_{j}^{\dagger}\right)=-\Tr\left(U\sum_{j=1}^{6}V_{j}^{\dagger}\right)\,.$
(8)

Now we use the property that any sum of $SU(2)$ matrices can be written as an
$SU(2)$ matrix times a non-negative real number, to write

$\sum_{j=1}^{6}V_{j}^{\dagger}=rW^{\dagger}\,,$ (9)

where $W\in SU(2)$ and $0<r\in{\mathbb{R}}$. It is now clear that we need to
minimize

$-r\Tr\left(UW^{\dagger}\right)\,.$ (10)

The minimum is $-r$ and is reached precisely when $U=W$. $W$ can be calculated
from the surrounding links using standard linear algebra, so the algorithm
replaces $U$ with $W$ then repeats this calculation for every link in the
lattice to complete a sweep.

###  2.3 Data generated and analysis techniques

With the intention of doing statistics on the space of configurations, 300
configurations were generated for three lattice shapes in both the $n_{03}=0$
and $n_{03}=1$ cases. The chosen lattice configurations were $24\times 6\times
6\times 24$, $24\times 6\times 6\times 6$, and $24\times 24\times 24\times 6$,
which correspond to the interesting cases
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$,
${\mathbb{R}}^{1}\times{\mathbb{T}}^{3}$, and
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, respectively. These are used to draw
a statistically meaningful bound on the size of parameter space that could
cool to points outside of the observed moduli spaces. In addition, a wide
variety of configurations with intermediate shapes and increased sizes were
also considered in order to deduce the effects of the finite lattice spacing
and to get a more fine grained understanding of the fractional instanton
solutions. Due to computational constraints444These computational constraints
were due to the fact that these calculations were performed on the author’s
personal computer, and could be easily overcome in follow-up studies with the
use of dedicated scientific computing resources., these solutions were not
generated in large numbers for each configuration. Instead, we treat them as
examples demonstrating the properties of the individual instantons.

To directly compare the solutions, it is useful to translate the fractional
instantons generated by different runs to align them to the same position on
the torus. Translations by $n$ spaces in the $i^{\text{th}}$ direction are
done in two steps. The first is to directly move all the values of the link
variables by $n$ lattice sites in the $i$-direction. This translates the
configuration, but also translates the location of the nonvanishing twists. To
accurately compare the configurations, the second step is to translate the
twists back to their original position. For a non-vanishing twist in the
$ij$-plane, it is translated $n$ plaquettes back along the $i$-direction by
multiplying all the link variables in the $j$-direction between the starting
and ending plaquettes by $-1$. See Figure 2 for a cartoon depicting this.

Figure 2: Cartoon example of the procedure for translating a twist in the
$03$-plane along the $x^{0}$ direction. The twist is marked by the blue "X".
From its initial location, we want to translate it down two plaquettes. The
procedure is to multiply the two red links (the links crossed by the twist
during the translation) by a $-1$.

Since there are twists in the $12$-plane, any translation in the $1$-direction
will result in some $2$-direction links being multiplied by $-1$, and vice
versa. The same will occur within the $03$-plane, when the $03$-twists are
included. One consequence of this translation procedure is that if there is a
twist in the $ij$-plane and the configuration is translated by $N_{i}$ in the
$i$-direction, then the first step will not change the configuration, but the
procedure for translating the twist will result in all the $j$-direction link
variables with a single value of $x^{i}$ flipping sign. This is exactly a
center symmetry transformation for loops winding around the $j$-direction.
Hence, on the twisted torus, center symmetry transformations are equivalent to
translations. This equivalence was previously reported in GarciaPerez:1992fj .

This is a very useful fact in considering the moduli space of solutions. For
example, it is known on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ that there
are two distinct types of monopole-instantons, the BPS and KK monopoles. These
instantons are however related by a center symmetry transformation of loops
winding the $3$-direction, so in our setup they should be connected by
translations by $N_{0}$ in the $0$-direction. Therefore, the distinct
configurations in the infinite volume continuum become part of a connected
moduli space on our periodic lattice.

##  3 General features of the solutions

This section is devoted to features of the $SU(2)$ solutions with $n_{12}=1$
that are not found to differ between different regions of parameter space,
namely the properties of the vacuum when $n_{03}=0$, the saturation of the BPS
bound for fractional instanton solutions with $n_{03}=1$, and the moduli space
of the instanton solutions.

###  3.1 True vacua

Upon removing the 03-twist from any torus (i.e. setting $n_{03}=0$, but
keeping $n_{12}=1$), the BPS bound from equation 4 is $S\geq 0$. The algorithm
finds four discrete classical vacua that saturate this bound to within
numerical error. To build more numerical support for this claim, 300 solutions
were generated at various torus sizes. Figure 3 shows how torus shape affect
the final action value. Notice in particular how the action stays below the
precision target of $1.0\times 10^{-6}$ (i.e. 1 ppm) for all configurations.
This suggests that the algorithm is converging towards a true minimal action
configuration that saturates the BPS bound.

Figure 3: Histogram of the final actions for minimal configurations with
$n_{03}=0$. The left plot includes all three datasets. The right plot zooms in
on just the $24\times 6\times 6\times 24$ and $24\times 6\times 6\times 6$
datasets, as they overlap and are more difficult to see in the left plot.

The fact that there are four such vacua can be seen from the holonomies in
these two solutions. The holonomies in each direction are constant across the
entire lattice. The solutions are (up to numerical noise) given by

$\begin{split}\frac{1}{2}\Tr W_{1}=&\,0\\\ \frac{1}{2}\Tr W_{2}=&\,0\\\
\frac{1}{2}\Tr W_{3}=&\pm 1\\\ \frac{1}{2}\Tr W_{0}=&\pm 1\end{split}$ (11)

Here the value of $\Tr W_{3}$ is independent of the value of $\Tr W_{0}$,
leading to the four distinct states. This counting of vacua is consistent with
Gonzalez-Arroyo:1997ugn  where it is shown that $SU(N)$ has $N^{2}$ vacua. Our
findings are also consistent with the center symmetry in the 0- and
3-directions being spontaneously broken, while the center symmetry in the 1-
and 2- directions is unbroken. To see that this holds generally, Figures 4, 5,
and 6 give histograms of the values of $\Tr W_{i}$ for $i=1,2,3,4$ which
include values from every lattice site for all 300 configurations at each
size. The four vacua are determined by the signs of $W_{0}$ and $W_{3}$, as
detailed in Table 1. The narrow peaks around the exact values mentioned above
suggest that there are no alternative vacua and that the vacua are consistent
across various lattice shapes.

Vacuum | $W_{0}$ | $W_{3}$
---|---|---
1 | + | +
2 | + | -
3 | - | +
4 | - | -
Table 1: Signs of $W_{0}$ and $W_{3}$ associated with the 4 distinct vacua in
Figures 4, 5, and 6. Figure 4: Histograms of the traces of the 4 holonomies
from every lattice site across the $24\times 24\times 24\times 6$ dataset with
$n_{03}=0$. Figure 5: Histograms of the traces of the 4 holonomies from every
lattice site across the $24\times 6\times 6\times 24$ dataset with $n_{03}=0$.
Figure 6: Histograms of the traces of the 4 holonomies from every lattice site
across the $24\times 6\times 6\times 6$ dataset with $n_{03}=0$.

Another interesting gauge invariant in these solutions is $\Tr W_{1}W_{2}$,
where the base points of the $W_{1}$ and $W_{2}$ coincide. Recall that, as
$SU(2)$ matrices, each $W_{i}$ can be written as
$\cos(\theta_{i})+i\sin(\theta_{i})\hat{n}\cdot\vec{\sigma}$. In the case
where $\Tr W_{i}=0$, we know that $\cos(\theta_{i})=0$ and
$\sin(\theta_{i})=1$, hence $W_{i}=\hat{n}_{i}\cdot\vec{\sigma}$. Therefore,
for these vacua solutions we have $\Tr
W_{1}W_{2}\propto\hat{n}_{1}\cdot\hat{n}_{2}$. Calculating $\Tr
W_{1}(x)W_{2}(x)$ for every lattice, $x$, in every configuration from the
large datasets, we find the data in Table 2. It is clear that our numerical
solutions are consistent with $\Tr W_{1}W_{2}=0$.

Dataset | Mean $\Tr W_{1}W_{2}$ | Minimal $\Tr W_{1}W_{2}$ | Maximal $\Tr W_{1}W_{2}$
---|---|---|---
$24\times 6\times 6\times 24$ | $0.0013\times 10^{-12}$ | $-1.4163\times 10^{-12}$ | $1.3476\times 10^{-12}$
$24\times 6\times 6\times 6$ | $-0.0055\times 10^{-13}$ | $-6.9555\times 10^{-13}$ | $7.1887\times 10^{-13}$
$24\times 24\times 24\times 6$ | $0.0022\times 10^{-6}$ | $-1.9887\times 10^{-6}$ | $2.3290\times 10^{-6}$
Table 2: Values of $\Tr W_{1}W_{2}$ calculated across the entirety of the
large datasets.

In the case where $L_{i}\ll\Lambda$ and $\frac{1}{2}\Tr W_{i}\neq\pm 1$, we
have that the vev of $W_{i}$ will higgs the gauge symmetry down to a $U(1)$
subgroup which aligns with $\hat{n}_{i}$ in color space. The strongest
breaking occurs for $\frac{1}{2}\Tr W_{i}=0$. Hence, in cases where both
$L_{1},L_{2}\ll\Lambda$, these vacua are maximally break $SU(2)$ into two
orthogonal $U(1)$ subgroups. This suggests that the gauge group is actually
broken to the ${\mathbb{Z}}_{2}$, since the ${\mathbb{Z}}_{2}$ center of
$SU(2)$ is the only possible intersection between orthogonal $U(1)$ subgroups.
Hence, the low energy theory around these vacua is a ${\mathbb{Z}}_{2}$ gauge
theory without propagating degrees of freedom.

All of these results for the vacua are in agreement with the known vacuum
solutions in the continuum. Consider a continuum Euclidean torus with constant
boundary conditions ($\Omega_{0}=\Omega_{3}=1$, $\Omega_{1}=\hat{\sigma}_{y}$,
and $\Omega_{2}=\hat{\sigma}_{z}$), then the solutions $A=0$ and
$A=-iT_{3}dT_{3}^{-1}$ are classical solutions with vanishing action555Here
$T_{3}$ is the large gauge transformation implementing a center symmetry
transformation on loops winding in the 3-direction.. One can directly check
that the gauge invariants all match with the numerical solutions described
above. The agreement between the numerical lattice solutions and the
analytical continuum solutions suggests that the numerical methods employed in
this paper are a reliable approach to extracting information about classical
Yang-Mills solutions.

###  3.2 Saturation of the BPS bound

As seen above, the vacuum states saturate the BPS bound for the case of
$Q_{top}=0$, where the BPS bound is $S_{Q_{top}=0}\geq 0$. A similar analysis
shows that the BPS bound is saturated for the fractional instanton
configurations with $Q_{top}=1/2$, where the BPS bound is $S_{Q_{top}=1/2}\geq
4\pi^{2}/g^{2}$. As mentioned above, we have set $g=1$ for finding classical
solutions, so we expect the action of our fractional instantons to be
$S=4\pi^{2}\approx 39.48$.

Figure 7 compares the actions of 300 configurations with $n_{03}=1$ at
different lattice shapes. It is clear that the values are close to the
expected value, but do not hit it exactly. Following GONZALEZARROYO1998273 ,
we can demonstrate that this discrepancy is a lattice effect by finding the
effect of varying the lattice sizes on the action. We focus on the three cases
$N_{L}\times N_{L}\times N_{L}\times
N_{S}\sim{\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, $N_{L}\times N_{S}\times
N_{S}\times N_{S}\sim{\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$, and
$N_{L}\times N_{S}\times N_{S}\times
N_{L}\sim{\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$, where $N_{L}$ is the
"large" value and $N_{S}$ is the "small" value. The results are collected in
Table 3. Given the 1 ppm convergence criteria, the values are expected to have
numerical uncertainty on the order of $4\times 10^{-5}$, so we present the
values with 5 decimal places. When $N_{L}=24$, solutions are relatively quick
to generate, so we consider the average over 300 (for $N_{S}=6$) or 10 (for
$N_{S}=9,12$) solutions. Comparing the minimum and maximum final actions to
the mean, we find a that the largest deviation is less than $8\times 10^{-5}$.
For larger values of $N_{L}$, we consider only a single solution.

Figure 7: Histograms showing the final actions of all configurations with in the three datasets with $n_{03}=1$. | $N_{L}\times N_{S}\times N_{S}\times N_{L}$
---|---
$N_{S}$ $N_{L}$ | 24 | 36 | 48
6 | $38.79690\begin{subarray}{c}+4\times 10^{-6}\\\ -3\times 10^{-6}\end{subarray}$ | 38.79688 | 38.79688
9 | $39.18044\begin{subarray}{c}+5\times 10^{-8}\\\ -7\times 10^{-8}\end{subarray}$ | 39.17992 | 39.17991
12 | $39.31421\pm 2\times 10^{-6}$ | 39.31145 | 39.31136
| $N_{L}\times N_{S}\times N_{S}\times N_{S}$
$N_{S}$ $N_{L}$ | 24 | 36 | 48
6 | $38.93490\pm 9\times 10^{-7}$ | 38.93486 | 38.93486
9 | $39.24188\begin{subarray}{c}+3\times 10^{-7}\\\ -5\times 10^{-7}\end{subarray}$ | 39.24081 | 39.24079
12 | $39.35070\begin{subarray}{c}+4\times 10^{-6}\\\ -6\times 10^{-6}\end{subarray}$ | 39.34574 | 39.34554
| $N_{L}\times N_{L}\times N_{L}\times N_{S}$
$N_{S}$ $N_{L}$ | 24 | 36 | 48
6 | $39.21615\pm 0.00007$ | 39.23172 | 39.23967
9 | $39.35266\begin{subarray}{c}+0.00004\\\ -0.00006\end{subarray}$ | 39.36367 | 39.36934
12 | $39.40310\begin{subarray}{c}+0.00005\\\ -0.00004\end{subarray}$ | 39.41032 | 39.41520
Table 3: Values of the final actions for different sizes and shapes of
lattices. Values with errors are averaged across a dataset of multiple runs at
the same lattice size. Errors are determined by the maximum and minimum values
across the dataset.

By changing the values of $N_{L}$ and $N_{S}$ separately, it is clear that the
dependence on $N_{S}$ is far stronger for all cases. The observed weak
dependence on $N_{L}$ is desirable, since we want our large dimension to be a
good approximation of infinite length limit. We want to quantify the lattice
effects on the minimal action. This would normally be done through the lattice
spacing, $a$, but in the classical case there are no physical scales that can
be used to meaningfully set the value of $a$. Any lattice spacing can be set
for any size of lattice and the solutions will not change, so just taking the
$a\rightarrow\infty$ limit is not useful. Instead, the limit we wish to
understand is the limit of infinite lattice sites in all directions where the
length of the larger dimensions tends towards infinity while the length of the
smaller directions remains finite. In other words, we want to consider
$N_{L},N_{S}\rightarrow\infty$ such that $aN_{L}\rightarrow\infty$ and
$aN_{S}\rightarrow L$ for some finite $L$. Hence, the lattice spacing should
scale like $L/N_{S}$. Since we are free to normalize distance scales to $L=1$,
$1/N_{S}$ as a good dimensionless substitute for the lattice spacing.

Since the action depends on the square of the lattice spacing, the leading
order dependence of the action on $N_{S}$ is

$S(N_{S})=S_{0}+\frac{c_{1}}{N^{2}_{s}}\,.$ (12)

Completing a least square error fit on the constants $S_{0}$ and $c_{1}$ for
each value of $N_{L}$, we find that the linear approximations given by the
dashed lines in Figures 8, 9, and 10. The infinite $N_{S}$ limits give a much
better agreement with the expected BPS bound.

Figure 8: Fits of the final actions for lattices with the shape, $N_{L}\times
N_{L}\times N_{L}\times N_{S}$. The dotted lines are linear fits with respect
to $1/N_{S}^{2}$ using the data with fixed $N_{L}$. The solid black line is
the fit of the $N_{L}\rightarrow\infty$ limit for each $N_{S}$, the
$N_{L}\rightarrow\infty$ is calculated by a linear fit with respect to
$1/N_{L}$. The solid red line is the value of the BPS limit in the continuum,
namely $4\pi^{2}$. Figure 9: Fits of the final actions for lattices with the
shape, $N_{L}\times N_{S}\times N_{S}\times N_{L}$. The dotted lines are
linear fits with respect to $1/N_{S}^{2}$ using the data with fixed $N_{L}$.
The solid black line is the fit of the $N_{L}\rightarrow\infty$ limit for each
$N_{S}$, the $N_{L}\rightarrow\infty$ is calculated by a linear fit with
respect to $1/N_{L}$. The solid red line is the value of the BPS limit in the
continuum, namely $4\pi^{2}$. Figure 10: Fits of the final actions for
lattices with the shape, $N_{L}\times N_{S}\times N_{S}\times N_{S}$. The
dotted lines are linear fits with respect to $1/N_{S}^{2}$ using the data with
fixed $N_{L}$. The solid black line is the fit of the $N_{L}\rightarrow\infty$
limit for each $N_{S}$, the $N_{L}\rightarrow\infty$ is calculated by a linear
fit with respect to $1/N_{L}$. The solid red line is the value of the BPS
limit in the continuum, namely $4\pi^{2}$.

However, taking the infinite $N_{S}$ limit at fixed $N_{L}$ is questionable.
Moreover, we see an slight improvement with increased $N_{L}$ that would be
nice to include. To properly take the dependence on $N_{L}$ into account, we
start by first taking the $N_{L}\rightarrow\infty$ limit. Assuming a rational
dependence on $N_{L}$, we fit the leading order dependence,

$S(N_{S},N_{L})=S_{0}(N_{S})+\frac{c_{2}}{N_{L}}\,,$ (13)

for each value of $N_{S}$. We can then fit the resulting $S_{0}(N_{S})$
according to 12. This fit gives the solid black lines in Figures 8, 9, and 10.
Here we see an even closer agreement between the continuum limiting value,
$S_{0}$, and the expected saturation of the BPS bound. This evidence suggests
that the discrepancy between the expected and calculated actions is a lattice
effect, and, therefore, that our solutions are approximating true BPS states.

###  3.3 Moduli space and antiinstantons

One advantage of being able to generate large datasets of configurations is
that it allows for statistically reliable statements regarding the moduli
space of configurations. We start by making an educated guess about moduli
space, then we generate 300 configurations from uniformly distributed random
initial configurations. If all 300 of the configurations end up within the
expected moduli space of solutions, then the "Rule of Three" ruleof3  dictates
that, with 95% confidence, the the probability for the algorithm to find a
solution outside of the expected moduli space is less than 1%. Since we are
uniformly (with respect to the Haar measure) randomly sampling the space of
initial configuration, this is equivalent to the statement that less than 1%
of the configuration space of the lattice will minimize out of the expected
moduli space, with 95% confidence. This gives us a statistically rigorous way
of stating how confident we are that other fractional instanton solutions do
not exist666The space of initial configurations is very high dimensional, so
1% still represents a large unexplored subset of the parameter space where
alternative solutions might exist. However, if other fractional instanton
solutions exist, they must have the unusual property of being heavily
disfavoured by this algorithm. Without any clear reason for bias in the
cooling algorithm, it seems safe to assume that we are not missing any regions
of moduli space..

To establish the expectation for the moduli space, consider the large Wilson
loops, $W_{i}$. The twist in the $12$-plane means that $W_{1}$ is antiperiodic
in $x^{2}$ and $W_{2}$ is antiperiodic in $x^{1}$. Similarly, the twist in the
$03$-plane means that $W_{0}$ is antiperiodic in $x^{3}$ and $W_{3}$ is
antiperiodic in $x^{0}$. This fact is equivalent to the statement that a shift
by $N_{i}$ in the $i^{\text{th}}$ direction is equivalent to a center symmetry
transformation in the $j^{\text{th}}$ direction, if $n_{ij}\neq 0$. Also, as
we demonstrate in later sections, the $W_{i}$ are not identically zero for any
$i$ or for any lattice shape. Hence, translating the configuration in any
direction will change the gauge invariant observables, but not the action.
Moreover, a translation of $2N_{i}$ (or a multiple thereof) is required in the
$i$-direction to return to the initial configuration. This suggests that the
moduli includes a 4-torus that is twice the size of the system in all
directions. Hence, in the continuum, we expect the moduli space to be

$S^{1}_{2L_{0}}\times S^{1}_{2L_{1}}\times S^{1}_{2L_{2}}\times
S^{1}_{2L_{3}}\,,$ (14)

where the $L_{i}$ is the length of the $i$-direction in spacetime. Index
theorems tell us that in the continuum limit, there should only be 4
continuous directions to the moduli space, suggesting that this torus is all
of moduli space. Of course, there might be other disconnected regions of
moduli, which is why we employ the statistical approach introduced above.

One important consideration is the fact that the algorithm can also produce
antiinstantons configurations. Antiinstantons have the same action as the
instantons but with the opposite sign of topological charge, and they should
have an identical moduli space. As mentioned above, the twists guarantee only
that our configurations have half integer topological charge. Hence,
fractional antiinstanton configurations in $SU(2)$ are not ruled out by the
twist, so we should expect them to show up in the dataset of configurations.
This will appear like a disconnected region of moduli space in our statistical
analysis (see Figure 12). However, since we know that these configurations
will show up, we can still apply our statistical approach to rule out
configurations outside of the fractional instanton and fractional
antiinstanton torii. For brevity, we discuss this system as having a single
moduli space including both the instanton and antiinstanton moduli spaces as
disconnected regions.

We can accurately distinguish the instantons from the antiinstantons by
approximately calculating the topological charge. We do this by calculating
the following gauge invariant quantity:

$Q_{top}=-\frac{1}{32\pi^{2}}\sum_{x}\varepsilon^{\mu\nu\rho\lambda}\Tr\left[\square_{\mu\nu}(x)\square_{\rho\lambda}(x)\right]$
(15)

where summation is implied over the indices. This is simply the naive discrete
version of the continuum definition of topological charge.

More sophisticated methods for determining topological charge from a lattice
configuration are known in the literature777See woit1985topcharge  for one of
the earliest examples and Alexandrou:2017hqw  for a comparison of modern
techniques., but this naive approach is powerful enough to distinguish the
instantons from the antiinstantons. Figure 11 demonstrates that the values are
clearly clustered near the values of $\pm 1/2$.

Figure 11: Histogram of the topological charges (calculated according to 15)
of all the final configurations in the large datasets with $n_{03}=1$. The
values were separated into positive and negative values before generating the
histograms. The black lines indicate the ideal continuum values of $\pm 1/2$.

In implementing our statistical strategy, it would be wildly impractical to
compare all 300 configurations by hand, so instead we develop an automated
approach. The idea is to align all the configurations as best as possible, by
translating each configuration such that the peaks of the action density are
at the same point and that the signs of the $W_{i}$ observables match at their
peaks. This fixes a particular location in each connected component of moduli
space. If the configurations belong to the same connected region of moduli,
all of their gauge invariant quantities should agree (to within reasonable
numerical error) after these translations are applied.

To test this, we use a gauge-invariant metric that quantifies the differences
between configurations. The metric involves summing the squared differences of
a large number of gauge-invariant quantities. This means that the metric can
only approach 0 when all these gauge invariant quantities are similar between
the configurations. The physical configuration space has $9N$ degrees of
freedom888Here $N=N_{0}N_{1}N_{2}N_{3}$, the total number of lattice sites.
There are $4N$ links, each with 3 degrees of freedom associated with an
$SU(2)$ matrix, so the total configuration space has $12N$ degrees of freedom.
However, gauge transformations are associated with $SU(2)$ matrices at each
lattice site, so the space of gauge transformations is $3N$-dimensional.
Hence, the number of physical degrees of freedom is $12N-3N=9N$., so the
metric must compare at least $9N$ gauge-invariants to properly distinguish
different configurations.

To start, consider a metric defined using all possible plaquettes:

$d^{2}_{P}\left(U,U^{\prime}\right)=\frac{1}{N}\sum_{x}\sum_{0\leq\mu<\nu\leq
3}\left(\Tr\square_{\mu\nu}(x)-\Tr\square^{\prime}_{\mu\nu}(x)\right)^{2}$
(16)

where we included a factor of $\frac{1}{N}$ as a practical measure to keep the
resulting numbers easy to work with. This is a useful starting point for our
metric, but it only compares $6N$ gauge-invariants, hence there are distinct
configurations that are not distinguished by this metric. For example,
consider Figure 12: the left column shows the evaluation of the metric,
$d_{P}$, between 300 configurations including instanton and antiinstanton
solutions. It is clear that $d_{P}$ is unable to distinguish instantons from
antiinstantons, even though we know they can be distinguished by gauge-
invariants such as the topological charge. We must compare additional gauge-
invariant quantities to make a proper metric.

Since the plaquette observables describe the local physics around a given
site, it is natural to look for non-local observables to round out the metric.
A first choice might be to include the holonomies around each direction:

$d^{2}_{W}\left(U,U^{\prime}\right)=\frac{1}{N}\sum_{x}\sum_{0\leq\mu\leq
3}\left(\Tr W_{\mu}(x)-\Tr W^{\prime}_{\mu}(x)\right)^{2}\leavevmode\nobreak\
.$ (17)

Unfortunately, since the holonomy in the $i$-direction is independent of the
value of $x^{i}$ where it starts, there are fewer than $N$ quantities999The
total number of independent gauge-invariants in $d_{W}$ is
$N_{W}=N_{0}N_{1}N_{2}+N_{0}N_{1}N_{3}+N_{0}N_{2}N_{3}+N_{1}N_{2}N_{3}$. If
$N_{i}>4$ for all directions (as it is for all the configurations we consider
here), then
$\begin{split}4N_{W}=&N_{0}N_{1}N_{2}4+N_{0}N_{1}4N_{3}+N_{0}4N_{2}N_{3}+4N_{1}N_{2}N_{3}\\\
<&N_{0}N_{1}N_{2}N_{3}+N_{0}N_{1}N_{2}N_{3}+N_{0}N_{1}N_{2}N_{3}+N_{0}N_{1}N_{2}N_{3}\\\
=&4N.\end{split}$ (18) Hence, $N_{W}<N$. compared in $d_{W}$. To remedy this
lack of information, we can also consider products of holonomies in orthogonal
directions:

$d^{2}_{WW}\left(U,U^{\prime}\right)=\frac{1}{N}\sum_{x}\sum_{0\leq\mu<\nu\leq
3}\left(\Tr\left[W_{\mu}(x)W_{\nu}(x)\right]-\Tr\left[W^{\prime}_{\mu}(x)W^{\prime}_{\nu}(x)\right]\right)^{2}\leavevmode\nobreak\
.$ (19)

This introduces another $6N$ gauge-invariant quantities. Importantly, it
distinguishes configurations that could not be distinguished with just $d_{P}$
and $d_{W}$, namely it distinguishes the instantons from the antiinstantons.

Using the above quantities, we can define a suitable metric via

$d^{2}=d_{P}^{2}+d_{W}^{2}+d_{WW}^{2}\leavevmode\nobreak\ .$ (20)

However, it is instructive to plot each part of this metric separately.

It is important to note that we have not proven that this collection of gauge-
invariants is strong enough to distinguish every distinct configuration. It
compares more than $12N$ gauge-invariant quantities, which implies the values
of these gauge-invariants cannot be completely independent from each other
(since they are measuring a $9N$-dimensional configuration space). It is
possible, in principle, that the true number of independent quantities
measured by $d$ is actually less than the necessary $9N$; however, this seems
unlikely as this would require a deep correlation between these quantities,
which is unknown to the author. The fact that $d_{WW}$ contains information
that is not contained in $d_{P}$ or $d_{W}$ (as evidenced by its ability to
distinguish instantons from antiinstantons) suggests there is a large degree
of independence between the considered quantities.

The results of comparing all 300 configurations for the $24\times 24\times
24\times 6$, $24\times 6\times 6\times 24$, and $24\times 6\times 6\times 6$
lattice sizes are given in Figure 12. This suggests that there are no
populations of solutions significantly differing from the proposed instanton
and antiinstanton. As stated above, this does not rule out alternative
solutions, but severely constrains them. The volume of parameter space that
minimizes down to a different solution is less than 1% of the total volume of
parameter space at the 95% confidence level. Moreover, at 99.99994% confidence
level, the volume is constrained to be less than 3.33% of the total space.
This suggests that the moduli space for instantons is the torus described in
14, and the same for the antiinstantons.

Figure 12: Comparison of the three distance metrics, $d_{P}$, $d_{W}$, and
$d_{WW}$, across the three large datasets of fractional instanton
configurations. The rows correspond to the datasets with lattice
configurations in the shape $24\times 6\times 6\times 24$, $24\times 6\times
6\times 6$, and $24\times 24\times 24\times 6$, respectively. The columns
correspond to the different distance metrics. The left column shows the
results for $d_{P}$. The central column shows results for $d_{W}$. The right
column shows results for $d_{WW}$. Distances are measured between every
possible pairing of the 300 configurations, divided based on whether they
compare instantons or antiinstantons, and plotted as a histogram. The x-axis
gives the distance value and the y-axis is the frequency with which that value
occurs in the dataset.

Given that this holds in the three extremal cases presented above, it is
natural to assume that this also holds for all sizes of torii considered in
this paper. 101010There may be an issue with this, however, near the case of
the self-dual torus (i.e. $\Delta\approx 0$). This case was considered
recently in a calculation of the gaugino condensate in SYM Anber:2022qsz ,
where it was found that this double circumference torus gives a result that is
a factor of 2 greater than the infinite volume result. In Section 6, we give
some preliminary evidence that the same moduli space does agree with 14.

##  4 Center vortices on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$

In this section, we discuss the properties of the solutions with small $N_{1}$
and $N_{2}$ and large $N_{0}$ and $N_{3}$. This is our best approximation to
the spacetime ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$, where the
confinement mechanism is understood in terms of BPS center vortices appearing
in Tanizaki:2022ngt . The use of center vortices in Tanizaki:2022ngt  was
based on numerical evidence from GONZALEZARROYO1998273  which we are expanding
on and strengthening in this work. The key feature needed for the
semiclassical argument of confinement in
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ is that the center vortices are
localized in ${\mathbb{R}}^{2}$ (in our case in the 03-plane), link with
Wilson loops embedded in ${\mathbb{R}}^{2}$, and are surrounded by the vacua
shown in 11. We confirm all of these properties below.

With regards to localization, consider Figure 13 which shows the action
density of the resulting configurations at various lattice shapes. Two things
are clear from these plots: the solutions are localized in the 03-plane and
the size of the vortex depends on the size of the small 12-plane. This matches
the expectation for the infinite volume limit.

Figure 13: Plots of the action density across the $03$\- and $12$-planes that
cut through the point of maximal action density. Both the $48\times 12\times
12\times 48$ and $48\times 9\times 9\times 48$ configurations are shown here
to highlight how the size of the localized instanton changes with the size of
the small directions.

A peculiar feature of these solutions is the peak in the action density within
the 12-plane. This raises the concern that the solutions might localize in the
12-plane, which would imply that they cannot be center vortices, since center
vortices must be codim-2. However, we can confirm that our solutions are
center vortices by demonstrating that our solutions properly link with Wilson
loops embedded anywhere in the 12-plane. In the full continuum quantum theory,
linking between a Wilson loop operator, $\hat{W}(C)$, defined on a contour $C$
and a center vortex, $\hat{V}(\Sigma)$, defined on a surface $\Sigma$ is
defined by the relation:

$\hat{W}(C)\hat{V}(\Sigma)=\exp{i\frac{2\pi}{N}\text{Linking}(C,\Sigma)}\hat{V}(\Sigma)\hat{W}(C)$
(21)

where $\text{Linking}(C,\Sigma)$ is the linking number between $C$ and
$\Sigma$. In the case of semiclassical $SU(2)$, this relation simplifies to
the statement that a Wilson loop that links with the center vortex must have
opposite sign to a Wilson loop that does not link with it. In our solutions, a
Wilson loop in the 03-plane is linked by the center vortex if the cross
section of the center vortex in the 03-plane is contained within the Wilson
loop. Hence to test the linking property of our solutions, it is sufficient to
measure the values of large Wilson loops embedded parallel to the 03-plane in
various positions. These values is shown in Figure 14 for a rectangular Wilson
loop that is translated across the 03-plane at various points in the 12-plane.
Notice that the change in sign that indicates proper linking with the center
vortex occurs at all point in the 12-plane, even when the action density is
minimal. This indicates that the linking is robust and the solutions are truly
codim-2 center vortices.

Figure 14: This plot shows the trace of a rectangular Wilson loop embedded in
the 03-plane of a $48\times 12\times 12\times 48$ solution as the Wilson loop
is moved across the solution in the 3-direction. The Wilson loop is 46 by 16,
and the different lines show Wilson loops embedded at different point in the
$12$-plane. The dotted black lines represent the points where the edges of the
Wilson loop cross the peak of the center vortex. The x-axis gives the value of
$x^{3}$ at the edge with the lowest value of $x^{3}$.

To show that the vacua surrounding the vortices have the correct form, we
simply need to check the values of the holonomies in all four directions. The
results are collected in Figure 15. It is clear that away from the center of
the vortex, $\Tr W_{1}$ and $\Tr W_{2}$ vanish as expected. $\frac{1}{2}\Tr
W_{0}$ interpolates between +1 at $x^{3}=1$ to -1 at $x^{3}=N_{3}$, and
similarly $\frac{1}{2}\Tr W_{3}$ interpolates between -1 at $x^{0}=1$ to +1 at
$x^{3}=N_{0}$. We also verify that $\Tr W_{1}W_{2}$ tends to zero
asymptotically. Therefore, when the center vortex is centered in the 03-plane,
it splits the plane into quadrants, each of which contains one of the four
possible solutions from 11.

Figure 15: Traces of the holonomies of the $48\times 12\times 12\times 48$
lattice configuration. Holonomies are plotted along planes that intersect the
point of maximal action density. Discontinuities are due to the fact that the
holonomies are antiperiodic across twists, which are always contained in the
plaquettes at the origin.

These observations confirm the identity of the half-BPS instantons as center
vortices. Combined with the statistical bounds on other half-BPS states from
Section 3.3, this provides solid numerical evidence to back the semiclassical
calculations of confinement in Tanizaki:2022ngt .

##  5 Monopole instantons on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$

In this section we focus on solutions when the lattice has small $N_{3}$ and
large $N_{0}$, $N_{1}$, and $N_{2}$. This approximates the case of
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, where semiclassics relies on
monopole-instantons. The key features of monopole instantons are that they are
1-dimensional and localized in all directions except the small circle
direction, that the asymptotic vacua surrounding them will abelianize the
theory to a $U(1)$ subgroup, and that with respect to this $U(1)$ they have a
well-defined magnetic charge quantized to $4\pi$.

The localization of our classical solution is apparent in plot of the action
density presented in Figure 16. Notice that in the $12$-plane the excitation
is localized to a small disk and in the $03$-plane the excitation is contained
in a thin band along the 3-direction. The fact that this solution has constant
action density along the 3-direction is a good indication that these are the
expected 1-dimensional monopole-instantons.

Figure 16: Plots of the action density across the $03$\- and $12$-planes that
cut through the point of maximal action density. Both the $48\times 48\times
48\times 12$ and $48\times 48\times 48\times 9$ configurations are shown here
to highlight how the size of the localized instanton changes with the size of
the small directions.

The external vacua are characterized by the values of the holonomies shown in
Figure 17. In contrast to the center vortex case, where the asymptotic vacua
agreed with 11, the vacua around the monopole-like solutions follow

$\begin{split}\frac{1}{2}\Tr W_{1}=&\pm 1\\\ \frac{1}{2}\Tr W_{2}=&\pm 1\\\
\frac{1}{2}\Tr W_{3}=&0\\\ \frac{1}{2}\Tr W_{0}=&0\\\ \end{split}$ (22)

with $\Tr W_{0}W_{3}=0$. By symmetry, these are the vacua in the case where
there is a twist in the $03$-plane and not in the $12$-plane.

Figure 17: Traces of the holonomies of the $48\times 48\times 48\times 12$
lattice configuration. Holonomies are plotted along planes that intersect the
point of maximal action density. Discontinuities are due to the fact that the
holonomies are antiperiodic across twists, which are always contained in the
plaquettes at the origin.

Since we are approximating ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, $W_{3}$
controls whether the theory abelianizes. The asymptotic vacuum has $\Tr
W_{3}=0$, meaning the theory outside the localized excitation abelianizes to
$U(1)$. This is desirable for defining a magnetic field and characterizing the
instanton as a monopole (which we do below). However, in the true vacuum,
$\frac{1}{2}\Tr W_{3}=\pm 1$ suggests that $W_{3}$ is proportional to the
identity. Therefore, in perturbation theory around the true vacuum, the theory
does not abelianize. If the theory does not abelianize, then it must be
strongly coupled in the IR and it is hopeless to start a controlled
semiclassical calculation.

Interestingly, having the twist in the $02$-plane causes $\frac{1}{2}\Tr
W_{3}=\pm 1$ classically, whereas in the continuum theory on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, we have $\frac{1}{2}\Tr W_{3}=\pm 1$
due to the GPY potential GPY1981 , which is a purely quantum effect. In both
cases, this suggests a spontaneous breaking of the center symmetry that cannot
be restored by semiclassical effects. This is consistent with the high
temperature deconfined phase of YM.

On the other hand, the inclusion of a deformation potential or massless
periodic adjoint fermions can stabilize the center symmetry. This forces the
vacuum to agree with $\frac{1}{2}\Tr W_{3}=0$, and here the monopole-
instantons have been shown to be effective in generating confinement
(textcolorredcite bions papers). Our solutions are useful for validating
qualitative features of these monopole-instantons, though we expect that the
additional potential will impose a compressive force that decreases the volume
of the instantons. It would be interesting to study these minimal solutions in
the presence of a deformation potential, but this would require a significant
modification of the cooling algorithm and thus is left to a future study.

The fact that $\frac{1}{2}\Tr W_{3}=0$ means that the theory will abelianize
from $SU(2)$ to $U(1)$, with the specific $U(1)$ aligning with the direction
of $W_{3}$ in colour-space. Given that $\square_{\mu\nu}(x)\sim
e^{ia^{2}\vec{F}_{\mu\nu}(x)\cdot\vec{\sigma}}$ and $\Tr W_{3}=0$ which
implies $W_{3}=i\hat{n}\cdot\vec{\sigma}$ for some unit vector
$\hat{n}\in{\mathbb{R}}^{3}$, we can calculate the approximate magnetic field
as

$\frac{1}{2}\Tr W_{3}(x)\square_{ij}(x)\sim a^{2}\hat{n}\cdot F_{ij}\,.$ (23)

Here $i,j=0,1,2$. The continuum limit of this quantity is the magnetic flux
passing through the plaquette $\square_{ij}(x)$. Notice that this is a gauge
invariant because the holonomy and the plaquette both start and end at the
same point, $x$.

However, within finite distance of the instanton, $\Tr W_{3}$ will not exactly
vanish giving instead
$W_{3}=\cos{\theta}+i\sin{\theta}\hat{n}\cdot\vec{\sigma}$ for some
$\theta\in[0,2\pi)$. We want to cancel the dependence on $\theta$, so we
define our approximate magnetic flux111111While this quantity is technically
the magnetic flux through the plaquette, $\square_{ij}(x)$, it is also
directly proportional to the magnetic field near $x$. We will not be overly
careful in making the distinction between the flux through a small plaquette
and the field. as

$\varepsilon_{ijk}B_{k}(x)=\frac{1}{2}\left(\frac{\Tr
W_{3}(x)\square_{ij}(x)-\Tr W_{3}(x)}{\sqrt{1-\left(\Tr
W_{3}(x)\right)^{2}}}\right)\,.$ (24)

The values of this gauge invariant quantity are plotted in Figure 18. It is
clear that close to the center of the configuration, the magnetic field lines
are pointing away from the localized instanton, consistent with a postive
magnetic charge. The magnetic fields die off to zero in the $1$\- and
$2$-directions, but do not die off along the $0$-direction. This boundary
behavior is consistent with an image charge picture where the instanton is
surrounded by a lattice of image magnetic charges, with all like charges along
the $1$\- and $2$\- directions, but alternating charges along the
$0$-direction. The magnetic fields behave differently in the $0$-direction
because they are antiperiodic in $x^{0}$, whereas they are periodic in $x^{1}$
and $x^{2}$. This is a direct consequence of $W_{3}$ being antiperiodic in
$x^{0}$ but periodic in $x^{1}$ and $x^{2}$. In fact, by translating the
instanton by $N_{0}$ sites in the $0$-direction, the sign of the magnetic
charge is flipped.

Figure 18: Plotting the magnetic flux of the $48\times 6$ configuration
calculated according to Equation 24. The top left plot is the 1-component of
the magnetic field, $B_{1}$, (calculated via the flux through the
$02$-plaquettes) plotted in the $12$-plane. The top right plot is the
2-component of the magnetic field, $B_{2}$, plotted in the $12$-plane. The
bottom left plot is the 0-component of the magnetic field, $B_{0}$, plotted in
the $01$-plane. While $B_{1}$ and $B_{2}$ die off away from the central
magnetic charge, it is clear that the $B_{0}$ field does not vanish as it
approaches $x^{0}=1$ and $x^{0}=48$. The bottom left plot shows the magnetic
charge distribution calculated via Gauss’ Law.

The continuum $SU(2)$ theory on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ has
two distinct types of monopoles, called BPS and KK. They have the same
topological charges but opposite magnetic charges and are exchanged by a
center symmetry transformation. Both types are necessary for the magnetic bion
mechanism for confinement Unsal2009bions . In our lattice configurations, it
is clear that the translation by $N_{0}$ sites in the $0$-direction, which is
a center symmetry transformation and flips the sign of the magnetic fields,
exchanges the BPS and KK monopoles. Hence, on the lattice, the two flavours of
monopoles are related by translations and therefore part of the same connected
region of moduli space.

Using Gauss’ Law, we can measure the magnetic charge contained within each
cube of our lattice configuration. The results are presented in the lower
right plot in Figure 18. Again this is consistent with a small blob of
positive magnetic charge coincident with the instanton. The total amount of
charge (and simultaneously the total magnetic flux) is consistent with $4\pi$
(see Table 4 for how close we get at various lattice sizes). This is the
expectation for a single quantum of magnetic charge in the continuum theory,
so this is consistent with continuum monopole-instantons.

$N_{S}$ $N_{L}$ | 24 | 36 | 48
---|---|---|---
6 | $12.5662\begin{subarray}{c}+0.0008\\\ -0.0011\end{subarray}$ | 12.5662 | 12.5664
9 | $12.5532\begin{subarray}{c}+0.0018\\\ -0.0031\end{subarray}$ | 12.5658 | 12.5664
12 | $12.4885\begin{subarray}{c}+0.0031\\\ -0.0099\end{subarray}$ | 12.5597 | 12.5661
Table 4: Values of the total magnetic flux for various sizes of lattice.
Values with errors are averaged across a dataset of multiple runs at the same
lattice size. Errors are determined by the maximum and minimum values across
the dataset

Interestingly, the flux escapes out through the $12$-planes at $x^{0}=1$ and
$x^{0}=N$. This suggests that the instantons are interpolating between vacua
with $\Tr W_{3}=0$ and a magnetic flux of $\pm 2\pi$ though the $12$-plane.
These are exactly the vacua originating in Perez:2013aa  and Unsal:2020yeh
and considered in Poppitz2023  as the proper vacua for dYM on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ with large
${\mathbb{T}}^{2}$. This is further evidence that these flux vacua are the
true vacua after stabilizing $\Tr W_{3}=0$ and that these configurations are
useful for semiclassics in center-stabilized YM on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$.

##  6 Interpolating between the different regimes on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$

This section is devoted to understanding how the fractional instantons on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ interpolate between
center vortices in the large $S^{1}$ limit and monopole-instantons in the
large ${\mathbb{T}}^{2}_{*}$ limit. The investigation focuses on comparing
minimal configurations on different sizes of lattice, specifically
$N_{0}\times N_{T}\times N_{T}\times N_{S}$ where $N_{0}$ is kept large and
one of $N_{T}$ or $N_{S}$ is always kept small while the other is set to
intermediate values. Essentially, we test how the instantons change as we go
from the ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ limit to the femtouniverse
limit (with $N_{T}$ and $N_{S}$ both small) to the
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ limit. We find it convenient for
this discussion to introduce a dimensionless parameter measuring the asymmetry
between the $03$\- and $12$-planes121212Be warned that this definition of
$\Delta$ differs slightly from other definitions due to the way we have chosen
to label our indices.:

$\Delta=\frac{N_{1}N_{2}-N_{0}N_{3}}{\sqrt{N_{0}N_{1}N_{2}N_{3}}}\,.$ (25)

Figure 19: "Phase" diagram of the fractional instantons found through the
cooling algorithm. The solid shaded regions and solid lines represent the
regions of parameter space we explicitly test in this work. The hatched
regions and dashed lines represent the expected continuation of these results
throughout the rest of parameter space.

The parameter space considered and our broad strokes findings are sketched in
Figure 19. The main result is that the instantons are separated into two
distinct classes: center vortex-like solutions when $\Delta<0$ and monopole-
instanton-like solutions when $\Delta>0$. The key feature distinguishing these
regimes is the asymptotic vacuum away from the localized excitations. Recall
from Section 4 that the center vortex solutions are surrounded by vacua
satisfying

$\begin{split}\frac{1}{2}\Tr W_{1}=&\frac{1}{2}\Tr W_{2}=\frac{1}{2}\Tr
W_{1}W_{2}=\,0\\\ \frac{1}{2}\Tr W_{3}=&\pm 1\\\ \frac{1}{2}\Tr W_{0}=&\pm
1\,.\end{split}$ (26)

These asymptotic vacua are also found around all the configurations in the
region $\Delta<0$. Recall also that these are the vacuum states we find when
the $03$-twist is turned off.

In contrast, like the monopole-instantons considered in Section 5, all the
fractional instantons in the region with $\Delta>0$ are surrounded by
asymptotic vacua characterized by

$\begin{split}\frac{1}{2}\Tr W_{1}=&\pm 1\\\ \frac{1}{2}\Tr W_{2}=&\pm 1\\\
\frac{1}{2}\Tr W_{3}=&\frac{1}{2}\Tr W_{0}=\frac{1}{2}\Tr
W_{0}W_{3}=\,0\,.\end{split}$ (27)

By symmetry, these should be the vacuum states when considering the torus with
only a $03$-twist (i.e. if we turn off the twists in the $12$-plane).

When both twists are turned on, it may asymptotically approach either the
vacua in 26 or 27, but because of the twists, the parameters $W_{i}$ must all
be antiperiodic in one direction. Consider the case that that our
configuration has the asymptotic vacua 26. Then $\frac{1}{2}\Tr W_{0}$ must
vary between $-1$ and $+1$ across the 3-direction and $\frac{1}{2}\Tr W_{3}$
must vary between $-1$ and $+1$ across the 0-direction. If the vacua are
asymptotically 27, then $\Tr W_{1}$ and $\Tr W_{2}$ will similarly vary across
the $12$-plane. There is an energy cost associated with these variations, and
there is a larger energy cost if the variations occur across a smaller plane.
Therefore, if the $03$-plane is smaller than the $12$-plane (i.e. positive
$\Delta$), it is energetically favourable for the asymptotic vacua to be 27.
Similarly, the asymptotic vacua in 26 are energetically favourable if the
$12$-plane is smaller (i.e. negative $\Delta$). This heuristic argument agrees
with the transition we see in the numerical data.

The above argument raises the question of which vacuum wins when $\Delta=0$.
This case is known as the symmetric torus, and the continuum theory was
studied in depth by ’t Hooft in tHooft:1981nnx . He found analytical solutions
for the BPS states. Importantly, these states have constant field strength
and, therefore, constant action density. This implies that there is no
asymptotic vacuum in this case, so neither vacuum wins.

The overall picture is that the monopole-instantons spread out as $\Delta$ is
lowered toward zero, becoming maximally spread out at $\Delta=0$, then the
qualitatively different vortex-like solutions are formed as $\Delta$ becomes
negative. This is reminiscent of a phase transition, though this is not
sufficient evidence to claim a phase transition occurs at $\Delta=0$ for the
full quantum system.

It is important to note that this hard transition between vortex-like and
monopole-like fractional instantons is a consequence of our particular path
through parameter space and not a fundamental incompatibility between the two
solutions. For example, in hayashi2024unifying , they consider Yang-Mills on
${\mathbb{R}}^{2}\times S^{1}_{L_{3}}\times S^{1}_{L_{4}}$ with a twist in the
$S^{1}_{L_{3}}\times S^{1}_{L_{4}}$. By keeping $L_{4}$ small and varying
$L_{3}$, they argue that the monopole-instanton solutions at large $L_{3}$
smoothly deform to center vortices at small $L_{3}$. Their work focused on the
magnetic dual description of the effective theory on ${\mathbb{R}}^{2}\times
S^{1}_{L_{3}}$. Our algorithm is the ideal tool to provide evidence of this
smooth transition in the full 4d electric description, however, such an
investigation is beyond the scope of the current article.

In the following subsections, we present the configurations and discuss their
properties in each of the regions $\Delta<0$, $\Delta>0$, and $\Delta\approx
0$.

###  6.1 $\Delta<0$

In this region, the fractional instantons behave like the center vortices
considered in Section 4.

Consider the shape of the action density plots in Figure 20. In agreement with
the center vortices in the ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ limit,
there is localization in all 4 directions for all configurations in the
$\Delta<0$ regime, though the localized excitation spreads out as $\Delta$
approaches 0. Interestingly, some additional structure appears when
$\Delta\approx 0$ (see the bottom right plot in Figure 20). The nature of this
additional structure is not entirely clear, but it remains when the
configurations are generated from different initial seeds. Thus, it is a
robust feature of these configurations. Exploring this structure further is
beyond the scope of this article, but could be a rewarding exercise. This
structure should also be accessible analytically through the
$\Delta$-expansion presented in GarciaPerez:2000aiw .

Figure 20: Plots of action densities in configurations of 3 different
geometries with $\Delta<0$. The left hand column plots the $03$-plane that
intersects the peak in action density and the right hand column plots the
$12$-plane that intersects the peak in action density.

To confirm that these configurations are center vortices, consider the
holonomies of a representative example as shown in Figure 21. This Figure
considers the "femtouniverse" configuration where all three spatial directions
are kept small, but the results are characteristic of the qualitative
structure of the holonomies throughout the $\Delta<0$ region of parameter
space. The holonomies in Figure 21 confirm that the asymptotic vacua agree
with 26. Moreover, they also confirm that these configurations work like
center vortices in the $03$-plane since the $0$\- and $3$-holonomies change
sign when passing the configurations.

Figure 21: Traces of the holonomies of a lattice configuration with
$\Delta<0$, namely $48\times 12\times 12\times 12$. Holonomies are plotted
along planes that intersect the point of maximal action density.
Discontinuities are due to the fact that the holonomies are antiperiodic
across twists, which are always contained in the plaquettes at the origin.

Recall that the vacua will abelianize the theory to an effective 2D
${\mathbb{Z}}_{2}$ gauge theory. Since this theory has no propagating degrees
of freedom and the gauge field is constrained to be ${\mathbb{Z}}_{2}$, the
only real observable available is the $3$-holonomy and it only takes the
values $\pm 1$. Hence, perturbatively we would expect there to be two
degenerate ground states $\ket{W_{3}=+1}$ and $\ket{W_{3}=-1}$. This
degeneracy spontaneously breaks the center symmetry acting on loops that wind
around the $3$-direction. It appears that our center vortex-like fractional
instantons are exactly the instantons that interpolate between these classical
vacua. Armed with these instantons and evidence that no other effects exist to
cancel their effects, we could conclude that the instanton effects would split
the degeneracy and leave a single center-symmetric ground state. This restores
center symmetry and therefore confirms that the theory is in the confining
phase.

However, the interpretation of our solutions as instantons interpolating
between $\ket{W_{3}=+1}$ and $\ket{W_{3}=-1}$ is a little hasty. Recall that
we originally found 4 classical vacua, based on the values of $W_{3}$ and
$W_{0}$, and that our solutions interpolate between different signs of $W_{0}$
across the $3$-direction. When considering the $0$-direction as Wick rotated
time, $W_{0}$ isn’t a good observable on Hilbert space (which is defined on a
given timeslice); however, it might be that another observable exists to
distinguish these vacua.

If the asymptotic states at $x^{0}=1$ and $x^{0}=N_{0}$ interpolate between
two vacua across the $3$-direction, then we would expect a domain wall to
separate these vacua. Otherwise, if the asymptotic states are good
approximations of true vacua, we should not see any domain wall forming. Such
a domain wall would look like a localized bump in the action density along the
$3$-direction at $x^{0}=1$. Plotting this, it appears that a domain wall
exists (see the top line in Figure 22), extinguishing our hopes of using
center vortices in a simple instanton argument for confinement; however, we
are concerned with the zero temperature limit, where we take $N_{0}$ to
infinity before the other directions. Extending the $0$-direction and plotting
the action density for different $N_{0}$, we find that the peak of the "domain
wall" quickly decays away as demonstrated in Figure 22. This suggests that in
the zero temperature limit, the center vortex solutions in this regime indeed
interpolate between $\ket{W_{3}=+1}$ and $\ket{W_{3}=-1}$, restoring center-
symmetry and confinement.

Figure 22: Plots showing that the apparent domain wall vanishes as $N_{0}$
grows. The action density is plotted at $x^{0}=1$ with $x^{1}$ and $x^{2}$
coinciding with the point of maximal action density. All configurations
considered here are $N_{0}\times 6\times 6\times 24$. Notice that as $N_{0}$
is increased the value at the peak of these distributions decreases by orders
of magnitude and the distribution appears to flatten out.

Thus, there is a single cohesive confinement mechanism (based on center-
vortices) for YM on every ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$
with a sufficiently small ${\mathbb{T}}^{2}$ to force abelianization to
${\mathbb{Z}}_{2}$. Given the basis on center vortices, this mechanism is also
consistent with the confining mechanism in the semi-infinite volume limit,
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$.

###  6.2 $\Delta>0$

In this region, the instantons are monopoles like those described in Section
5. Figure 23 shows the action density of configurations in this regime. Notice
that the configurations are completely spread out along the $3$-direction with
no peak, just like the monopole solutions shown in Figure 16. The monopoles
spread out as $\Delta$ gets closer to 0.

Figure 23: Plots of action densities in configurations of 3 different
geometries with $\Delta>0$. The left hand column plots the $03$-plane that
intersects the peak in action density and the right hand column plots the
$12-plane$ that intersects the peak in action density.

Figure 24 shows the holonomies for a representative configuration with
$\Delta>0$. Here we can see that the asymptotic vacua agree with 27. As
mentioned in Section 5, this raises difficulties with using these
configurations for semiclassics in pure YM, but it also allows for us to
calculate the magnetic fields surrounding these instantons using Equation 24.

Figure 24: Traces of the holonomies of a lattice configuration with
$\Delta>0$, namely $24\times 18\times 18\times 6$. Holonomies are plotted
along planes that intersect the point of maximal action density.
Discontinuities are due to the fact that the holonomies are antiperiodic
across twists, which are always contained in the plaquettes at the origin.

The magnetic fields of a characteristic configuration within the $\Delta>0$
region are plotted in Figure 25. There are clear similarities with the
magnetic fields in Figure 18. To demonstrate that these are truly monopoles,
we test the quantization of magnetic charge by measuring the total flux. The
results are shown in Table 5. All the values are consistent with the fluxes
measured for similarly sized lattices in Section 5. As demonstrated
previously, these values are consistent with having a $4\pi$ flux in the
continuum limit. This is the flux required for continuum monopoles.

Figure 25: Magnetic field for the lattice configuration with $24\times 18\times 18\times 6$. Notice the qualitative similarity to Figure 18. $N_{1}=N_{2}$ | $\Delta$ | Total magnetic flux ($\Phi_{3}$)
---|---|---
14 | 0.3095 | 12.5718
16 | 0.5833 | 12.5680
18 | 0.8333 | 12.5665
20 | 1.0666 | 12.5665
22 | 1.2878 | 12.5670
24 | 1.5 | 12.5662
Table 5: Values of the total magnetic flux, $\Phi_{3}$, in the region
$\Delta>0$. $\Delta$ is varied by changing $N_{1}$ and $N_{2}$ while
maintaining $N_{1}=N_{2}$. The values of $N_{0}$ and $N_{3}$ are fixed at 24
and 6, respectively.

While the structure of the monopoles is interesting, it remains difficult to
use them for semiclassics in YM on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$. First, there is the
issue of the asymptotic vacuum differing from the true vacuum. Moreover, the
true vacuum does not abelianize the theory. These were already mentioned in
Section 5 where we discussed the ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$
limit. In the ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ limit,
there is another issue: $\Delta$ is never positive in the zero temperature
limit. Recall that the zero temperature limit involves taking
$N_{0}\rightarrow\infty$ first. As $N_{0}$ grows, $\Delta$ decreases and
eventually turns negative. Hence, there is no 4-dimensional zero temperature
limit131313One could consider keeping ${\mathbb{T}}^{2}_{*}$ finite and
$\Delta$ positive by making the $S^{1}$ vanishingly small, but the continuum
limit in this case would be the dimensional reduction on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}$. This might be interesting for
studying the twisted Polyakov model, but it is not useful for the present
article’s goal of studying the 4-dimensional case. consistent with a finite
${\mathbb{T}}^{2}_{*}$ and $\Delta>0$.

It is the author’s hope that all these difficulties can be overcome with the
introduction of a deformation potential that stabilizes the center symmetry in
the 3-direction. This deformation potential should make a vacuum with $\Tr
W_{3}=0$ energetically preferred and should hopefully stabilize monopole
instantons with similar qualitative features within the $\Delta<0$ region.
This should lead to another cohesive picture of confinement on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ whenever the
${\mathbb{T}}^{2}_{*}$ is too large to abelianize down to ${\mathbb{Z}}_{2}$,
but the ${\mathbb{S}}^{1}$ is small enough to abelianize to $U(1)$. However,
as mentioned before, numerics in the presence of the deformation potential are
significantly more involved, so the inclusion of this potential is outside the
scope of the present article.

###  6.3 $\Delta=0$

We now understand that the sign of $\Delta$ determines whether the BPS
fractional instantons in the theory are monopoles or center vortices, but this
leaves the question of what happens at $\Delta=0$. In our simulations, this
corresponds to a lattice size of $24\times 12\times 12\times 6$. This is
analogous to the case of the self-dual torus, where we have analytical
continuum solutions due to ’t Hooft tHooft:1981nnx .

To see the solutions consider a continuum ${\mathbb{T}}^{4}$ with lengths
$L_{0}$, $L_{1}$, $L_{2}$, and $L_{3}$ in each direction. Here $\Delta$ is
defined as

$\Delta=\frac{L_{1}L_{2}-L_{0}L_{3}}{\sqrt{L_{0}L_{1}L_{2}L_{3}}}\,.$ (28)

Hence, $\Delta=0$ implies $L_{1}L_{2}=L_{0}L_{3}$. In the continuum, the
$03$-plane and $12$-plane twists are most easily implemented with boundary
conditions. We denote the boundary conditions by $\Omega_{\mu}(x)$ such that

$A(x+L_{\mu}e_{\mu})=\Omega_{\mu}(A-id)\Omega_{\mu}^{\dagger}\,.$ (29)

The following boundary conditions implement the desired twist:

$\begin{split}\Omega_{0}(x)=&e^{-i\pi\frac{x^{3}}{L_{3}}\sigma^{3}}\\\
\Omega_{1}(x)=&1\\\ \Omega_{2}(x)=&e^{-i\pi\frac{x^{1}}{L_{1}}\sigma^{3}}\\\
\Omega_{3}(x)=&1\,.\end{split}$ (30)

With these boundary conditions, ’t Hooft’s self-dual solutions at $\Delta=0$
are easy to write down:

$\begin{split}A_{0}=&\frac{z_{0}}{L_{0}}\frac{\sigma^{3}}{2}\\\
A_{1}=&\left(\frac{2\pi
x^{2}}{L_{1}L_{2}}+\frac{z_{1}}{L_{1}}\right)\frac{\sigma^{3}}{2}\\\
A_{2}=&\frac{z_{2}}{L_{2}}\frac{\sigma^{3}}{2}\\\ A_{3}=&\left(\frac{2\pi
x^{0}}{L_{0}L_{3}}+\frac{z_{3}}{L_{3}}\right)\frac{\sigma^{3}}{2}\,.\end{split}$
(31)

Here the $z_{i}$ are dimensionless parameters that parameterize the moduli
space of these solutions.

In this gauge, the field strength tensor is constant, taking the values

$\begin{split}F_{03}=-F_{30}=&\frac{2\pi}{L_{0}L_{3}}\frac{\sigma^{3}}{2}\\\
F_{12}=-F_{21}=&\frac{2\pi}{L_{1}L_{2}}\frac{\sigma^{3}}{2}\\\ \end{split}$
(32)

with all other components identically zero. The constant field strength is not
a gauge invariant property, but it guarantees that gauge invariant observables
built out of the field strength, such as the action density, are constant. In
other words, the action density plots should show that the $\Delta=0$ solitons
are maximally spread out.

Figure 26 displays such plots. It is clear from the figure that the action
density is not completely uniform, there is a peak in each plane. However,
this peak is very small compared to the baseline value of the action density.

Figure 26: Plots of the action density in configurations with $\Delta=0$ and
near $\Delta=0$. The $24\times 12\times 12\times 6$ configuration has exactly
$\Delta=0$. The other configurations are included for comparison. As with all
previous action density plots, the left hand column plots the $03$-plane that
intersects the peak in action density and the right hand column plots the
$12-plane$ that intersects the peak in action density.

To gain a quantitative understanding of how much the action deviates from a
constant solution, we compare the average value of the action density with the
maximum, minimum and standard deviations. The results are collected in Table
6, along with a lattice configuration of size $24\times 14\times 14\times 6$
and $24\times 10\times 10\times 6$. The data clearly demonstrates that the
solution is very close to uniform, especially when compared to the monopole
and center vortex solutions. This suggests that we are approximating the
continuum constant field strength solutions reasonably well, so we can
reasonably blame the deviation away from the constant solution to the effects
of finite lattice spacing.

| $24\times 12\times 12\times 6$ | $24\times 14\times 14\times 6$ | $24\times 10\times 10\times 6$
---|---|---|---
| Value | % error | Value | % error | Value | % error
Mean | 0.001904 |  | 0.001395 |  | 0.002734 |
Standard deviation | 0.000070 | 3.7% | 0.001154 | 82.7% | 0.001573 | 57.5%
Minimum | 0.001691 | -11.2% | 0.000333 | -76.2% | 0.000301 | -89.0%
Maximum | 0.002009 | 5.5% | 0.009651 | 591.8% | 0.005614 | 105.4%
Table 6: Average and variation of the action density across configurations
with different geometries. The percent errors are calculated with respect to
the mean action density within the same geometry. The key point is that the
action density on the self-dual torus (i.e. the $24\times 12\times 12\times 6$
lattice) varies little compared to the mean, especially compared to fractional
instantons on lattices with $\Delta\neq 0$.

It is also interesting to consider what happens as $\Delta$ approaches 0 from
both sides. Figure 26 demonstrates how both the center vortices and the
monopoles spread out at small values of $\left|\Delta\right|$. Here it is
clear that the transition between the two regimes is sharp, since the unique
shapes of the configurations are recognizable in all cases except exactly at
$\Delta=0$.

Considering Equation 31 and including the boundary conditions, we can also
work out the holonomies in this solution:

$\begin{split}W_{0}=\,&e^{i\left(z_{0}-2\pi\frac{x^{3}}{L_{3}}\right)\frac{\sigma^{3}}{2}}\\\
W_{1}=\,&e^{i\left(z_{1}-2\pi\frac{x^{2}}{L_{2}}\right)\frac{\sigma^{3}}{2}}\\\
W_{2}=\,&e^{i\left(z_{2}-2\pi\frac{x^{1}}{L_{1}}\right)\frac{\sigma^{3}}{2}}\\\
W_{3}=\,&e^{i\left(z_{3}-2\pi\frac{x^{0}}{L_{0}}\right)\frac{\sigma^{3}}{2}}\,.\end{split}$
(33)

It is clear from these equations that shifting the $z_{i}$ can be reabsorbed
into a shift of the coordinates $x^{i}$. Hence, the moduli space consists of
translations, just as for the other configurations we studied. It also means
we can shift the solutions such that the $z_{i}$ vanish.

Considering Equation 33 with $z_{i}=0$. In this case, the gauge invariant
traces of the holonomies in each of the four directions should interpolate
between $-1$ and $+1$ across some other direction. We can see the agreement
with this in Figure 27. This gives further evidence that we are approximating
’t Hooft’s solutions with our numerical solutions, and suggests that the
moduli space on the self-dual torus agrees with Equation 14.

Figure 27: Traces of the holonomies of the lattice configuration with
$\Delta=0$, namely $24\times 12\times 12\times 6$. By comparison with Equation
33, it is clear that this configuration has been translated such that the
$z_{\mu}$ vanish. Holonomies are plotted along planes that intersect the point
of maximal action density. Discontinuities are due to the fact that the
holonomies are antiperiodic across twists, which are always contained in the
plaquettes at the origin.

##  7 Conclusion and future directions

We used a numerical cooling algorithm to investigate the fractional instantons
that exist on ${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$,
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, and
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$. We considered lattices
with sizes $N_{0}\times N_{1}\times N_{2}\times N_{3}$. For all
configurations, $N_{0}$ was kept large to approximate Euclidean time at low
temperature, $N_{1}$ and $N_{2}$ were kept equal and a twist included in the
$12$-plane to make an approximate ${\mathbb{T}}^{2}_{*}$, and $N_{3}$ was
varied to approximate different sizes of the $S^{1}$ direction. $N_{1}$,
$N_{2}$, and $N_{3}$ were varied at fixed $N_{0}$ to approximate the various
continuum limits. To find fractional instantons we included an additional
twist in the $03$-plane to force a fractional topological charge. The
parameter

$\Delta=\frac{N_{1}N_{2}-N_{0}N_{3}}{\sqrt{N_{0}N_{1}N_{2}N_{3}}}$ (34)

was useful for distinguishing different regimes in the parameter space.

The solutions came in three varieties: center vortices when $\Delta$ was
negative, monopole instantons when $\Delta$ was positive, and ’t Hooft’s
uniform solutions when $\Delta$ exactly vanished. We confirmed that these
identifications were accurate by calculating characteristic gauge invariant
quantities in each regime, such as confirming linking with Wilson loops for
the center vortices and correct magnetic flux quantization in the monopole
instantons.

This numerical evidence demonstrates that the center vortices have the correct
properties to ensure confinement on
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}_{*}$ and
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ whenever the
${\mathbb{T}}^{2}_{*}$ is sufficiently small. This strengthens previous
numerical evidence and expands the validity to a larger class of geometries,
namely ${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ with any size of
circle.

While the monopole instantons require additional center stabilization to be
effective in explaining confinement, we showed that there is a sharp
transition between monopoles instantons on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ and the center vortices on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$. If we allow for finite
temperature, then we find monopole instantons on
$S^{1}_{\beta}\times{\mathbb{T}}^{2}_{*}\times S^{1}_{L}$ for large
${\mathbb{T}}^{2}_{*}$ and the sharp transition occurs at some finite size of
${\mathbb{T}}^{2}_{*}$. The fractional instantons at the point of transition
are given by ’t Hooft’s constant field strength solutions. The details of this
sharp transition are, as far as the author knows, new to this work.

There is rich physics hiding in these classical solutions and their
semiclassical effects on the full quantum theory. Several extensions and
further uses for the algorithm were discussed in the text. One promising
direction that would not require any modifications to the algorithm would be
to investigate the electric description of the setup from hayashi2024unifying
, which should have a smooth crossover between the monopole instantons and the
center vortices, in contrast to the sharp transition we found. This
investigation could be carried out by changing the twists to live in the
$01$\- and $23$-planes, and investigating configurations of the form
$N_{L}\times N_{L}\times N_{I}\times N_{S}$, where $N_{L}$ is a large value,
$N_{S}$ is a small value, and $N_{I}$ interpolates between the large and small
values.

In addition, there might be some interesting physics to be found in the
additional structure observed when $-1\ll\Delta<0$. The nature of this
structure should agree with the analytical $\Delta$-expansion solutions on
nearly self-dual torii developed in GarciaPerez:2000aiw .

An important generalization of this algorithm would be the inclusion of a
deformation potential. The deformation potential necessitates a more involved
algorithm, but it would allow for a direct numerical test of the continuity
between regimes discussed in Poppitz2023  and guvendik2024metamorphosis . This
is because the theory with deformation potential would energetically favour
the $F_{12}\neq 0$ vacua these papers require to discuss semiclassics.
Moreover, it should allow monopole instantons at zero temperature on
${\mathbb{R}}\times{\mathbb{T}}^{2}_{*}\times S^{1}$ with finite
${\mathbb{T}}^{2}_{*}$. It might even smooth out the sharp transition we
observed in pure YM theory, confirming the crossover hypothesis in
guvendik2024metamorphosis .

Acknowledgements:

I would like to thank Erich Poppitz for hours of helpful discussion and for
asking all the difficult questions that lead to the most interesting results
in this work.

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