aainstitutetext: Centre for Particle Theory, Department of Mathematical
Sciences, Durham University, South Road, Durham DH1 3LE, UKbbinstitutetext:
Department of Physics, University of Toronto, 60 St George St., Toronto, ON
M5S 1A7, Canada

# The gaugino condensate from asymmetric four-torus with twists

Mohamed M. Anber b Erich Poppitz<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We calculate the gaugino condensate in $SU(2)$ super Yang-Mills theory on an
asymmetric four-torus $\mathbb{T}^{4}$ with ’t Hooft’s twisted boundary
conditions. The $\mathbb{T}^{4}$ asymmetry is controlled by a dimensionless
detuning parameter $\Delta$, proportional to $L_{3}L_{4}-L_{1}L_{2}$, with
$L_{i}$ denoting the $\mathbb{T}^{4}$ periods. We perform our calculations via
a path integral on a $\mathbb{T}^{4}$. Its size is taken much smaller than the
inverse strong scale $\Lambda$ and the theory is well inside the semi-
classical weak-coupling regime. The instanton background, constructed for
$\Delta\ll 1$ in GarciaPerez:2000aiw , has fractional topological charge
$Q=\frac{1}{2}$ and supports two gaugino zero modes, yielding a non-vanishing
bilinear condensate, which we find to be $\Delta$-independent. Further, the
theory has a mixed discrete chiral/$1$-form center anomaly leading to double
degeneracy of the energy eigenstates on any size torus with ’t Hooft twists.
In particular, there are two vacua, $|0\rangle$ and $|1\rangle$, that are
exchanged under chiral transformation. Using this information, the
$\Delta$-independence of the condensate, and assuming further that the semi-
classical theory is continuously connected to the strongly-coupled
large-$\mathbb{T}^{4}$ regime, we determine the numerical coefficient of the
gaugino condensate: $\langle 0|\mbox{tr}\lambda\lambda|0\rangle=|\langle
1|\mbox{tr}\lambda\lambda|1\rangle|=32\pi^{2}\Lambda^{3}$, a result equal to
twice the known $\mathbb{R}^{4}$ value. We discuss possible loopholes in the
continuity approach that may lead to this discrepancy.

##  1 Introduction

Strongly coupled gauge theories have been under intense study over the past
few years, thanks to the recent developments of generalized global symmetries
Gaiotto:2014kfa . These are operations that implement the group multiplication
laws via topological constructions such that the symmetry operations are
supported on topological surfaces that are robust under small deformations. An
ordinary $0$-form symmetry $G$, which acts on point-like particles, is
implemented by operators supported on co-dimension $1$ surfaces. These
surfaces obey the group-multiplication laws via fusion rules and give rise to
phases valued in $G$ when they cross the charged objects. Likewise, a $1$-form
symmetry acts on $1$-dimensional objects and is implemented via co-dimension
$2$ surfaces. For example, Wilson lines in $4$-D $SU(N)$ pure or super Yang-
Mills theories are charged under $\mathbb{Z}_{N}^{(1)}$ 1-form symmetry. The
$1$-form symmetry is implemented via topological $2$-dimensional surfaces that
obey the $\mathbb{Z}_{N}$ group multiplication laws and give rise to
$\mathbb{Z}_{N}$ phases when they link with Wilson lines. Like ordinary
$0$-form symmetries, $1$-form symmetries organize the spectrum of a theory
into representations, satisfy Ward identities, and may become anomalous if one
tries to gauge them. In particular, ’t Hooft anomaly matching conditions (or
’t Hooft anomalies for short), which impose stringent constraints on quantum
field theory (QFT), can be generalized to include anomalies of $1$-form
symmetries. Detecting the anomaly in $4$-D requires defining a given QFT on
manifolds with nontrivial $2$-cycles, the typical example being the $4$-torus
$\mathbb{T}^{4}$. Generalized ’t Hooft anomalies provide a framework for
classifying QFT and its phases. Many recent works entertained this generalized
framework to shed light on a plethora of asymptotically free gauge theories,
including vector-like Gaiotto:2017yup ; Tanizaki:2018wtg ; Anber:2019nze ;
Bolognesi:2019fej ; Anber:2020gig ; Sulejmanpasic:2020zfs ; Anber:2019nfu ;
Lohitsiri:2022jyz  and chiral theories Bolognesi:2020mpe ; Anber:2021iip .

Assuming a gauge theory generates a mass gap in the infrared (IR), one way to
match its ’t Hooft anomalies is via the formation of condensates. A famous
example, and the subject of this paper, is the formation of gaugino
condensates in ${\cal N}=1$ $SU(N)$ super Yang-Mills theory (SYM), which are
needed to match a generalized ’t Hooft anomaly. This anomaly can be understood
as follows. In addition to the $\mathbb{Z}_{N}^{(1)}$ $1$-form symmetry that
acts on the Wilson lines, SYM enjoys a $\mathbb{Z}_{2N}^{d\chi}$ $0$-form
global discrete chiral, or $R$, symmetry. We may gauge the $1$-form symmetry
by turning on a background gauge field of $\mathbb{Z}_{N}^{(1)}$, which is
done either by coupling SYM to a $\mathbb{Z}_{N}$ TQFT Kapustin:2014gua  or by
activating a ’t Hooft flux tHooft:1979rtg ; tHooft:1981nnx . The latter is a
field configuration on $\mathbb{T}^{4}$ that carries a fractional flux $B\in
H^{2}(\mathbb{T}^{4},\mathbb{Z}_{N})$ (i.e. the flux piercing $2$-cycles obeys
the quantization rule
$\int_{\mathbb{T}^{2}\subset\mathbb{T}^{4}}B\in\frac{2\pi\mathbb{Z}}{N}$) and
fractional topological charge
$Q=\frac{1}{8\pi^{2}}\int_{\mathbb{T}^{4}}B\wedge B\in\frac{\mathbb{Z}}{N}$.
In mathematical language, we consider the $PSU(N)\equiv SU(N)/\mathbb{Z}_{N}$
bundle instead of the $SU(N)$ bundle. The former has a non-trivial Brauer
class $w\in H^{2}(BPSU(N),\mathbb{Z}_{N})$ that obstructs the lifting of
$PSU(N)$ to $SU(N)$, and we set $B=w$. In practice, the $PSU(N)$ flux is
turned on by imposing twisted boundary conditions on the fields that live on
$\mathbb{T}^{4}$. Next, we examine the partition function ${\cal Z}[PSU(N)]$
of SYM in this background by performing a $\mathbb{Z}_{2N}^{d\chi}$ rotation.
We find ${\cal Z}[PSU(N)]\rightarrow e^{i\frac{2\pi}{N}}{\cal Z}[PSU(N)]$, and
thus, the theory stops being invariant under the action of
$\mathbb{Z}_{2N}^{d\chi}$ once we gauge the $1$-form symmetry. This is the
sought generalized ’t Hooft anomaly—a mixed anomaly between the
$\mathbb{Z}_{N}^{(1)}$ $1$-form symmetry and the $\mathbb{Z}_{2N}^{d\chi}$
$0$-form discrete chiral symmetry. The anomaly indicates that the IR theory
cannot be trivially gapped. Assuming confinement, the theory breaks its
$\mathbb{Z}_{2N}^{d\chi}$ symmetry and forms $N$ vacua separated by domain
walls. The order parameter of $\mathbb{Z}_{2N}^{d\chi}$ symmetry is the
bilinear gaugino condensate $\langle\mbox{tr}\lambda^{2}\rangle$ or higher-
order condensates $\langle(\mbox{tr}\lambda^{2})^{n}\rangle$, $n>1$ and
$n\,\mbox{mod}\,N\neq 0$. The existence of $N$ vacua is also in accordance
with the Witten index Witten:1982df .

###  1.1 Gaugino condensate: a historical prelude

Generally speaking, the matching conditions can only provide kinematical
constraints and do not, by themselves yield insights into the details of the
IR dynamics of a gauge theory. One needs an extra guide if, at all, there is a
hope to understand the dynamics, e.g., the condensates. Thanks to
supersymmetry, such studies are possible in SYM. These date back to the early
eighties Novikov:1983ee ; Rossi:1983bu ; Amati:1984uz ; Affleck:1983rr ;
Novikov:1985ic  (see Amati:1988ft ; Shifman:1999mv ; Shifman:1999kf ;
Dorey:2002ik ; Vandoren:2008xg ; Terning:2006bq  for reviews). The condensate
calculations are based on the Belavin-Polyakov-Schwartz-Tyupkin (BPST)
instanton calculus Belavin:1975fg : BPST instantons are (anti) self-dual Yang-
Mills configurations that violate the non-renormalization theorems of SYM and
hence, give a non-zero vacuum expectation value to the condensates. On
dimensional-analysis grounds one can write
$\langle\frac{\mbox{tr}\lambda^{2}}{16\pi^{2}}\rangle=c\Lambda^{3}$, where
$\Lambda$ is the strong-coupling scale and $c$ is a numerical factor. The
exact value of $c$ was a controversial issue that caused many debates in the
past. Generally, there are two methods to compute the bilinear gaugino
condensate in $4$-D SYM: the strong-coupling and the weak-coupling instanton
methods.

In the first method, we start directly from the $4$-D SYM in its strong-
coupling regime and do instanton calculus, as in Novikov:1983ee ; Rossi:1983bu
. A single $SU(N)$ BPST instanton carries integer topological charge
$Q\in\mathbb{Z}$, and the configuration with the lowest topological charge
$Q=1$ admits $2N$ gaugino zero modes. The saturation of the zero modes in the
$Q=1$ instanton background gives a nonzero value to the $2N$-point function
$\langle(\mbox{tr}\lambda^{2})^{N}\rangle$, from which one can naively extract
the value of the $2$-point function
$\langle(\mbox{tr}\lambda^{2})\rangle=\left[\langle(\mbox{tr}\lambda^{2})^{N}\rangle\right]^{1/N}$.
A detailed calculation, keeping track of all numerical coefficients, gives
$\langle(\mbox{tr}\lambda^{2})\rangle=2((N-1)!(3N-1))^{-1/N}(16\pi^{2}\Lambda^{3})e^{i\frac{2\pi
k}{N}}$, with $k=0,1,...,N-1$. The complex phase results from taking the $N$th
root of unity, in accordance with the expectation that the theory admits $N$
distinct vacua needed to match the generalized ’t Hooft anomaly.

In the weak-coupling instanton method, we consider super QCD with $N-1$
fundamental flavors $\Phi_{i}$, $i=1,..,N-1$, where $\Phi_{i}$ is a chiral
superfield, and give all the flavors small masses $m$. We work in the limit
$|\Phi_{i}|\gg\Lambda_{Q}$, where $\Lambda_{Q}$ is the strong scale in the
presence of quarks. Since there are $N-1$ flavors, the gauge group fully
abelianizes and we are well inside the weak-coupling regime. The total
superpotential of this theory takes the from ${\cal
W}=m_{j}^{i}\bar{\Phi}^{j}\Phi_{i}+\frac{\Lambda_{Q}^{2N+1}}{\mbox{Det}(\bar{\Phi}\Phi)}$,
where the second term is the Affleck-Dine-Seiberg (ADS) superpotential
Affleck:1983rr . The ADS term is nonperturbative in nature and is based on
holomorphy and the symmetry structure of super QCD. It also results from
saturating the quarks’ zero modes in the BPST instanton background (the
numerical coefficient was obtained111The comparison between the weak-coupling
and strong-coupling instanton methods in $SU(2)$ was first performed in
Novikov:1985ic , where the correct ratio between the two methods was given. in
Cordes:1985um , and corrected in Finnell:1995dr ). Since we are in a weak-
coupling limit, the instanton calculations are reliable. Minimizing the
energy, we obtain the supersymmetric vacuum
$\bar{\Phi}^{j}\Phi_{i}=\left(m^{-1}\right)_{i}^{j}\left[\Lambda_{Q}^{(2N+1)}\mbox{Det}m\right]^{1/N}$.
Finally, we substitute this result back into ${\cal W}$ to find ${\cal
W}=N\left[\Lambda_{Q}^{2N+1}\mbox{Det}m\right]^{1/N}$. We then decouple the
quarks by taking $m\gg\Lambda_{Q}$, thus, leaving the weak-coupling regime.
Using holomorphy, we can write ${\cal
W}_{\scriptsize\mbox{eff}}=N\Lambda^{3}$, where $\Lambda$ is the strong scale
at the mass threshold, and it exactly coincides with SYM strong scale at the
decoupling limit. Recalling that one can write the holomorphic strong scale as
$\Lambda=\mu e^{2\pi i\tau/3N}$, with $\tau=\frac{4\pi i}{g^{2}_{h}(\mu)}$
($g_{h}$ is the holomorphic gauge coupling, running at one loop only and $\mu$
is some arbitrary energy scale) and that
$\langle\mbox{tr}\lambda^{2}\rangle=-8\pi i\frac{\partial{\cal
W}_{\scriptsize\mbox{eff}}}{\partial\tau}$, one obtains
$\langle\mbox{tr}\lambda^{2}\rangle=16\pi^{2}\Lambda^{3}e^{i\frac{2\pi k}{N}}$
in the $k$-th vacuum.222The definition of the strong coupling scale we follow
in this paper is given by
$\Lambda^{3}=\mu^{3}\frac{e^{-8\pi^{2}/Ng^{2}}}{g^{2}}$, the one used in
Davies:2000nw ; Dorey:2002ik ; Vandoren:2008xg .

Having two different methods that yield two different answers resulted in many
debates in the literature about the validity of both methods. It was earlier
understood that the strong-coupling instanton method is in tension with the
cluster decomposition principle (CDP). Consider the correlator
$\langle\mbox{tr}\lambda^{2}(x)\mbox{tr}\lambda^{2}(x^{\prime})\rangle$. In
the limit $|x-x^{\prime}|\rightarrow\infty$ we expect
$\langle\mbox{tr}\lambda^{2}(x)\mbox{tr}\lambda^{2}(x^{\prime})\rangle=\langle\mbox{tr}\lambda^{2}\rangle^{2}$.
However, since a BPST instanton cannot saturate $2$ gaugino zero modes, one
finds $\langle\mbox{tr}\lambda^{2}\rangle=0$, contradicting CDP. A possible
resolution of this puzzle was proposed in Kovner:1997im . It was hypothesized
that SYM admits an extra phase with vanishing bilinear condensate
$\langle\mbox{tr}\lambda^{2}\rangle=0$ and that averaging over the chirally
symmetric and non-symmetric phases gives the result of the strong-coupling
instanton method. This hypothesis was carefully examined in Ritz:1999mq  by
considering a softly broken ${\cal N}=2$ Seiberg-Witten theory down to SYM. It
was shown that the chirally-symmetric phase is absent, casting doubt on the
averaging hypothesis. Further considerations in Cachazo:2002ry  excluded the
symmetric vacuum. We also note that the anomaly-matching argument of
Csaki:1997aw  also precludes such a phase. It was further shown in
Hollowood:1999qn  that strong-coupling multi-instanton calculations are
inconsistent with CDP.

Extra support to the weak-coupling calculations came from studying SYM on
$\mathbb{R}^{3}\times\mathbb{S}^{1}_{L}$, where $\mathbb{S}^{1}_{L}$ is a
small spatial circle with circumference $L\ll(N\Lambda)^{-1}$, and both gauge
fields and gauginos are given periodic boundary conditions on
$\mathbb{S}^{1}_{L}$ Davies:1999uw ; Davies:2000nw . Compactification over a
small circle causes this theory to fully abelianize and enter its weakly-
coupled regime.333The $4$-D coupling constant on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}_{L}$ ceases to run at scale $\sim
1/NL$, roughly the $W$-boson mass. Taking $L\ll(N\Lambda)^{-1}$, we stay in
the weakly-coupled regime. The theory on
$\mathbb{R}^{3}\times\mathbb{S}^{1}_{L}$ admits monopole-instantons, the
microscopic constituents of calorons.444A $SU(N)$ caloron with a unit
topological charge is composed of $N$ monopole instantons. Calorons are Yang-
Mills (anti) self-dual configurations defined on
$\mathbb{R}^{3}\times\mathbb{S}^{1}$ with integral topological charges and
non-trivial holonomy (the expectation value of the Polyakov’s loop) along
$\mathbb{S}^{1}$. A BPST instanton, in contrast, (or more precisely, the
Harrington-Shepard solution Harrington:1978ve  defined on
$\mathbb{R}^{3}\times\mathbb{S}^{1}$) has a trivial holonomy. Calorons were
discovered by Kraan and van Baal Kraan:1998pm  using the Atiyah-Drinfeld-
Hitchin-Manin (ADHM) construction Atiyah:1978ri  and independently by Lee and
Yi Lee:1997vp  and Lee and Lu Lee:1998bb  in the context of D-branes. Let $L$
and $P$ be the size of $\mathbb{S}^{1}$ and the holonomy (in units of $1/L$).
If $LP\gtrsim 1$, then the monopole constituents are well-separated in space,
and one can make sense of them in a semi-classical treatment, as in the case
of SYM on $\mathbb{R}^{3}\times\mathbb{S}_{L}^{1}$. In particular, in the
supersymmetric vacuum, which preserves the $0$-form $\mathbb{Z}_{N}$ center
symmetry of the theory, the constituent monopoles are of equal action
$S=\frac{8\pi^{2}}{Ng^{2}}$. In the opposite limit, $LP\ll 1$, the monopoles
hide inside the caloron core. At the supersymmetric vacuum, the monopoles have
topological charges of $1/N$. Using the index theorem, we deduce that a single
monopole can saturate $2$ gaugino zero modes giving rise to the bilinear
condensate. Detailed calculations give
$\langle\frac{\mbox{tr}\lambda^{2}}{16\pi^{2}}\rangle=\Lambda^{3}$, the exact
same result from $4$-D weak-coupling instanton calculations. The advantage of
the compactified theory over the $4$-D theory at weak coupling is that in the
former, one can understand the dynamics responsible for the condensate
formation without relying on the miracle of holomorphy. As a bonus, the
proliferation of the monopoles causes the compactified theory to generate a
mass gap and confine, a result that is prohibitively difficult to understand
in $4$-D.

The continuity of confinement and condensate between the small and large $L$
limits may imply that the fractional instantons are responsible for the
dynamics on $\mathbb{R}^{4}$, even though one may not make analytical sense of
them in a strongly-coupled setup. The continuity conjecture was taken
seriously over the past decade in supersymmetric and nonsupersymmetric
theories; see Dunne:2016nmc ; Poppitz:2021cxe  for reviews. The conjecture
withstood many tests, but we are still far from a firm conclusion about the
role of fractional microscopic objects in strongly-coupled phenomena. This
continuity was mainly tested on $\mathbb{R}^{3}\times\mathbb{S}^{1}_{L}$, and
one wishes to examine whether it holds in other geometries.

One such geometry is $\mathbb{T}^{4}$, very natural from the point of view of
lattice practicalities. Right after ’t Hooft presented his twisted solutions
(solutions with twisted boundary conditions, i.e. $PSU(N)$ bundle solutions,
with fractional topological charges) tHooft:1981nnx , van Baal studied their
mathematical properties vanBaal:1982ag ; vanBaal:1984ar . Later, it was argued
in a series of works Gonzalez-Arroyo:1995ynx ; Gonzalez-Arroyo:1995isl ;
Gonzalez-Arroyo:1996eos , see also the review GarciaPerez:2009mmu , that
(anti) self-dual ’t Hooft fractional instantons can be seen in realistic
simulations of $4$-D pure Yang-Mills theory and that such configurations could
be utilized to explain confinement.555Even before these studies, a program
known as the “femtouniverse” utilized the Hamiltonian formalism on
${\mathbb{R}}\times{\mathbb{T}}^{3}$, to study Yang-Mills theories at small
volumes Luscher:1982ma ; see vanBaal:2000zc  for a review. Further, it was
argued in GarciaPerez:1999hs , via extended lattice simulations, that ’t Hooft
fractional instantons on $\mathbb{T}^{4}$ are ultimately connected to monopole
instantons in the infinite $3$-volume limit and finite time666Notice that
these are simulations in pure Yang-Mills theory, and thus, unlike SYM, there
is no distinction between thermal and spatial circles. direction. In
particular, it was shown that an exact $SU(2)$ caloron solution with a unit
topological charge and equal-action constituent monopoles (see Footnote 4) can
be constructed on the twisted $\mathbb{T}^{4}$ by gluing two twisted
solutions, each carrying $Q=\frac{1}{2}$ charge, along the space
directions.777In fact, there is an obstruction to the existence of $Q=1$
(anti) self-dual caloron on $\mathbb{T}^{4}$ with untwisted boundary
conditions Braam:1988qk . Yet, in practice, one can find a very good
approximate self-dual solution even in the absence of twists.

###  1.2 Gaugino condensate on asymmetric $\mathbb{T}^{4}$: a summary of the
procedure and results

The possible connection between ’t Hooft’s solutions and monopole-instantons
GarciaPerez:1999hs  calls for a serious examination of this finding. Since
both the $\mathbb{R}^{3}\times\mathbb{S}_{L}^{1}$ monopole-instanton and $4$-D
weak-coupling instanton methods give the same gaugino condensate, one wonders
whether the same result can be obtained in the ’t Hooft flux background.888We
interchangeably use the term “’t Hooft flux background” and “’t Hooft twists.”
The latter is more precise, since it refers to the twist of the boundary
conditions, which does not always lead to nonzero “flux,” i.e. nonzero gauge
field strength. Whether such field strength is present or not is a dynamical
issue, see e.g. Witten:1982df . We hope that our abuse of terminology does not
lead to confusion.  This is especially timely after the advent of the new
generalized anomalies, which can be easily detected when a QFT is put on
$\mathbb{T}^{4}$ with a $PSU(N)$ bundle (or ’t Hooft twisted boundary
conditions). Recently, Ünsal advocated that a refinement of the instanton sum
in the partition functions of $SU(N)$ theories has to be considered: while
fractional instantons of $PSU(N)$ bundles contribute to observables like the
gaugino condensate and vacuum energy, the sum over the fractional objects has
to be constrained to yield the integer topological charges of the $SU(N)$
bundle Unsal:2020yeh  (see also vanBaal:2000zc  for an earlier assertion).

In this paper, we put this proposal under scrutiny and perform detailed
calculations of the gaugino condensate on $\mathbb{T}^{4}$ and a $PSU(2)$
bundle.999The gaugino condensate on a symmetric $\mathbb{T}^{4}$ in the
background of ’t Hooft flux was first considered long ago in Cohen:1983fd .
These calculations were solely based on supersymmetry transformations along
with dimensional analysis, and no attempt to determine the numerical
coefficient was made. Our conclusion is that the condensate in any of the two
vacua is given by $\langle 0|\mbox{tr}\lambda^{2}|0\rangle=|\langle
1|\mbox{tr}\lambda^{2}|1\rangle|=32\pi^{2}\Lambda^{3}$, with a coefficient
that is twice that obtained via the weak-coupling instanton method on
$\mathbb{R}^{4}$ and the semi-classical calculations on
$\mathbb{R}^{3}\times\mathbb{S}^{1}_{L}$. The extra factor of $2$ is
unexpected and calls for further serious examinations of the role of the
$PSU(N)$ bundles in $SU(N)$ gauge theories and of the continuity conjecture.

The calculations that lead to this puzzling result are surprisingly rich. The
simplest $SU(2)$ pure Yang-Mills (i.e. Yang-Mills equations with zero source)
fractional instanton with topological charge $Q=\frac{1}{2}$ and action
$S_{0}=\frac{4\pi^{2}}{g^{2}}$ was constructed by ’t Hooft in his seminal work
tHooft:1981nnx . This solution is abelian in nature, i.e. the gauge field
components are along the Cartan direction. The solution must be (anti) self-
dual, otherwise, the fluctuations in this background will have negative modes
signaling instability. Let $L_{1},L_{2},L_{3},L_{4}$ be the lengths of the
periods in $\mathbb{T}^{4}$. Then, the self-duality of the abelian solution is
guaranteed if and only if $L_{1}L_{2}=L_{3}L_{4}$. We dub the $\mathbb{T}^{4}$
that obeys this relation as the self-dual torus, the simplest one being the
symmetric $\mathbb{T}^{4}$ with $L_{1}$=$L_{2}$=$L_{3}$=$L_{4}$. Given the
simplicity of the abelian solution on the symmetric $\mathbb{T}^{4}$, one is
tempted to use it as a source for the gaugino condensate. According to the
index theorem, this background must saturate $2$ gaugino zero modes.
Nevertheless, the direct solution of the Dirac equation yields $6$ zero modes
(four “undotted” and two “dotted” ones). To make things worse, the extra zero
modes source the super Yang-Mills equations of motion, hinting that such an
abelian solution might not be a consistent background. The resolution of both
puzzles could have been achieved had we been able to show that higher-order
corrections (including loops) lift the extra gaugino zero modes. Although we
believe that this should be the case, we were not able to show it in a
satisfactory way.

In order to circumvent this difficulty, we chose to depart from the symmetric
self-dual to the non-self-dual torus. This is both a blessing and a curse. The
curse is the technical difficulty of the problem, while the blessing is that
the asymmetric torus enables us to take various interesting limits, e.g.
${\mathbb{R}}\times{\mathbb{T}}^{3}$ and
${\mathbb{R}}^{2}\times{\mathbb{T}}^{2}$. If the relation
$L_{1}L_{2}=L_{3}L_{4}$ is violated, the abelian solution must be modified to
include non-abelian pieces, ensuring that the solution persists to be (anti)
self-dual. To date, there exists no analytical solution with $Q=\frac{1}{2}$
on $\mathbb{T}^{4}$ with arbitrary shape. However, a systematic procedure to
deal with non-self-dual tori was developed in GarciaPerez:2000aiw . This
method gives an approximate analytical self-dual solution, with charge
$|Q|=\frac{1}{2}$ and action $S_{0}=\frac{4\pi^{2}}{g^{2}}$, as an expansion
in a “detuning” parameter
$\Delta\equiv(L_{3}L_{4}-L_{1}L_{2})/\sqrt{L_{1}L_{2}L_{3}L_{4}}$ measuring
the deviation from a self-dual $\mathbb{T}^{4}$.

Solving the Dirac equation on a small asymmetric $\mathbb{T}^{4}$ with
$\Delta\ll 1$ and limiting our explicit treatment to the lowest order in
$\Delta$, we find exactly $2$ gaugino zero modes as per the index theorem. We
also verified that the explicit solutions of the fermion zero modes are
consistent with supersymmetry transformations. These modes do not source the
Yang-Mills equations of motion, implying that our approximate solution is a
consistent background. The gauge field fluctuations admit $4$ bosonic zero
modes (also in accordance with the index theorem), interpreted as
translational moduli. Supersymmetry guarantees the cancelation between the
bosonic and fermionic excited states, leaving only the bosonic and fermion
zero mode integrals to deal with.

The contributions of zero modes to the path integral are taken into account
using the method of collective coordinates, and therefore, we need to
integrate over the moduli space ${\cal M}$. To correctly identify the shape
and size of ${\cal M}$, we carefully examine all gauge invariant observables
in the background of the fractional instanton on $\mathbb{T}^{4}$. This
includes both local gauge-invariant densities and Wilson loops. While gauge-
invariant densities are invariant under translations over a period on
$\mathbb{T}^{4}$, a Wilson loop acquires a $\mathbb{Z}_{2}$
phase.101010Effectively, a Wilson loop measures the $\mathbb{Z}_{2}$ flux of
the twisted solution. Therefore, we find ${\cal M}\sim\mathbb{T}^{4}$ with
double the periods of the physical torus to account for the gauge inequivalent
classes. This result is further supported by investigating the Hamiltonian
formalism of the theory. Interestingly, the metric on ${\cal M}$ is found to
be proportional to the background action and is, hence, independent of
$\Delta$. Putting the pieces together, we finally obtain
$\langle\mbox{tr}\lambda^{2}\rangle=64\pi^{2}\Lambda^{3}$, with a coefficient
four times the expected number on ${\mathbb{R}}^{4}$.

To understand the significance of this result, we need to interpret the
expectation value $\langle\mbox{tr}\lambda^{2}\rangle$ using the Hamiltonian
formalism. Due to the mixed discrete chiral/one-form center anomaly, the
energy eigenstates are doubly degenerate in the background of a ’t Hooft flux
on any torus size: there are $2$ vacua that are exchanged under the operation
of chiral transformation, which explains the extra factor of $2$, while the
relative phase in the condensate in the two vacua is compensated by the ’t
Hooft twist in the Euclidean time direction Cox:2021vsa . Thus, restricting
the condensate to one vacuum, upon taking the limit of large
${\mathbb{T}}^{4}$, we obtain $\langle
0|\mbox{tr}\lambda^{2}|0\rangle=|\langle
1|\mbox{tr}\lambda^{2}|1\rangle|=32\pi^{2}\Lambda^{3}$. This coefficient is
twice the known value on $\mathbb{R}^{4}$ (using the weak-coupling instanton
method) or on $\mathbb{R}^{3}\times\mathbb{S}_{L}^{1}$.

We emphasize that our calculations are performed on a small $\mathbb{T}^{4}$,
compared to the strong scale of the theory, and thus, we are well inside the
semi-classical regime. The calculations in this regime are under analytic
control, thanks to the smallness of the coupling constant. However, to make a
connection to the result on $\mathbb{R}^{4}$, we made a few assumptions. The
invalidation of any of these assumptions can explain the discrepancy between
our result and the expected one. We now list the assumptions that led to our
result:

  1. 1.

We assume that there is a unique fractional instanton on the asymmetric
$\mathbb{T}^{4}$ with topological charge $Q=\frac{1}{2}$. This solution is
nonabelian in nature and is obtained from ’t Hooft’s abelian solution on a
symmetric $\mathbb{T}^{4}$ as an expansion in the detuning parameter $\Delta$
GarciaPerez:2000aiw . However, we were not able to prove the
uniqueness111111To avoid possible confusion, we note that the series in
$\Delta$ determines a unique self-dual configuration for fixed values of the
moduli (see the proof in Appendix A.1, near eqn. (• ‣ A.1)). The possibility
of non-uniqueness mentioned here refers to existence of a genuinely nonabelian
$Q={1\over 2}$ solution not connected to ’t Hooft’s abelian solution. The
numerical study of GarciaPerez:2000aiw  appears to support uniqueness in this
sense, but does not constitute a proof. of the solution. Instead, we rely on
the numerical evidence in GarciaPerez:2000aiw .

  2. 2.

Another lingering issue is the radius of convergence of the expansion, which
is yet to be understood. Although our final result does not depend on
$\Delta$, one needs to be cautious when interpreting the result. In order to
interpret the result and compare with the condensate on $\mathbb{R}^{4}$, we
had to make use of the Hamiltonian formalism. Here, first, one puts the theory
on a small spatial torus $\mathbb{T}^{3}$, with a ’t Hooft twist, and with
periods of length $L\ll\Lambda^{-1}$, and then takes the limit of a large
Euclidean time direction, $L_{4}\gg\Lambda^{-1}$. This is the limit
$\Delta\sim\sqrt{\frac{L_{4}}{L}}\gg 1$, well outside the small-$\Delta$
regime used to compute the condensate.

  3. 3.

After taking $\Delta$ large, one needs to take the size of the remaining
${\mathbb{T}}^{3}$ periods beyond the inverse strong-coupling scale. Thus, we
necessarily leave the semi-classical regime, yet we assume no extra instantons
contribute to the condensate in this limit. (However, we can not help but note
that this is similar, at least in spirit, to the extrapolation of the
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}_{L}$ semiclassical calculation of the
gaugino condensate to ${\mathbb{R}}^{4}$.)

  4. 4.

In carrying out our calculations, we assume that quantum corrections due to
bosonic, fermionic, and ghost fluctuations cancel exactly in our
supersymmetric background, to all loop orders, and for arbitrary
$\mathbb{T}^{4}$ size. Although the cancellation of the determinants to any
loop order is well-established in $\mathbb{R}^{4}$, one may need to examine
this assumption more carefully in the fractional instanton background of
$\mathbb{T}^{4}$.

In view of the above discussion, in the bulk of the paper, we present our
calculations in sufficient detail to help the interested readers follow all
numerical factors and enable them to dwell on the procedures used—and on
improving the interpretation of our result.

###  1.3 Future directions

Here, in lieu of a conclusion, we point out that our study warrants a few
expansion directions:

  1. 1.

An immediate step would be generalizing the $SU(2)$ result to $SU(N\geq 3)$.
Self-dual instantons with topological charge $\frac{1}{N}$, $N\geq 3$, are
necessarily non-abelian solutions of Yang-Mills equations tHooft:1981nnx .
This makes the treatment more involved, especially as we deviate from the
self-dual torus.121212The definition of a self-dual torus in $SU(N)$ is
different from the $SU(2)$ case tHooft:1981nnx . Fortunately, a systematic
analysis to deal with this problem, a generalization of the method in
GarciaPerez:2000aiw , appeared in Gonzalez-Arroyo:2019wpu . One can follow the
same line of thought in our work to compute the condensate in $SU(N)$.

  2. 2.

It is tempting to study the condensates in other gauge groups, e.g., $Sp(N)$
and $Spin(2N)$ groups. These groups have small center groups: $Sp(N)$ has a
$\mathbb{Z}_{2}$ and $Spin(2N)$ has a $\mathbb{Z}_{4}$ or a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ center group depending on whether $N$ is
odd or even. Owing to the small centers, ’t Hooft fluxes (or fractional
instantons) of these groups will, in general, support more than $2$ gaugino
zero modes. This situation is different from the calculations on
$\mathbb{R}^{3}\times\mathbb{S}^{1}_{L}$, where a monopole-instanton always
supports $2$ zero modes for all gauge groups, even those with no center
symmetry. Therefore, at first, it is not clear how the twists and monopoles
are connected. Investigating this problem is important for the continuity
program.

  3. 3.

It is also interesting to carry out our procedure to higher orders in
$\Delta$. The convergence of the series in $\Delta$ may shed more light on the
problem of continuity. In this regard, we note that the closely related work
Gonzalez-Arroyo:2004fmv  in the two-dimensional abelian Higgs model was able
to carry the $\Delta$-expansion to 51-st order, with the results indicating
convergence to the infinite volume limit.

  4. 4.

Another interesting geometry is to consider SYM on
$\mathbb{T}^{2}\times\mathbb{R}^{2}$ with a ’t Hooft flux turned on
$\mathbb{T}^{2}$. This setup was considered previously in Tanizaki:2022ngt .
It was shown via dimensional reduction from
$\mathbb{T}^{2}\times\mathbb{R}^{2}$ to $\mathbb{R}^{2}$ that the theory
admits $\mathbb{Z}_{N}$ vortices and that the gaugino condensate forms in the
$2$-D effective theory $\langle\mbox{tr}\lambda^{2}\rangle\sim\Lambda^{3}$, as
expected. However, the authors did not attempt to compute the numerical
coefficient of the condensate. Such a calculation would mandate a more careful
treatment of the dimensionally reduced $2$-D theory, presumably using the
power of supersymmetry. We also note that $\mathbb{T}^{2}\times\mathbb{R}^{2}$
is the limiting geometry of our asymmetric $\mathbb{T}^{4}$ after taking
$\Delta$ to infinity.

###  1.4 Outline

Our paper is organized as follows:

In Section 2, we formulate the theory and spell out all the necessary
ingredients to define the partition function and the condensate on
$\mathbb{T}^{4}$ with twisted boundary conditions. Further, we explain that a
self-dual torus gives rise to extra, unexpected, zero modes. Then, we present
the solution on the asymmetric $\mathbb{T}^{4}$ in Section 3 and discuss both
the fermion and bosonic zero modes. In Section 4, we calculate the measures of
the bosonic and fermion zero modes. In Section 5, we deviate to the
Hamiltonian formalism to discuss two important aspects. First, we argue that
the moduli space of the bosonic zero modes is isomorphic to $\mathbb{T}^{4}$
with a period size twice the size of the physical period. Next, we recall that
the $0$-form/$1$-form mixed anomaly implies that the energy eigenstates on
${\mathbb{T}}^{3}$ are doubly degenerate and the theory admits $2$ degenerate
vacua. These features are important for the interpretation of our calculation
of the gaugino bilinear. Finally, in Section 6, we put all the pieces together
to obtain our result of the gaugino condensate.

Owing to the mismatch between our and both $\mathbb{R}^{4}$ and
$\mathbb{R}^{3}\times\mathbb{S}^{1}_{L}$ results for the gaugino condensate,
and in order to offer the reader the opportunity to catch mistakes, if any, we
present our rather detailed calculations of various quantities in two
appendices.

In Appendix A, we review in great detail the construction of
GarciaPerez:2000aiw  of the fractional instanton to order $\Delta$, with
emphasis on the dependence on the collective coordinates. We also provide
expressions of the field strengths and Wilson loops in the background of the
leading-order solution.

In Appendix B, we construct the fermion zero modes by directly solving the
Weyl equation as well as by using supersymmetry transformation, and we prove
that both methods yield the same result. Then, we construct the bosonic zero
modes to any order in $\Delta$, first by employing the fermionic zero modes
and then by taking derivatives of the classical background solution w.r.t. the
collective coordinates modulo gauge transformation (we also determine these
gauge transformations). Finally, we determine the Jacobian of the bosonic zero
modes moduli space needed to complete the calculations.

##  2 Fractional instantons on the symmetric torus

We consider the $SU(2)$ SYM theory on $\mathbb{T}^{4}$ with periods of lengths
$L_{1},L_{2},L_{3},L_{4}$. The Euclidean action of the theory is given
by131313The Euclidean action, supersymmetry transformations, and the matrices
$\sigma_{n}$, $\bar{\sigma}_{n}$, $\sigma_{mn}$, $\bar{\sigma}_{mn}$, are as
in Dorey:2002ik , except that we use hermitean gauge fields, necessitating the
replacement $A^{\text{that ref.}}=iA^{\text{this paper}}$. See also Appendices
A.1 and B.2.

$\displaystyle
S_{SYM}=\frac{1}{g^{2}}\int_{\mathbb{T}^{4}}\mbox{tr}\left[{1\over
2}F_{mn}F_{mn}+{2}(\partial_{n}\bar{\lambda}_{\dot{\alpha}}+i[A_{n},\bar{\lambda}_{\dot{\alpha}}])\bar{\sigma}_{n}^{\dot{\alpha}\alpha}\lambda_{\alpha}\right]\,,$
(1)

and $\lambda$ is a left-handed adjoint Weyl fermion, the gaugino.
$D_{n}=\partial_{n}+i[A_{n},\,]$ is the covariant derivative,
$\sigma_{n}\equiv(i\vec{\sigma},1)$,
$\bar{\sigma}_{n}\equiv(-i\vec{\sigma},1)$, $\vec{\sigma}$ are the Pauli
matrices, and the Latin letters $n,m$ run over $1,2,3,4$. The field strength
is given by $F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}+i[A_{m},A_{n}]$. This
action is invariant under the supersymmetry transformations

$\displaystyle\delta A_{n}$ $\displaystyle=$
$\displaystyle\zeta^{\alpha}\;\sigma_{n\;\alpha\dot{\alpha}}\;\bar{\lambda}^{\dot{\alpha}}+\bar{\zeta}_{\dot{\alpha}}\;\bar{\sigma}_{n}^{\dot{\alpha}\alpha}\;\lambda_{\alpha}\,,\quad\delta\lambda_{\alpha}=-\sigma_{mn\;\alpha}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\beta}\;\zeta_{\beta}\;F_{mn}\,,\quad\delta\bar{\lambda}^{\dot{\alpha}}=-\bar{\sigma}^{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \dot{\alpha}}_{mn\;\;\leavevmode\nobreak\
\leavevmode\nobreak\ \dot{\beta}}\;\bar{\zeta}^{\dot{\beta}}\;F_{mn}\,,$

where the spinors obey $\xi^{1}=\xi_{2},\xi^{2}=-\xi_{1}$ and likewise for the
dotted ones. The equations of motion that result from the variation of
$S_{SYM}$ are

$\displaystyle(D_{m}F_{mn})^{A}=-i\;\mbox{tr}\;\bar{\lambda}\bar{\sigma}_{n}[T^{A},\lambda]\,,\quad\bar{\sigma}_{n}^{\dot{\alpha}\alpha}D_{n}\lambda^{A}_{\alpha}=0\,,\quad\sigma_{n\;\alpha\dot{\alpha}}D_{n}\bar{\lambda}^{A\dot{\alpha}}=0,$
(3)

where $A=1,2,3$ labels the color group generators $T^{A}=\tau^{A}/2$ with
$\tau^{A}$ the Pauli matrices. We shall consider SYM with twisted boundary
conditions on $\mathbb{T}^{4}$. Without loss of generality, we can use the
following transition functions:

$\Omega_{2}(x)=e^{-i2\pi{x_{1}\over L_{1}}{\tau^{3}\over
2}},\;\;\Omega_{4}(x)=e^{-i2\pi{x_{3}\over L_{3}}{\tau^{3}\over
2}},\leavevmode\nobreak\ \text{while}\leavevmode\nobreak\
\Omega_{1}=\Omega_{3}=1.$ (4)

$\Omega_{2}$ and $\Omega_{4}$ implement the twists along the $x_{2}$ and
$x_{4}$ directions, while the transition functions along the $x_{1}$ and
$x_{3}$ directions are trivial. The transition functions obey the cocycle
conditions

$\Omega_{i}(x+L_{j}\hat{e}_{j})\;\Omega_{j}(x)=e^{i\pi
n_{ij}\tau_{3}}\;\Omega_{j}(x+L_{i}\hat{e}_{i})\;\Omega_{i}(x),\leavevmode\nobreak\
i,j=1,2,3,4,\;\forall x\in{\mathbb{R}}^{4},$ (5)

where $\hat{e}_{n}$ is a unit vector in the $x_{n}$ direction,
$n_{12}=n_{34}=-n_{21}=-n_{43}=1$, and the rest of $n_{ij}$ are zeros. The
periodicity condition on the gauge fields and gaugino in ${\mathbb{R}}^{4}$
defines the ${\mathbb{T}}^{4}$ fields:

$\displaystyle A(x+\hat{e}_{n}L_{n})$ $\displaystyle=$
$\displaystyle\Omega_{n}(x)(A(x)-id)\Omega_{n}^{-1}(x),\leavevmode\nobreak\
n=1,2,3,4,\leavevmode\nobreak\ \forall x\in{\mathbb{R}}^{4},$
$\displaystyle\lambda(x+\hat{e}_{n}L_{n})$ $\displaystyle=$
$\displaystyle\Omega_{n}(x)\lambda(x)\Omega_{n}^{-1}(x)\,,$ (6)

where we denoted $A(x)=\sum\limits_{n=1}^{4}A_{n}(x)dx_{n}$. Let the $SU(2)$
non-abelian field $\Phi$ denote either the gauge field or the gaugino and
expand $\Phi$ using the Cartan-Weyl basis:

$\displaystyle\Phi(x)=\Phi^{3}\frac{\tau^{3}}{2}+\Phi^{+}\tau^{+}+\Phi^{-}\tau^{-}\,.$
(7)

The Pauli matrices $\tau^{3},\tau^{\pm}$ are the generators of $SU(2)$ in this
basis. Using the commutation relations $[\tau^{3},\tau^{\pm}]=\pm\tau^{3}$,
$[\tau^{+},\tau^{-}]=\tau^{3}$ along with the Baker-Campbell-Hausdorff
formula, one finds that the boundary conditions on $\Phi$ are satisfied if and
only if:

$\displaystyle\Phi^{3}(x_{i}+L_{i})$ $\displaystyle=$
$\displaystyle\Phi^{3}(x)+{\cal I}\,,$
$\displaystyle\Phi^{\pm}(x_{1}+L_{1},x_{2},x_{3},x_{4})$ $\displaystyle=$
$\displaystyle\Phi^{\pm}(x_{1},x_{2},x_{3},x_{4})\,,$
$\displaystyle\Phi^{\pm}(x_{1},x_{2}+L_{2},x_{3},x_{4})$ $\displaystyle=$
$\displaystyle e^{\mp i\frac{2\pi
x_{1}}{L_{1}}}\Phi^{\pm}(x_{1},x_{2},x_{3},x_{4})\,,$ (8)

where ${\cal I}$ is a nonhomogeneous term that contributes to the gauge field
and appears upon shifting the $x_{2}$ coordinate:
$A_{n}^{3}(x+\hat{e}_{2}L_{2})=A_{n}^{3}(x)+\frac{2\pi}{L_{1}}\delta_{n,1}$.
Similar boundary conditions hold in the 3-4 plane. As we shall show, there is
a classical solution to the equations of motion (3), which satisfies the
boundary conditions (8) and has a topological charge $Q=\frac{1}{2}$.

The Euclidean partition function of our system is given by the path integral:

$\displaystyle{\cal
Z}_{SYM}[n_{12}=1,n_{34}=1]=\sum_{\nu\in\mathbb{Z}}\int[DA_{\mu}][D\lambda][D\bar{\lambda}]e^{-S_{SYM}-i(\nu+{1\over
2})\theta}|_{n_{12}=1,n_{34}=1}\,,$ (9)

where we have emphasized that this path integral is to be computed with the
twists in the 1-2 and 3-4 plane, as per the transition functions (4), and
thus, all the fields need to satisfy the boundary conditions (8). Notice that
one also needs to sum over the integer topological sectors
$\nu\in{\mathbb{Z}}$ in order for ${\cal Z}_{SYM}$ to respect locality and
unitarity.141414If $\nu\neq 0$, the transition functions (4) (but not the
cocycle conditions (5)) have to be appropriately modified. The vacuum angle
$\theta$ can be transformed away by applying an axial rotation on the
gauginos. Our main aim is to find the bilinear gaugino condensate starting
from (9).

The solution of ${\cal Z}_{SYM}$, as well as the condensate, will proceed by
using the semi-classical techniques, which amount to computing the path
integral as the sum of paths of small fluctuations in the background of
instantons. The semiclassical approach is justified in the limit of small
volume.

Notice, however, that this partition function vanishes identically, which can
be understood in two different ways. First, as we shall show, the theory
exhibits $2$ fermion zero modes. These are the zero modes that are saturated
in the twisted background and give rise to the fermion condensate. We can also
argue that ${\cal Z}_{SYM}=0$ because of the mixed anomaly between the
$0$-form discrete chiral and $1$-form center symmetries, as was previously
shown in Cox:2021vsa .

The gaugino condensate calculations can proceed by inserting the gaugino
bilinear $\,{\rm tr}\,\lambda\lambda$ in the path integral (9), where the
trace is taken in the color space. We define the expectation value of the
gaugino condensate as:

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle={\cal
N}^{-1}\sum_{\nu\in\mathbb{Z}}\int[DA_{\mu}][D\lambda][D\bar{\lambda}]\;\,{\rm
tr}\,\lambda\lambda\;e^{-S_{SYM}-i(\nu+{1\over
2})\theta}|_{n_{12}=1,n_{34}=1}\,,$ (10)

with normalization constant ${\cal N}$. Only the $\nu=0$ sector (in the
presence of the twists) can contribute to the bilinear condensate, and thus,
we restrict our calculations to this sector. The standard way of normalizing
the expectation value of an operator is to divide by the partition function.
Since ${\cal Z}_{SYM}=0$, we need to search for another appropriate way to
normalize our physical observables. We choose to divide by the partition
function with twists only in the 1-2 plane. Explicitly,

$\displaystyle{\cal
N}=\sum_{\nu\in\mathbb{Z}}\int[DA_{\mu}][D\lambda][D\bar{\lambda}]\;e^{-S_{SYM}-i\nu\theta}|_{n_{12}=1}\,.$
(11)

This partition function is saturated by the $\nu=0$ term. In the
small-${\mathbb{T}}^{4}$ limit, it can be evaluated using a semiclassical
expansion around two zero-action classical saddle point configurations,
related by the ${\mathbb{Z}}_{2}$ center symmetry in the $x_{3}$ direction,
generated by the “improper gauge transformation” $T_{3}$. In a gauge with
constant transition functions151515These zero-action configurations can also
be exhibited in a gauge where $\Omega_{2}$ is as in (4), with
$\Omega_{1,3,4}=1$, but details are to be given elsewhere. in the
$x_{1}$-$x_{2}$ plane, these saddle points are $A=0$ and
$A=-iT_{3}dT_{3}^{-1}$ Witten:1982df . Each of these saddle points gives an
identical contribution to ${\cal N}$. There are no zero modes (see
GonzalezArroyo:1987ycm  for calculations of the spectrum) and all
determinants, fermionic and bosonic, are the ones computed in the $A=0$
background (we note that this is the normalization of instanton transition
amplitudes already taken in tHooft:1976snw ). The partition function (11), up
to a normalization factor that is expected to cancel the numerical factor in
the numerator of (10), is equal to the Witten index $\mbox{Tr}[(-1)^{F}]$,
which counts the number of the ground states. Thus, we take ${\cal N}=2$. See
Section 5 for a Hamiltonian treatment.

As a first trial, let us find a self-consistent $Q=1/2$ fractional-instanton
solution to the equations of motion (3) with the boundary conditions (8). It
is easy to see that the abelian gauge configuration:

$\displaystyle\bar{A}_{1}^{3}$ $\displaystyle=$ $\displaystyle{2\pi x_{2}\over
L_{1}L_{2}}+{z_{1}\over L_{1}}\,,\quad\bar{A}_{2}^{3}={z_{2}\over L_{2}}\,,$
$\displaystyle\bar{A}_{3}^{3}$ $\displaystyle=$ $\displaystyle{2\pi x_{4}\over
L_{3}L_{4}}+{z_{3}\over L_{3}}\,,\quad\bar{A}_{4}^{3}={z_{4}\over L_{4}}\,,$
(12)

obeys (8). The bar is introduced here to serve a later convenience, and the
constants $z_{m}$ are the collective coordinates, which will be set to zero in
this section without loss of generality. This field configuration was first
found by ’t Hooft, and it solves the Pure Yang-Mills equations
$D_{m}F_{mn}=0$, owing to its abelian nature and constant field strength. It
carries a topological charge $Q=\frac{1}{2}$ and its action is half the action
of a BPST instanton: $S=S_{0}\equiv\frac{4\pi^{2}}{g^{2}}$. This solution must
be self-dual or anti-self-dual, $F_{mn}=\pm\frac{1}{2}\epsilon_{mnpq}F_{pq}$,
which guarantees that the bosonic and fermionic determinants in this
background do not yield negative zero modes, a sign of instability of the
solution on $\mathbb{T}^{4}$. Self-duality of the $Q=1/2$ solution (12)
implies the condition

$\displaystyle L_{1}L_{2}=L_{3}L_{4}\,.$ (13)

A torus that satisfies the relation (13) is said to be self-dual, and a simple
choice that satisfies the condition (13) is the symmetric torus with
$L_{1}=L_{2}=L_{3}=L_{4}$. The classical solution (12) will also hold in SYM
provided that the right-hand side of the first equation in (3) vanishes. To
check that, we first need to solve the Weyl equations (second and third
equations in (3)) in the background (12), along with the boundary conditions
(8). We find $6$-independent zero modes: $4$ modes for $\lambda$ and $2$ modes
for $\bar{\lambda}$:

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle\frac{\tau^{3}}{2}\left[\begin{array}[]{c}\xi_{1}\\\
\xi_{2}\end{array}\right]+\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over
L_{3}})}h_{0}^{12}(x_{2}-n_{1}L_{2})\;h_{0}^{34}(x_{4}-n_{3}L_{4})\tau^{+}\left[\begin{array}[]{c}0\\\
\xi_{3}\end{array}\right]$ (21)
$\displaystyle\quad\quad\quad\quad+\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over
L_{3}})}h_{0}^{12}(x_{2}-n_{1}L_{2})\;h_{0}^{34}(x_{4}-n_{3}L_{4})\tau^{-}\left[\begin{array}[]{c}\xi_{4}\\\
0\end{array}\right]\,,$ $\displaystyle\bar{\lambda}$ $\displaystyle=$
$\displaystyle\frac{\tau^{3}}{2}\left[\begin{array}[]{c}\xi_{5}\\\
\xi_{6}\end{array}\right]\,,$ (24)

where $\xi_{i}$ are Grassmann numbers. The functions $h_{0}^{12}$ and
$h_{0}^{34}$ are the ground eigenstates of the simple harmonic oscillator, of
frequencies $\omega_{12}={2\pi\over L_{1}L_{2}}$ and $\omega_{34}={2\pi\over
L_{3}L_{4}}$, respectively; see Appendices A.1 and B.2.3. Owing to the abelian
nature of the classical background, the Weyl equation yields the solutions in
the $\tau^{3}$ directions. Substituting the fermion zero mode solutions into
the right-hand side of the bosonic equation of motion in (3), one easily finds
that the source term is nonzero. This simple exercise shows that the abelian
background (12) is either inconsistent or that some gaugino zero modes must be
lifted by higher-order corrections in order to have
$\mbox{tr}\bar{\lambda}\bar{\sigma}^{n}[T^{A},\lambda]=0$. Presumably, higher-
order corrections will gap $4$ of the fermion zero modes, and in the end, we
will have a consistent story. We were not able to show that this is the case
in a satisfactory way.161616For completeness, we note that an analogous
problem occurs with the bosonic zero modes around the solution (12) on the
self-dual torus: there are 8 (four real and two complex), rather than 4,
bosonic zero modes, as was shown long ago by van Baal vanBaal:1984ar . We also
stress that the existence of the four undotted and two dotted fermionic zero
modes (21) is consistent with the index theorem, which only determines their
difference.

Instead, we chose to detune the self-dual torus, i.e., to relax the condition
$L_{1}L_{2}=L_{3}L_{4}$, and modify the abelian background to include non-
abelian pieces that are needed to guarantee the self-duality of the fractional
instanton on an asymmetric ${\mathbb{T}}^{4}$. The detuned solution does not
suffer from any of the above-mentioned problems. A further motivation for
considering the asymmetric ${\mathbb{T}}^{4}$ connects to interesting
semiclassical limits that have been considered.

##  3 Fractional instantons and gaugino zero modes on the asymmetric torus

In order to circumvent the problems of the self-dual $\mathbb{T}^{4}$, we
instead search for a self-dual instanton on an asymmetric torus using a
perturbation technique introduced in GarciaPerez:2000aiw . The gauge field
$A_{n}$ can be written in the general form

$\displaystyle A_{n}^{cl}(x,z)=(\bar{A}_{n}^{3}(x,z)+S_{n}(x,z)){\tau^{3}\over
2}+W_{n}(x,z)\tau^{+}+W_{n}^{*}(x,z)\tau^{-}\,,$ (25)

where we have split the $\tau^{3}$ component into two parts. The first part
$\bar{A}_{n}^{3}(x,z)$ is the abelian background (12) that solves the
sourceless Yang-Mills equations on the symmetric $\mathbb{T}^{4}$. The
functions $S_{n}(x,z)$ and $W_{n}(x,z)$ will be determined perturbatively. We
also introduce the dimensionless detuning parameter
$\Delta\equiv\left(L_{3}L_{4}-L_{1}L_{2}\right)/\sqrt{L_{1}L_{2}L_{3}L_{4}}$,
which measures the deviation from the self-dual torus, and take $\Delta>0$.
The solution of the sourceless Yang-Mills equations on the asymmetric
$\mathbb{T}^{4}$ is obtained by imposing the self-duality condition
$F_{mn}=\frac{1}{2}\epsilon_{mnpq}F_{pq}$, where $\epsilon_{1234}=1$, etc. In
order to reduce the gauge redundancy, one also imposes the background gauge
condition $\partial_{n}A_{n}^{cl}+i[\bar{A}_{n},A_{n}^{cl}]=0$.

The details of the construction of the self-dual solution in an expansion in
powers of $\Delta$ is given in Appendix A.1. The presentation there follows
GarciaPerez:2000aiw  but for completeness we present it in detail using our
notation, with an emphasis on the dependence of the solution of the $z_{n}$
variables. Here, we only give an idea of the construction of the fractional
instanton and present its main features. To simplify the equations, it is
convenient to use a quaternion notation. Thus, we introduce the matrices:

$\displaystyle w=\sigma^{n}W_{n}\,,\quad w_{c}=Cw^{*}C\,,\quad
s=\sigma^{n}S_{n}\,,$ (26)

where $\sigma^{n}=(i\vec{\sigma},1_{2\times 2}),\leavevmode\nobreak\
\bar{\sigma}^{n}=(\sigma^{n})^{\dagger}$ and
$C=\left(\begin{array}[]{cc}0&-i\cr i&0\end{array}\right)$. Self-duality and
the background gauge conditions (see (101)) then yield the following
equations:

$\displaystyle\bar{\sigma}^{n}\partial_{n}s={2\pi{\Delta}\over\sqrt{V}}i\tau^{3}-i(w_{c}^{\dagger}w_{c}-w^{\dagger}w)\,,\quad\bar{\sigma}^{n}(\partial_{n}+i\bar{A}_{n}^{3})w=-{i\over
2}(s^{\dagger}w-w_{c}^{\dagger}s)\,.$ (27)

These equations are subject to the boundary conditions (8). The solutions to
(27) is found as series expansions in $\sqrt{\Delta}$:

$\displaystyle w=\Delta^{1/2}\sum_{j=0}^{\infty}w_{j}\Delta^{j}\,,\quad
s=\sum_{j=1}^{\infty}s_{j}\Delta^{j}\,.$ (28)

The symmetry structure of the self-duality equations (27), as well as their
solution to the leading order in $\Delta$, is discussed in Appendix A.1. The
final answer for the fractional instanton solution, giving the order
$\sqrt{\Delta}$ terms in (25), reads:

$\displaystyle W_{1}(x,z,\alpha)$ $\displaystyle=$ $\displaystyle-{i\over
2}\sqrt{\Delta}F(x,z)e^{i\alpha}{\sqrt{2\pi}\over
V^{1/4}}+{\cal{O}}(\Delta^{3/2})$ $\displaystyle W_{2}(x,z,\alpha)$
$\displaystyle=$ $\displaystyle{1\over
2}\sqrt{\Delta}F(x,z)e^{i\alpha}{\sqrt{2\pi}\over
V^{1/4}}+{\cal{O}}(\Delta^{3/2})=iW_{1}+{\cal{O}}(\Delta^{3/2}),$
$\displaystyle W_{3}$ $\displaystyle=$
$\displaystyle{\cal{O}}(\Delta^{3/2})\,,\quad
W_{4}={\cal{O}}(\Delta^{3/2})\,,\quad S_{n}={\cal{O}}(\Delta)\,.$ (29)

We stress that the ${\cal{O}}(\Delta)$ contribution to $S_{n}$ is determined
by the ${\cal{O}}(\sqrt{\Delta})$ terms shown above, in a manner described in
Appendix A.1, but explicit expressions will not be needed. The arbitrary phase
$\alpha$ is due to the gauge freedom to rotate around the $\tau^{3}$ isospin
direction, and the function $F(x,z)$ is given by the expression:

$\displaystyle F(x,z)$ $\displaystyle=$
$\displaystyle\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i{z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})-i{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4})}$ (30)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \times\leavevmode\nobreak\
\leavevmode\nobreak\ h_{0}^{12}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2})\;h_{0}^{34}(x_{4}-(n_{3}-{z_{3}\over
2\pi})L_{4})\leavevmode\nobreak\ ,$

with the normalization $\int_{\mathbb{T}^{4}}|F|^{2}=\prod_{i=1}^{4}L_{i}=V$.
The functions $h_{0}^{12,34}$ are the same harmonic oscillator ground state
wave functions appearing in eqn. (21) and described there.

It remains to check that this solution obeys the equations of motion (3), and
in particular, that the fermionic source in the bosonic equation of motion
vanishes identically. To this end, we need to find the fermion zero modes,
which we do in Section 3.1. For this purpose, we will need to use the field
strength, a computation carried out in detail in Appendix A.2. We find that to
order ${\cal O}(\sqrt{\Delta})$:

$\displaystyle F_{12}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{L_{1}L_{2}}\,,\quad
F_{34}=\frac{2\pi}{L_{3}L_{4}}\,,$ $\displaystyle F_{13}$ $\displaystyle=$
$\displaystyle-i\gamma^{*}e^{i\alpha}\sqrt{2\pi\over
2L_{3}L_{4}}G(x,z)\tau^{+}+i\gamma e^{-i\alpha}\sqrt{2\pi\over
2L_{3}L_{4}}G^{*}(x,z)\tau^{-},$ $\displaystyle F_{14}$ $\displaystyle=$
$\displaystyle\gamma^{*}e^{i\alpha}\sqrt{2\pi\over
2L_{3}L_{4}}G(x,z)\tau^{+}+\gamma e^{-i\alpha}\sqrt{2\pi\over
2L_{3}L_{4}}G^{*}(x,z)\tau^{-}\,,$ (31)

where $\gamma={i\over 2}\sqrt{2\pi\Delta\over\sqrt{V}}$ and the function
$G(x,z)$ is

$\displaystyle G(x,z)$ $\displaystyle=$
$\displaystyle\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i{z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})-i{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4})}$ (32)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \times\leavevmode\nobreak\
\leavevmode\nobreak\ h_{0}^{12}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2})\;h_{1}^{34}(x_{4}-(n_{3}-{z_{3}\over 2\pi})L_{4})\,.$

The function $h_{1}$ is the first excited state of the simple harmonic
oscillator, and just like $F(x,z)$, $G$ is dimensionless and similarly
normalized $\int_{\mathbb{T}^{4}}|G|^{2}=\prod_{i=1}^{4}L_{i}=V$. To find the
field strength to order ${\cal O}(\Delta)$ and establish the self-duality of
the solution, one needs to solve for the functions $S_{n}$, see (161). In
Appendix A.1, we discuss the relations obeyed by the field strength of $S_{n}$
(explicit expressions will not be needed in our work). These are important for
establishing the self-duality of the solution, and thus of the fact that its
action $S_{0}$ saturates the BPS bound, $S_{0}={4\pi^{2}\over g^{2}}$.

Finally, we stress that the solutions, whose construction via a small-$\Delta$
expansion was described above, have been subjected to comparison with “exact”
solutions obtained by numerically minimizing the lattice Yang-Mills theory
action with an $n_{12}=n_{34}=1$ twist. A good qualitative (and in some cases
quantitative) agreement of the gauge invariants charactering the solution,
computed analytically171717This is also done in our Appendix A.3, where the
$z_{n}$ dependence of the local and non-local gauge invariants characterizing
the solution is discussed in detail. to ${\cal{O}}(\Delta)$, with the
numerical approximation of the exact solution was found, for $\Delta$ in the
range $0.02-0.09$ for various lattice sizes, see GarciaPerez:2000aiw  for a
detailed discussion.

The existence of a consistent solution in the form (28) to all orders in
$\Delta$ has been shown in GarciaPerez:2000aiw , but explicit calculations
beyond the leading order have not yet been performed. The radius of
convergence in $\Delta$ is also not known. In what follows, we assume that the
all-order $\Delta$-expansion gives rise to a unique self-dual solution of
action $S_{0}={4\pi^{2}\over g^{2}}$, at least for sufficiently small
$\Delta$. This is supported by the consistency of (28) and the agreement with
numerical tests GarciaPerez:2000aiw .

###  3.1 Fermion zero modes

The fermion zero modes can be found by solving the Weyl equations from (3),
$D_{n}\bar{\sigma}_{n}\lambda=0\,,D_{n}\sigma_{n}\bar{\lambda}=0$ in the
background (25). We perform these explicit computations in Appendix B.2.2.

The fermion zero modes can also be obtained via the supersymmetry
transformations (LABEL:susyhermitean1bulk), with the result agreeing with the
direct solution of the Weyl equations. Consider the effect of the
supersymmetry transformation (LABEL:susyhermitean1bulk) in the background of
the bosonic solution (25), with fermions set to zero,
$\bar{\lambda}=\lambda=0$. Since our solution is self-dual, i.e. obeys
$\bar{\sigma}^{mn}F_{mn}=0$, the SUSY transformation only produces
$\lambda_{\alpha}$ variations:

$\displaystyle\delta\lambda=-\sigma_{mn}F_{mn}\zeta\,,\quad\delta\bar{\lambda}=0\,,$
(33)

which is true to all orders in $\Delta$. Computing $\delta\lambda$ we obtain,
see Appendix B.2.4, to ${\cal O}(\sqrt{\Delta})$:

$\displaystyle\left(\begin{array}[]{c}\delta\lambda_{1}\cr\delta\lambda_{2}\end{array}\right)$
$\displaystyle=$
$\displaystyle-2iF_{12}^{(0)}\left(\begin{array}[]{c}\zeta_{1}\cr-\zeta_{2}\end{array}\right)+2F_{13}^{(1)}\left(\begin{array}[]{c}\zeta_{2}\cr-\zeta_{1}\end{array}\right)-i2F_{14}^{(1)}\left(\begin{array}[]{c}\zeta_{2}\cr\zeta_{1}\end{array}\right)$
(42) $\displaystyle=$
$\displaystyle\left[\left(\begin{array}[]{c}\eta_{1}\cr\eta_{2}\end{array}\right){\tau^{3}\over
2}+{V^{1\over 4}\over\pi^{1\over
2}}\gamma^{*}e^{i\alpha}G(x,z)\left(\begin{array}[]{c}\eta_{2}\cr
0\end{array}\right)\tau^{+}+{V^{1\over 4}\over\pi^{1\over 2}}\gamma
e^{-i\alpha}G^{*}(x,z)\left(\begin{array}[]{c}0\cr\eta_{1}\end{array}\right)\tau^{-}\right]\,,$
(49)

where in going from the first to second line we used the definitions
$\eta_{1}\equiv 4\pi V^{-1/2}\zeta_{1}$, $\eta_{2}\equiv-4\pi
V^{-1/2}\zeta_{2}$. Thus, as expected, we find $2$ fermion zero modes in
accordance with the index theorem. One can easily see that
$\mbox{tr}\bar{\lambda}\bar{\sigma}_{n}[T^{A},\lambda]=0$, and hence, the
self-dual instanton (25) on the asymmetric $\mathbb{T}^{4}$ solves the
sourceless equations of motion $D_{m}F_{mn}=0$ and is a consistent background
for SYM.

###  3.2 Bosonic zero modes

For every fermionic zero mode

$\displaystyle\phi_{\alpha}^{\;(\beta)\;A}=-(\sigma_{mn})_{\alpha}^{(\beta)}F_{mn}^{A},\beta=1,2,A=1,2,3,$
(51)

there are two bosonic zero modes. Thus, in total, there are four independent
bosonic zero modes. The advantage of the discussion that follows is that the
bosonic zero modes automatically obey the background gauge condition and,
furthermore, their construction holds to arbitrary orders in $\Delta$.

The four-vector expressions for the bosonic zero modes are denoted by
$Z_{n}^{(\beta)\;A}$ and $Z_{n}^{(\beta\;^{\prime})\;A}$, where
$\beta,\beta^{\prime}={1,2}$. These modes are determined as described in e.g.
Vandoren:2008xg ; Dorey:2002ik : from each zero mode $\phi_{\alpha}^{(\beta)}$
of the undotted Dirac equation one builds two four-vector bosonic zero modes,
denoted by $Z_{n}^{(\beta)}$ and $Z_{n}^{(\beta\;^{\prime})}$ (from here on,
we suppress the Lie-algebra index $A$ to reduce clutter). Their four-vector
components are then (see Appendix B.3.2, eqn. (266)):

$\displaystyle Z_{n}^{(\beta)}$ $\displaystyle=$
$\displaystyle\left\\{\Im\phi_{2}^{(\beta)},-\Re\phi_{2}^{(\beta)},\Im\phi_{1}^{(\beta)},\Re\phi_{1}^{(\beta)}\right\\},\leavevmode\nobreak\
Z_{n}^{(\gamma\;^{\prime})}=\left\\{\Re\phi_{2}^{(\gamma)},\Im\phi_{2}^{(\gamma)},\Re\phi_{1}^{(\gamma)},-\Im\phi_{1}^{(\gamma)}\right\\}.$

Using the expression of $\phi_{\alpha}^{(\beta)}$ in terms of $F_{mn}$, we can
also express $Z_{n}^{(\beta)}$ and $Z_{n}^{(\gamma\;^{\prime})}$ in terms of
$F_{mn}$:

$\displaystyle Z_{n}^{(1)}$ $\displaystyle=$
$\displaystyle\left\\{\Im\phi_{2}^{(1)},-\Re\phi_{2}^{(1)},\Im\phi_{1}^{(1)},\Re\phi_{1}^{(1)}\right\\}=\left\\{-2F_{14},2F_{13},-2F_{12},0\right\\},$
$\displaystyle Z_{n}^{(2)}$ $\displaystyle=$
$\displaystyle\left\\{\Im\phi_{2}^{(2)},-\Re\phi_{2}^{(2)},\Im\phi_{1}^{(2)},\Re\phi_{1}^{(2)}\right\\}=\left\\{2F_{12},0,-2F_{14},2F_{13}\right\\},$
$\displaystyle Z_{n}^{(1\;^{\prime})}$ $\displaystyle=$
$\displaystyle\left\\{\Re\phi_{2}^{(1)},\Im\phi_{2}^{(1)},\Re\phi_{1}^{(1)},-\Im\phi_{1}^{(1)}\right\\}=\left\\{-2F_{13},-2F_{14},0,2F_{12}\right\\},$
$\displaystyle Z_{n}^{(2\;^{\prime})}$ $\displaystyle=$
$\displaystyle\left\\{\Re\phi_{2}^{(2)},\Im\phi_{2}^{(2)},\Re\phi_{1}^{(2)},\Im\phi_{1}^{(2)}\right\\}=\left\\{0,2F_{12},2F_{13},2F_{14}\right\\}\,.$
(52)

It is a simple exercise to check $D_{n}Z_{n}^{(\beta)}=0$ and
$D_{n}Z_{n}^{(\gamma\;^{\prime})}=0$, and thus, these zero modes solve the
classical equations of motion as expected.

Further, one can write down $Z_{n}^{(\beta)}$ and $Z_{n}^{(\gamma^{\prime})}$
as the derivatives with respect to the collective coordinates $\\{z_{i}\\}$
modulo gauge transformations. To this end, we define new zero modes
$Y_{n}^{(i)}$, $i=1,2,3,4$, by relabeling (3.2) as follows

$\displaystyle Y_{n}^{(3)}\equiv Z_{n}^{(1)}\,,\quad Y_{n}^{(1)}\equiv-
Z_{n}^{(2)}\,,\quad Y_{n}^{(4)}\equiv-Z_{n}^{(1\;^{\prime})}\,,\quad
Y_{n}^{(2)}\equiv-Z_{n}^{(2\;^{\prime})}\,,$ (53)

and show in Appendix B.3.3 that

$\displaystyle Y_{n}^{(k)}=\frac{4\pi L_{k}}{\sqrt{V}}\frac{\partial
A_{n}}{\partial z_{k}}+D_{n}(A^{cl})\Lambda^{(k)}\,.$ (54)

There, we also explicitly find the expressions for the background-gauge
restoring gauge transformations $\Lambda^{(k)}$ to order ${\sqrt{\Delta}}$ and
show that they obey boundary conditions preserving the transition functions.
For later convenience, we also define the inner product of the zero-mode wave
functions (or moduli space metric)

$U^{kl}={2\over g^{2}}\int_{\mathbb{T}^{4}}\,{\rm tr}\,Y_{n}^{(k)}Y_{n}^{(l)}$
(55)

which in diagonal basis is simply $U^{kl}=\delta_{lk}u_{l}$. Using (3.2), see
Appendix B.3.2 for details, we readily obtain

$U^{kl}=\frac{4\delta_{kl}}{g^{2}}\int_{\mathbb{T}^{4}}\,{\rm
tr}\,F_{mn}\tilde{F}_{mn}=4\delta_{kl}S_{0}=\delta^{kl}\;{16\pi^{2}\over
g^{2}}\,.$ (56)

This result is valid to all orders in $\Delta$, since the action of a
fractional instanton does not depend on the size or shape of $\mathbb{T}^{4}$.

##  4 The path integral: bosonic and fermionic measures

The contributions from the non-zero modes of fermions, bosons, and ghosts
cancel in the path integral, thanks to supersymmetry. Thus, in our subsequent
discussion, we only discuss the contribution from zero modes to the bosonic
and fermionic measures.

###  4.1 Fermionic measure

The fermion zero modes measure will be inferred from the one for the non-zero
modes. We expand the fermions as eigenfunctions of the second order Hermitian
operators

$\displaystyle D\bar{D}$ $\displaystyle=$ $\displaystyle
D^{2}+iF_{mn}\sigma^{mn}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-(D\bar{D})_{\alpha}^{\leavevmode\nobreak\
\beta}\lambda_{\beta}=\omega^{2}\lambda_{\beta}\leavevmode\nobreak\ $ (57)
$\displaystyle\bar{D}D$ $\displaystyle=$ $\displaystyle
D^{2}+iF_{mn}\bar{\sigma}^{mn}\leavevmode\nobreak\ =D^{2},\leavevmode\nobreak\
\leavevmode\nobreak\
-D^{2}\bar{\lambda}^{\dot{\beta}}=\omega^{2}\bar{\lambda}^{\dot{\beta}}\leavevmode\nobreak\
,$

where, in the second line, we used the self-duality of the background.

To discuss the measure, we begin by considering the contribution of a single
nonzero eigenvalue $\omega$ to the fermion path integral. Let
$-(D\bar{D})_{\alpha}^{\;\;\beta}\phi_{\beta}^{i}=\omega^{2}\phi_{\alpha}^{i}$,
where $i$ labels the different eigenfunctions, the commuting functions
$\phi_{\alpha}^{i}$, with the same eigenvalue $\omega$ (we note that there are
at least two of them). We expand the nonzero-mode part of the fermion field
(for brevity, denoting it with the same letter $\lambda$, $\bar{\lambda}$)

$\displaystyle\lambda_{\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i}\chi^{i}\;\phi^{i}_{\alpha}$ (58)
$\displaystyle\bar{\lambda}^{\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\sum_{i}\bar{\chi}^{i}\;{1\over\omega}\bar{D}^{\dot{\alpha}\alpha}\phi_{\alpha}^{i},$

where we used the fact that the nonzero eigenfunctions of $D\bar{D}$ and
$\bar{D}D$ are related as shown and we attach the spinor index to the bosonic
solution of the 2nd order equation and not to the Grassmann variable,
$\chi^{i}$ or $\bar{\chi}^{i}$ (the fact that there is more than a single
solution for every $\omega$ is accounted by the index $i$). We also indicate
that the $\lambda$ and $\bar{\lambda}$ expansions have each their separate
Grassmann variables $\chi^{i},\bar{\chi}^{i}$.

Plugging (58) into the fermionic action (1), we obtain after integration by
parts and using the fact that $\phi$ is an eigenvector of $D\bar{D}$:

$\displaystyle S_{F}={2\over g^{2}}\,{\rm
tr}\,(D_{n}\bar{\lambda}_{\dot{\alpha}}\bar{\sigma}_{n}^{\dot{\alpha}\alpha}\lambda_{\alpha})$
$\displaystyle=$ $\displaystyle\sum_{ij}\bar{\chi}^{j}\chi^{i}\omega\;({2\over
g^{2}}\int_{\mathbb{T}^{4}}\,{\rm tr}\,\phi^{i\alpha}\phi^{j}_{\alpha})$
$\displaystyle=$
$\displaystyle\omega\sum_{ij}\bar{\chi}^{j}\chi^{i}\;U_{F}^{ij}\leavevmode\nobreak\
,\leavevmode\nobreak\ $

where the fermion mode inner product matrix is

$\displaystyle U_{F}^{ij}={2\over g^{2}}\int_{\mathbb{T}^{4}}\,{\rm
tr}\,(\phi^{i}_{2}\phi^{j}_{1}-\phi^{i}_{1}\phi^{j}_{2}),\leavevmode\nobreak\
U_{F}^{ij}=-U_{F}^{ji}.$ (60)

Then we define the fermion nonzero mode path integral so that it produces
$\omega$ (the minimal number of eigenfunctions with the same eigenvalue is
two, i.e. $i,j=1,2$, with $U_{F}$ generically a $2\times 2$ matrix)

$\int\prod_{i}d\chi^{i}d\bar{\chi}^{i}\;(\det
U_{F})^{-1}e^{-S_{F}}=\omega\leavevmode\nobreak\ .$ (61)

When all nonzero eigenvalues are taken into account, we obtain the square root
of the product over all nonzero eigenvalues of $\bar{D}D$ (or $D\bar{D}$).

The integrals over the fermion zero modes are defined in the same manner via
the same mode normalization matrix, $U_{F}^{ij}$, defined in (60). Recalling
that only the undotted spinors $\lambda_{\alpha}$ have zero modes, we
expand181818Here and below, we use $\phi^{i}_{\alpha}$ to denote the zero-mode
solutions of $D\bar{D}$, obeying
$(D\bar{D})_{\alpha}^{\;\;\beta}\phi_{\beta}^{i}=0$. The reader should forgive
us for using the same letter as in the non-zero mode discussion near (206).

$\lambda_{\alpha}=\sum_{i}\eta^{i}\phi^{i}_{\alpha}+\text{nonzero modes},$
(62)

where we use $\eta^{i}$ to denote the zero-mode Grassmann variable. The
fermion zero-mode measure is then taken to be the “square root” of
(61):191919This definition ensures that, upon perturbing with a zero-mode
lifting mass term, $\delta S_{m}={m\over g^{2}}\,{\rm
tr}\,\lambda^{\alpha}\lambda_{\alpha}$, one obtains $m$ for the zero-mode
contribution to the path integral.

$d\mu_{F}=\prod_{i}d\eta^{i}\;(\det U_{F})^{-1/2}\leavevmode\nobreak\
=\prod_{i}d\eta^{i}\;(\text{Pf}U_{F})^{-1}\leavevmode\nobreak\ .$ (63)

In Appendix B.3.1, we calculate the Pfaffian, using the zero-mode wave
functions (51), and show that
$\text{Pf}U_{F}=-U_{F}^{12}=4\times{4\pi^{2}\over g^{2}}$, see eqn. (259).
Notice that this result is valid to all orders in $\Delta$ (thus including the
nonabelian part of the zero modes), since $\text{Pf}U_{F}$ involves
integrating the square of the field strength $F_{mn}$ over $\mathbb{T}^{4}$,
which is proportional to the action, a $\Delta$-independent quantity.

Furthermore, it follows from (62) that $\,{\rm
tr}\,\lambda^{\alpha}\lambda_{\alpha}={1\over
2}\eta^{\alpha}\eta^{\beta}\phi^{(\alpha)A}_{\gamma}\phi^{(\beta)A}_{\delta}\epsilon^{\delta\gamma}+$(nonzero
modes). Thus, combining with (63), one finally finds:

$\displaystyle\int d\mu_{F}\,{\rm tr}\,\lambda\lambda(x)$ $\displaystyle=$
$\displaystyle{g^{2}\over 16\pi^{2}}\int d\eta_{1}d\eta_{2}\,{\rm
tr}\,[\lambda^{\alpha}\lambda_{\alpha}(x)]$ (64) $\displaystyle=$
$\displaystyle{g^{2}\over 16\pi^{2}}{1\over
2}\;(\phi^{(2)A}_{\gamma}\phi^{(1)A}_{\delta}-\phi^{(1)A}_{\gamma}\phi^{(2)A}_{\delta})[x]\epsilon^{\delta\gamma}\leavevmode\nobreak\
.$

We next note that all gauge invariant quantities in the fractional instanton
background depend on the combinations of $x_{i}$ with the dimensionless
collective coordinates $z_{l}$, see eqn. (98) in Appendix A.1. Thus, to
calculate the condensate, in eqn. (64) one should replace

$\displaystyle x_{1}\rightarrow x_{1}-{L_{1}z_{2}\over 2\pi},x_{2}\rightarrow
x_{2}+{L_{2}z_{1}\over 2\pi},x_{3}\rightarrow x_{3}-{L_{3}z_{4}\over
2\pi},x_{4}\rightarrow x_{4}+{L_{4}z_{3}\over 2\pi}.$ (65)

That all gauge invariant quantities depend on (65) follows from the actions of
translations in our background and is explained in Appendix A.1 (see Footnote
30 there).

###  4.2 Bosonic measure

We express the bosonic field $A_{n}(x)$, to be integrated over in the path
integral, as a sum of the classical solution $A_{n}^{cl}(x,z)$ (25), the zero
mode fluctuations normalized as in (331) of Appendix B.3.3, and the nonzero
modes of the fluctuation operator denoted by $Z_{n}^{q}$ (of eigenvalues
$\omega_{q}$) using the same notation as in Appendix B.1, see the discussion
after eqn. (B.1):

$\displaystyle A_{n}(x)$ $\displaystyle=$ $\displaystyle
A_{n}^{cl}(x,z)+\sum\limits_{k=1}^{4}\zeta_{k}^{(0)}{\sqrt{V}\over 4\pi
L_{k}}\leavevmode\nobreak\ Y_{n}^{(k)}(x,z)+\sum_{q}\zeta_{q}Z_{n}^{(q)}\,.$
(66)

Using the inner product defined in (56), the measure of the bosonic zero modes
takes the form

$\displaystyle d\mu_{B}=\left(\det{V\over
16\pi^{2}L_{k}L_{l}}\;U^{kl}\right)^{1\over
2}\;\prod\limits_{k=1}^{4}\left[{d\zeta_{k}^{(0)}\over\sqrt{2\pi}}\right]={V\over
g^{4}}\prod\limits_{k=1}^{4}{d\zeta_{k}^{(0)}\over\sqrt{2\pi}}\,.$ (67)

In Appendix B.3.4, we show how we change the variables $\zeta_{k}^{(0)}$ to
the collective coordinates $\\{z_{m}\\}$ by inserting a unity, á la Faddeev-
Popov gauge-fixing method. The resulting expression is

$\displaystyle d\mu_{B}$ $\displaystyle=$ $\displaystyle{V\over
g^{4}}\prod\limits_{k=1}^{4}{dz_{k}\over\sqrt{2\pi}}\leavevmode\nobreak\ .$
(68)

##  5 The Hamiltonian formalism, Wilson loops, and the moduli space

Up to this point, we have all pieces to compute the gaugino condensate except
for the shape and size of the moduli space ${\cal M}$, or in other words, the
range of integration over the collective coordinates $\\{z_{m}\\}$.
Determining ${\cal M}$ will force us to deviate, for now, from the path
integral to the Hamiltonian formalism. We refer the reader to Cox:2021vsa  for
a detailed description of the Hilbert space, while here, we only provide a
synopsis needed to study the Wilson loops and moduli space.

###  5.1 Pure Yang-Mills theory

The moduli space is determined in the absence of fermions, and thus, we start
by studying the Hamiltonian formalism of pure Yang-Mills theory with vacuum
angle $\theta$ on $\mathbb{T}^{3}$ with a unit ’t Hooft magnetic flux.

Consider $L_{1,2,3}$ of $\mathbb{T}^{3}$ as space of volume $V_{3}$ and
$L_{4}$ as Euclidean time, where $V_{3}$ is to be taken much smaller than the
inverse of $\Lambda^{3}$, the strong-coupling scale, while $L_{4}$ can be
varied from small to large w.r.t. $\Lambda^{-1}$. Recall that we apply the
twists $m_{3}\equiv n_{12}=1$, this is a “magnetic flux” piercing the 1-2
plane, and $k_{3}\equiv n_{34}=1$. The latter is related to a twist of the
partition function by a centre symmetry transformation, to be described below.
The physical Hilbert space lives along the constant time slices and, thus, is
in the $m_{3}=1$ “magnetic flux” background. Let $|\psi\rangle$ denote a state
in the physical Hilbert space on ${\mathbb{T}}^{3}$ with twisted boundary
conditions, ${\cal H}_{m_{3}=1}$.

We introduce the operator $\hat{T}_{3}$ as the generator of the
$\mathbb{Z}^{(1)}_{2}$ $1$-form center symmetry in the $L_{3}$ direction. The
generator $\hat{T}_{3}$ commutes with the Hamiltonian,
$[\hat{T}_{3},\hat{H}]=0$, and thus, they can be diagonalized in the same
basis. It can be shown that the action of $\hat{T}_{3}$ on its eigenstates,
$|\psi\rangle=|e_{3}\rangle$, in the physical Hilbert space is
$\hat{T}_{3}|e_{3}\rangle=e^{i\pi e_{3}-i\frac{\theta}{2}}|e_{3}\rangle$,
where $e_{3}\in\\{0,1\\}$ is the $\mathbb{Z}_{2}$ “electric flux” of the
state. This terminology can be explained as follows. Since $\hat{T}_{3}$ is a
center-symmetry generator, it acts on Wilson loops winding around the $L_{3}$
direction as $\hat{T}_{3}\hat{W}_{3}=-\hat{W}_{3}\hat{T}_{3}$. Then, one
readily finds
$\hat{T}_{3}(\hat{W}_{3}|e_{3}\rangle)=e^{i\pi(e_{3}+1)-i\frac{\theta}{2}}(\hat{W}_{3}|e_{3}\rangle)$,
i.e., the action of $\hat{W}_{3}$ on a state increases the $\mathbb{Z}_{2}$
flux of this state by one unit. The fact that $\hat{W}_{3}$ can be thought of
as creating a winding electric flux tube explains the terminology. We conclude
that $\hat{T}_{3}$ measures the center flux of a given state in Hilbert space.
The generators of the center symmetries in the other two spatial directions,
$x_{1}$ and $x_{2}$, $\hat{T}_{\ell}$, $\ell=1,2$, act similarly on Wilson
loops winding in the corresponding directions. They also commute with the
Hamiltonian and thus all eigenstates of $\hat{H}$ are also labeled by
$e_{1}\in\\{0,1\\}$ and $e_{2}\in\\{0,1\\}$.202020The action of $\hat{T}_{1}$
on states of electric flux $e_{1}$ is $\hat{T}_{1}|e_{1}\rangle=e^{i\pi
e_{1}}|e_{1}\rangle$ (and similar for $\hat{T}_{2}$), without the $\theta/2$
factor in the action of $\hat{T}_{3}$ (which is due to the $m_{3}=1$ twist).

The twisted partition function of pure Yang-Mills theory, the one we have been
studying in the absence of fermions, is the one with an insertion of a
$\hat{T}_{3}$ twist:

$\displaystyle{\cal Z}_{YM}[n_{12}=1,n_{34}=1]$ $\displaystyle=$
$\displaystyle{\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[e^{-L_{4}\hat{H}}\hat{T}_{3}\right]$
(69) $\displaystyle=$ $\displaystyle\sum_{E(e_{3}),\,e_{3}=\\{0,1\\}}\langle
E(e_{3}),e_{3}|e^{-L_{4}E(e_{3})}\hat{T}_{3}|E(e_{3}),e_{3}\rangle$
$\displaystyle=$ $\displaystyle\sum_{E(e_{3}),\,e_{3}=\\{0,1\\}}\langle
E(e_{3}),e_{3}|e^{-L_{4}E(e_{3})}|E(e_{3}),e_{3}\rangle e^{i\pi
e_{3}-i\frac{\theta}{2}}\,,$

where in going from the first to the second line, we summed over a complete
set of eigenstates in the physical Hilbert space ${\cal H}_{m_{3}=1}$ that
diagonalize $\hat{H}$ and $\hat{T}_{3}$ simultaneously (for brevity, omitting
the summation over $e_{1,2}$). The expectation value of the Wilson loop
operator $\hat{W}_{3}$ is212121We ignore the normalization of Wilson loops. We
will come back to normalization in calculating the gaugino condensate.
$\langle
W_{3}\rangle={\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[e^{-L_{4}\hat{H}}\hat{W}_{3}\hat{T}_{3}\right]$.
Using the relation $\hat{T}_{3}\hat{W}_{3}=-\hat{W}_{3}\hat{T}_{3}$ and the
fact that $\hat{T}_{3}$ (but not $\hat{W}_{3}$) commutes with $\hat{H}$, we
immediately find $\langle\hat{W}_{3}\rangle=-\langle\hat{W}_{3}\rangle$, and
hence, $\langle\hat{W}_{3}\rangle=0$.

One can also show that the expectation value of the Wilson loop winding in any
other direction vanishes identically. Since $\hat{T}_{\ell}$, $\ell=1,2,3$ and
$\hat{H}$ form a set of commuting operators, they can be diagonalized
simultaneously. Let $\\{|E(e_{1},e_{2},e_{3}),e_{1},e_{2},e_{3}\rangle\\}$ be
a set of orthonormal eigenstates of the set of the $4$ operators
$\hat{H},\\{\hat{T}_{\ell},\ell=1,2,3\\}$, where $e_{\ell}\in\\{0,1\\}$ is the
$\mathbb{Z}_{2}$ electric flux in the $L_{\ell}$ direction, and we emphasized
that, in general, the energy of the state depends on these fluxes. The
expectation value of a general Wilson loop $\hat{W}_{p}$ wrapping the $L_{p}$
direction and computed using the $\hat{T}_{3}$-twisted partition function (69)
reads

$\displaystyle\langle\hat{W}_{p}\rangle$ $\displaystyle=$
$\displaystyle\sum_{E(e_{1},e_{2},e_{3}),\,e_{i}=\\{0,1\\}}e^{-L_{4}E}\langle
E(e_{1},e_{2},e_{3}),e_{1},e_{2},e_{3}|\hat{W}_{p}\hat{T}_{3}|E(e_{1},e_{2},e_{3}),e_{1},e_{2},e_{3}\rangle$
$\displaystyle=$
$\displaystyle\sum_{E(e_{1},e_{2},e_{3}),\,e_{i}=\\{0,1\\}}e^{-L_{4}E}\langle
E(e_{1},e_{2},e_{3}),e_{1},e_{2},e_{3}|\hat{W}_{p}|E(e_{1},e_{2},e_{3}),e_{1},e_{2},e_{3}\rangle
e^{i\pi e_{3}-i\frac{\theta}{2}}\,.$

However, the insertion of $\hat{W}_{p}$ increases the electric flux by one
unit in the $L_{p}$ direction and produces a sum over different energy
eigenstates. Thus, its diagonal matrix element in the
$|E(e_{1},e_{2},e_{3}),e_{1},e_{2},e_{3}\rangle$ state vanishes. We
immediately conclude that a Wilson loop that wraps $L_{1}$, $L_{2}$, or
$L_{3}$ must vanish on $\mathbb{T}^{4}$ with twisted boundary conditions. The
vanishing of $\langle\hat{W}_{4}\rangle$ follows by applying a $90$-degree
rotation to any spatial Wilson loop, i.e. upon considering a different
${\mathbb{T}}^{4}$ direction as time. Thus, we conclude that for any winding
Wilson loop $\hat{W}_{p}$,

$\displaystyle\langle\hat{W}_{p}\rangle={\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[e^{-L_{4}\hat{H}}\hat{W}_{p}\hat{T}_{3}\right]=0.$
(71)

We spent so much time explaining the expected result (71) because its
consistency with semiclassics is one of our main criteria used to determine
the moduli space of the fractional instanton.

Thus, we now contrast the general result (71) with the computation of the
expectation value of a Wilson loop in the path integral formalism, in the
semiclassical approximation. Consider a Wilson loop
$W(C_{n_{1},n_{2},n_{3},n_{4}})$, with $C$ beginning at an arbitrary point $x$
in $\mathbb{T}^{4}$ and winding $n_{\ell}$ times around each direction
$L_{\ell}$:

$\displaystyle W(x,C_{n_{1},n_{2},n_{3},n_{4}})=\,{\rm
tr}\,\left({\cal{P}}e^{i\int_{C_{n_{1},n_{2},n_{3},n_{4}}}{A}_{k}(x^{\prime})dx^{\prime}_{k}}\;\Omega_{2}^{n_{2}}(x)\;\Omega_{4}^{n_{4}}(x)\right),$
(72)

where we inserted $\Omega_{2}^{n_{2}}(x)\;\Omega_{4}^{n_{4}}(x)$ to enforce
the gauge invariance of $W(x,C_{n_{1},n_{2},n_{3},n_{4}})$. Using the
classical self-dual background (25),(29), we show in eqn. (A.3) in the
Appendix, that $W$ to order $\Delta$ is:

$\displaystyle W(x,C_{n_{1},n_{2},n_{3},n_{4}})$ (73) $\displaystyle=$
$\displaystyle 2\cos\left[{1\over 2}\left(n_{1}(z_{1}+{2\pi x_{2}\over
L_{2}})+n_{2}(z_{2}-{2\pi x_{1}\over L_{1}})+n_{3}(z_{3}+{2\pi x_{4}\over
L_{4}})+n_{4}(z_{4}-{2\pi x_{4}\over L_{2}})\right)\right].$
$\displaystyle\times\left[1+\Delta{\cal F}(x,z)\right]\,.$

The ${\cal{O}}(1)$ and ${\cal{O}}(\Delta)$ contributions come from the abelian
and nonabelian components of (25). The cosine function has $4\pi$ periodicity
in $\\{z_{i}\\}$, while the ${\cal{O}}(\Delta)$ piece ${\cal F}(x,z)$ is a
periodic and even function of $\\{z_{1}+{2\pi x_{2}\over L_{2}},z_{2}-{2\pi
x_{1}\over L_{1}},z_{3}+{2\pi x_{4}\over L_{4}},z_{4}-{2\pi x_{4}\over
L_{2}}\\}$ with periodicity $2\pi$ for $\\{z_{i}\\}$.

Now, using the results from the previous section and limiting our discussion
to pure Yang-Mills theory and ignoring issues of normalization, the
expectation value of a general Wilson loop is

$\displaystyle\langle
W(x,C_{n_{1},n_{2},n_{3},n_{4}})\rangle=\sum_{\nu}\int[DA_{\mu}]W(x,C_{n_{1},n_{2},n_{3},n_{4}})e^{-S_{YM}-i\theta(\nu+{1\over
2})}|_{n_{12}=1,n_{34}=1}$ $\displaystyle\sim e^{-S_{0}-i{\theta\over
2}}{V\over g^{4}}\int_{\cal
M}\prod\limits_{k=1}^{4}{dz_{k}\over\sqrt{2\pi}}\;W(x,C_{n_{1},n_{2},n_{3},n_{4}})+e^{-S_{0}+i{\theta\over
2}}{V\over g^{4}}\int_{\cal
M}\prod\limits_{k=1}^{4}{dz_{k}\over\sqrt{2\pi}}\;\tilde{W}(x,C_{n_{1},n_{2},n_{3},n_{4}})\,,$

where in going from the first to the second line, we ignored the quantum loops
and used the bosonic zero-mode measure (68). We limited the r.h.s. to the
contributions of the $\nu=0$ sector, with total topological charge
$Q=\frac{1}{2}$, and the $\nu=-1$ sector, with total topological charge
$Q=-\frac{1}{2}$. Ignoring higher-$Q$ contributions is justified in the limit
when semiclassics holds, i.e. for a small ${\mathbb{T}}^{4}$.

Furthermore, in writing the above expression, we assumed that the fractional
instantons (and antinstantons222222$\tilde{W}$ in the second term is the
Wilson loop (72), but evaluated in the anti-instanton background, whose
explicit form is similar to (73) and shall not be needed.) we obtained are the
only $|Q|={1\over 2}$ saddle points in the path integral, contributing to the
first and second terms above, respectively. While we have no proof of
uniqueness, the favourable comparison of “exact” (i.e. obtained by numerically
minimizing the action) fractional instantons on an asymmetric
${\mathbb{T}}^{4}$ with the solutions obtained by small-$\Delta$ expansion
gives support in favor of this assumption, at least for small enough $\Delta$.
In this regard, we note that ref. GarciaPerez:2000aiw  found good agreement
for $\Delta$ as large as $0.08$.

Our point now is that the $d^{4}z$ integral in (LABEL:wilsonlooplimits)
vanishes, and is thus consistent with (71), for all values of $\theta$ and for
all $x$, if and only if the limits of integration are taken
$z_{i}\in[0,4\pi)$. This leads us to conclude that the moduli space ${\cal
M}\sim\mathbb{T}^{4}$ with period $4\pi$ in every direction.232323Moreover, it
is easy to check that changing the limits of integration to any other values
yields a position-dependent Wilson loop. For example, consider $\langle
W(x,C_{n_{1}=1,n_{2}=0,n_{3}=0,n_{4}=0})\rangle$ and take the range of
$z_{i}\in[0,2\pi)$, which yields $\langle
W(x,C_{n_{1}=1,n_{2}=0,n_{3}=0,n_{4}=0})\rangle\sim-\sin\left(\pi\frac{x^{2}}{L}\right)+{\cal
O}(\Delta)$. This result should not be expected on physical grounds since our
background has constant field strength on $\mathbb{T}^{4}$ (to leading order
in $\Delta$); the expectation value of the Wilson loop should, at most, be a
constant (to leading order in $\Delta$).

A further argument in favour of this identification of the moduli space is
that the classical fractional instanton field configurations with $z_{n}$
differing by $2\pi$ are distinguished by the gauge invariant winding Wilson
loop operators, while those differing by $2\pi$ are not distinguished. Thus
one expects that $z_{n}$ and $z_{n}+2\pi$ are not to be identified. In
contrast, no local gauge invariant operators can distinguish between $z_{n}$
and $z_{n}+2\pi$, see Section A.3 in the Appendix for detail.242424We note,
without discussing the details, that a virtually identical argument for
extending the limits of $z_{n}$ integration holds also in the $SU(N)$ case in
the background of the $Q=1/N$ solutions found by ’t Hooft tHooft:1981nnx ,
recently extended in the framework of the small-$\Delta$ expansion Gonzalez-
Arroyo:2019wpu .

###  5.2 SYM theory

Before turning to the calculations of the gaugino condensate using the path
integral method, we pause here to discuss SYM theory using the Hamiltonian
formalism, akin to our discussion of pure Yang-Mills of Section 5.1. The
partition function of SYM with twisted boundary conditions in the 1-2 and 3-4
planes, identical for bosons and fermions, is given by

$\displaystyle{\cal Z}_{SYM}[n_{12}=1,n_{34}=1]=\mbox{tr}_{{\cal
H}_{m_{3}=1}}\left[(-1)^{F}e^{-L_{4}\hat{H}}\hat{T}_{3}\right]\,.$ (75)

The insertion of the fermion number $(-1)^{F}$ guarantees that both fermions
and bosons obey periodic boundary conditions in the time direction. Let
$\hat{X}$ be the $\mathbb{Z}_{4}^{d\chi}$ discrete chiral symmetry generator.
Then, it can be shown that the symmetry operators $\hat{T}_{3}$ and $\hat{X}$
obey the algebra:

$\displaystyle\left[\hat{H},\hat{T}_{3}\right]=0\,,\quad\left[\hat{H},\hat{X}\right]=0\,,\quad\hat{T}_{3}\hat{X}=e^{i\pi}\hat{X}\hat{T}_{3}\,,$
(76)

where the last relation is the result of the mixed anomaly between the
$0$-form discrete chiral and $1$-form center symmetries Cox:2021vsa .252525The
same algebra arises in the 2-D massless charge-$2$ Schwinger model
Anber:2018jdf , due to a similar anomaly. Since $\hat{H}$ commutes with
$\hat{T}_{3}$, we can as before, label the physical states in ${\cal
H}_{m_{3}=1}$ by $|E(e_{3}),e_{3}\rangle$. The algebra (76) requires that
$\hat{X}|E(e_{3}),e_{3}\rangle=|E(e_{3}),e_{3}-1\rangle$, and thus, it is
easily seen the states $|E(e_{3}),e_{3}=1\rangle$ and
$|E(e_{3}),e_{3}=0\rangle$ are degenerate:
$\hat{H}|E,e_{3}=0\rangle=E(e_{3}=0)|E,e_{3}=0\rangle$,
$\hat{H}|E,e_{3}=1\rangle=E(e_{3}=0)|E,e_{3}=1\rangle$, at any size of the
spatial ${\mathbb{T}}^{3}$, as a consequence of the anomaly.

Now, we can readily calculate the partition function (75) to find

$\displaystyle{\cal Z}_{SYM}[n_{12}=1,n_{34}=1]$ $\displaystyle=$
$\displaystyle{\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[(-1)^{F}e^{-L_{4}\hat{H}}\hat{T}_{3}\right]$
(77) $\displaystyle=$
$\displaystyle\sum\limits_{E,\,e_{3}=\\{0,1\\}}(-1)^{F}(-1)^{e_{3}}\langle
E,e_{3}|e^{-EL_{4}}|E,e_{3}\rangle$ $\displaystyle=$
$\displaystyle\sum\limits_{E,\,e_{3}=\\{0,1\\}}(-1)^{F}(-1)^{e_{3}}e^{-EL_{4}}=0\leavevmode\nobreak\
.$

This equation provides one way to see that the twisted partition function of
SYM vanishes, by noting that all states in Hilbert space are doubly degenerate
and their contributions cancel each other in the twisted partition function
(77).

Now we insert the gaugino condensate, the $\,{\rm tr}\,\lambda^{2}$ operator,
where the trace is taken in the color space. The operator is inserted at
$x_{4}=0$ and some position on ${\mathbb{T}}^{3}$:

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle$ $\displaystyle=$
$\displaystyle{\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[(-1)^{F}e^{-L_{4}\hat{H}}\,{\rm
tr}\,\lambda^{2}\;\hat{T}_{3}\right]\over{\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[(-1)^{F}e^{-L_{4}\hat{H}}\right]$
(78) $\displaystyle=$ $\displaystyle{\cal
N}^{-1}\sum\limits_{E,\,e_{3}=\\{0,1\\}}(-1)^{F}(-1)^{e_{3}}e^{-EL_{4}}\langle
E,e_{3}|\,{\rm tr}\,\lambda^{2}|E,e_{3}\rangle\,.$

Here, we took the normalization constant ${\cal N}$ to be the Witten index:
${\cal N}={\text{tr}}_{{\cal{H}}_{m_{3}=1}}\left[(-1)^{F}\right]=2$, given by
the partition function on ${\mathbb{T}}^{3}$ with an $n_{12}$ twist, but
without a $\hat{T}_{3}$ twist in the time direction (recall (11) in the path
integral framework).

Next, we know that $\hat{X}|E,e_{3}\rangle=|E,e_{3}+1\rangle$ from the anomaly
(76) and that $\hat{X}$ acts on the condensate as $\hat{X}\,{\rm
tr}\,\lambda^{2}=-\,{\rm tr}\,\lambda^{2}\hat{X}$. Thus, because

$\langle E,0|\hat{X}^{\dagger}\hat{X}\,{\rm
tr}\,\lambda^{2}|E,0\rangle=-\langle E,1|\,{\rm
tr}\,\lambda^{2}|E,1\rangle\,,$ (79)

$\,{\rm tr}\,\lambda^{2}$ has opposite expectation values in the two
degenerate states $e_{3}=0$ and $e_{3}=1$. Thus, we find

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle$ $\displaystyle=$
$\displaystyle{\cal
N}^{-1}\sum\limits_{E,\,e_{3}=\\{0,1\\}}(-1)^{F}e^{-EL_{4}}\langle E,0|\,{\rm
tr}\,\lambda^{2}|E,0\rangle_{V_{3},n_{12}}$ (80) $\displaystyle=$
$\displaystyle\sum\limits_{E}(-1)^{F}e^{-EL_{4}}\langle E,0|\,{\rm
tr}\,\lambda^{2}|E,0\rangle_{V_{3},n_{12}}\,,$

where the sum is over only the half of the Hilbert space with $e_{3}=0$ and we
are reminded that the condensate is being computed in a small $V_{3}$ theory
Hilbert space with boundary conditions twisted by $n_{12}$.

If $L_{4}$ is small compared to the energy scales in the problem, there is no
reason to assume that any particular values of $E$ dominate. Thus, eqn. (80)
is the Hamiltonian expression of the expectation value we have computed for
general $L_{4}$. However, we can try take the $L_{4}\rightarrow\infty$ limit
to obtain262626Equating the result of our calculation, eqn. (84) below, to the
r.h.s. of (81) is only possible if we assume that the semi-classical treatment
holds as $L_{4}$ is varied between small and large, and, in particular, there
are no new contributions that contribute to the l.h.s. This can be only
justified by understanding the convergence radius of the $\Delta$ expansion.

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle=\langle 0,0|\,{\rm
tr}\,\lambda^{2}|0,0\rangle_{V_{3},n_{12}}\,,$ (81)

where we used that supersymmetry is unbroken and the small-$V_{3}$ ground
state is bosonic, $(-1)^{F}=1$.272727We note that a similar equation holds in
$SU(N)$ theories for one of the $N$ vacua of the twisted, small-$V_{3}$
theory. There ${\cal{N}}=N$ and there is $N$-fold degeneracy of all energy
states on ${\mathbb{T}}^{3}$ with $n_{12}=1$.

In conclusion, eqn. (81) shows that in the large-$L_{4}$ limit, the
calculation of $\langle\,{\rm tr}\,\lambda^{2}\rangle$ using the
$\hat{T}_{3}$-twisted parition function and normalized by the Witten index, as
in (78), yields the gaugino condensate in one of the two degenerate ground
states of the small-$V_{3}$ theory with a unit ’t Hooft flux $n_{12}$. The
state $|0,0\rangle_{V_{3},n_{12}}$ is further expected, as
$V_{3}\rightarrow\infty$, to become one of the two degenerate ground states of
the $SU(2)$ SYM theory on ${\mathbb{R}}^{4}$.

##  6 The gaugino condensate

Finally, we put pieces together to read out the condensate. We use the fact
that non-zero modes in (10) cancel between the bosons, fermions, and ghosts.
Then, we combine the integral of $\,{\rm tr}\,\lambda^{2}$ over the zero mode
measure for the fermions from eqn. (64) with the boson zero mode measure (68).
We also recall that we normalize by the Witten index ${\cal N}=2$ to obtain:

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle$ $\displaystyle=$
$\displaystyle{M_{PV}^{3}e^{-\frac{4\pi^{2}}{g^{2}}}\over{\cal N}}\;{V\over
g^{4}}$
$\displaystyle\times\int_{z_{i}\in[0,4\pi]}\prod\limits_{k=1}^{4}{dz_{k}\over\sqrt{2\pi}}\;{g^{2}\over
16\pi^{2}}{1\over
2}\;\epsilon^{\delta\gamma}(\phi^{(2)A}_{\gamma}\phi^{(1)A}_{\delta}-\phi^{(1)A}_{\gamma}\phi^{(2)A}_{\delta})[\\{x\\}]\bigg{|}_{x_{1}\rightarrow
x_{1}-{z_{2}L_{1}\over 2\pi},\;\text{see eqn.}\;(\ref{replace})}$
$\displaystyle=$ $\displaystyle{\Lambda^{3}\over 2}{V\over 4\pi^{2}}{1\over
16\pi^{2}}\int\limits_{-2\pi}^{2\pi}d^{4}z\;(\phi^{(2)A}_{2}\phi^{(1)A}_{1}-\phi^{(1)A}_{2}\phi^{(2)A}_{1})[-{z_{2}L_{1}\over
2\pi},{z_{1}L_{2}\over 2\pi},-{z_{4}L_{3}\over 2\pi},{z_{3}L_{4}\over 2\pi}].$
(82)

To obtain the last line, we recalled the substitution of (65), the fact that
all local gauge invariants are periodic functions of $\hat{z}_{k}$ (defined in
(65)) of period $2\pi$, and used shifts of the $z_{n}$ variables to set the
$x_{k}$-coordinates (of period $L_{k}$) to zero. Further, Pauli-Villars
regularization has been utilized to renormalize the theory, and the scale
$M_{PV}$ is the Pauli-Villars mass. The factor $M_{PV}^{3}$ comes from
regularizing the boson and fermion determinants, with each zero-mode
contributing an appropriate factor of $M_{PV}$: the bosonic determinant gives
$M_{PV}^{4}$ and the fermionic determinant gives $M_{PV}^{-1}$ (we do not
describe this in detail, as it is a standard procedure in supersymmetric
instanton calculations Davies:1999uw ; Davies:2000nw ; Dorey:2002ik ;
Vandoren:2008xg ). In going from the first to the second line of (82), we used
the fact that $\Lambda^{3}={M_{PV}^{3}\over g^{2}}e^{-\frac{4\pi^{2}}{g^{2}}}$
in the scheme often used for gaugino condensate calculations Dorey:2002ik ;
Vandoren:2008xg , notably, in the weakly-coupled
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ set up and its comparison with the
weak-coupling calculation on ${\mathbb{R}}^{4}$ Davies:2000nw .

Next, we rescale the $z_{k}$ variables in (82) by appropriate factors of
$2\pi\over L_{p}$, contributing an overall Jacobian ${(2\pi)^{4}\over V}$.
Taking the liberty to again call the rescaled variables $x_{k}$, we obtain for
the condensate

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle$ $\displaystyle=$
$\displaystyle{\Lambda^{3}\over 2}{1\over
4}\prod_{k}\int\limits_{-L_{k}}^{L_{k}}dx_{k}\;(\phi^{(2)A}_{2}\phi^{(1)A}_{1}-\phi^{(1)A}_{2}\phi^{(2)A}_{1})[-x_{1},x_{2},-x_{3},x_{4}]$
(83) $\displaystyle=$ $\displaystyle{\Lambda^{3}\over 2}{2^{4}\over
4}\int\limits_{{\mathbb{T}}^{4}}d^{4}x\;(\phi^{(2)A}_{2}\phi^{(1)A}_{1}-\phi^{(1)A}_{2}\phi^{(2)A}_{1})[x]={\Lambda^{3}\over
2}{2^{4}\over 4}{g^{2}U_{F}^{21}}\leavevmode\nobreak\ .$

As indicated, the integrand on the second line is the same as the one
appearing in $U_{F}^{21}$ from (60), safe for the absence of a factor of
$1/g^{2}$. Thus, recalling that $U_{F}^{21}={16\pi^{2}\over g^{2}}$, as per
the discussion after (63) (see also (259)), we finally obtain for (80):

$\displaystyle\langle\,{\rm tr}\,\lambda\lambda\rangle$ $\displaystyle=$
$\displaystyle 32\pi^{2}\Lambda^{3}.$ (84)

Further, assuming that the result extends to $L_{4}\rightarrow\infty$, as
described in the previous section, we use (81) and (84) to conclude

$\displaystyle\langle 0,0|\,{\rm
tr}\,\lambda^{2}|0,0\rangle_{V_{3},n_{12}}=2\times
16\pi^{2}\Lambda^{3}\leavevmode\nobreak\ .$ (85)

Thus, the condensate in one of the two degenerate vacua of the
small-$V_{3},n_{12}=1$ theory has twice the value calculated on
${\mathbb{R}}^{4}$ as described in the Introduction.282828In the spirit of
Footnote 24, without discussing the details, we remark that a similar result
is obtained, upon taking the large-$L_{4}$ limit of ’t Hooft’s $SU(N)$
solutions with $Q={1\over N}$, constructed via the $\Delta$ expansion in
Gonzalez-Arroyo:2019wpu . One finds, instead of (85), that the condensate in
one of the $N$ degenerate vacua on ${\mathbb{T}}^{3}$ with an $n_{12}=1$
twist, $\langle 0,0|\,{\rm tr}\,\lambda^{2}|0,0\rangle_{V_{3},n_{12}}=N\times
16\pi^{2}\Lambda^{3}$, i.e. equals $N$ times the ${\mathbb{R}}^{4}$ result.

Acknowledgments:

E.P. is grateful to Antonio Gonzalez-Arroyo for many conversations on
fractional instantons and the $\Delta$-expansion during the “Fluxtube-22”
Workshop at KITP during the Winter of 2022, supported in part by the National
Science Foundation under Grant No. NSF PHY-1748958. We also thank F. David
Wandler for comments on the manuscript. Both authors would like to thank the
Simons Centre for Geometry and Physics, where this work started. E.P.
acknowledges the hospitality of Durham University during work on this project
and M.A. acknowledges the hospitality of the University of Toronto, where this
work was completed. M.A. is supported by STFC through grant ST/T000708/1. E.P.
is supported by a Discovery Grant from NSERC.

##  Appendix A Constructing instantons on the asymmetric ${\mathbb{T}}^{4}$
with twists

Our $SU(2)$ Lie algebra convention is as follows. We denote the generators in
the Cartan as $\tau^{3}\over 2$ and the non-Cartan as $\tau^{\pm}$, where
$\tau^{+}=\left(\begin{array}[]{cc}0&1\cr
0&0\end{array}\right)=(\tau^{-})^{\dagger}$. We expand the Hermitean gauge
field as

$A_{n}=A_{n}^{3}{\tau^{3}\over
2}+A_{n}^{+}\tau^{+}+A_{n}^{-}\tau^{-},\leavevmode\nobreak\
A^{-}=(A^{+})^{\dagger},\leavevmode\nobreak\ n=1,2,3,4.$ (86)

An identical expression (but without the reality condition) also holds for the
adjoint representation fermions $\lambda_{\alpha}$,
$\bar{\lambda}_{\dot{\alpha}}$ ($\alpha,\dot{\alpha}=1,2$), which are
independent variables in Euclidean space, to be discussed in more detail
later.

The fields are smooth and defined on ${\mathbb{R}}^{4}$, with coordinates
$x_{n}$, $n=1,2,3,4$. The restriction to a ${\mathbb{T}}^{4}$ with periods
$L_{n},$ and with ’t Hooft twists is accomplished by imposing periodicities
defined by transition functions $\Omega_{n}(x)$, $x\in{\mathbb{R}}^{4}$. As
explained in the main text, we choose a gauge where the ${\mathbb{T}}^{4}$
transition functions are trivial in the $x_{1}$ and $x_{3}$ directions and
nontrivial in the $x_{2}$ and $x_{4}$ directions

$\Omega_{2}(x)=e^{-i2\pi{x_{1}\over L_{1}}{\tau^{3}\over
2}},\;\;\Omega_{4}(x)=e^{-i2\pi{x_{3}\over L_{3}}{\tau^{3}\over
2}},\leavevmode\nobreak\ \text{while}\leavevmode\nobreak\
\Omega_{1}=\Omega_{3}=1_{2\times 2},\leavevmode\nobreak\ \forall
x\in{\mathbb{R}}^{4}.$ (87)

Thus, there are two nontrivial twists, $n_{12}=n_{34}=1$, since (87) obey the
cocycle conditions

$\Omega_{i}(x+L_{j}\hat{e}_{j})\;\Omega_{j}(x)=e^{i\pi
n_{ij}}\;\Omega_{j}(x+L_{i}\hat{e}_{i})\;\Omega_{i}(x),\leavevmode\nobreak\
i,j=1,2,3,4,\;\forall x\in{\mathbb{R}}^{4},$ (88)

where $\hat{e}_{n}$ is a unit vector in the $x_{n}$ direction. The periodicity
condition on the gauge fields in ${\mathbb{R}}^{4}$ define the
${\mathbb{T}}^{4}$ fields:

$A(x+\hat{e}_{n}L_{n})=\Omega_{n}(x)(A(x)-id)\Omega_{n}^{-1}(x),\leavevmode\nobreak\
n=1,2,3,4,\leavevmode\nobreak\ \forall x\in{\mathbb{R}}^{4},$ (89)

where we denoted $A(x)=\sum\limits_{n=1}^{4}A_{n}(x)dx_{n}$. The Euclidean
path integral is performed over fields obeying (89), with fixed transition
functions $\Omega_{n}(x)$. Gauge transformations $g(x)\in
SU(2),x\in{\mathbb{R}}^{4}$ act on the gauge field the usual way292929 Our
convention for gauge transformations, $A(x)\rightarrow
g(x)(A(x)-id)g^{-1}(x)$, implies
$F_{mn}=\partial_{m}A_{n}-\partial_{n}A_{m}+i[A_{m},A_{n}]$ and
$D_{m}\phi=\partial_{m}\phi+i[A_{m},\phi]$ for any adjoint $\phi$. Adjoint
fermions obey the same periodicity conditions as (89), but without the
inhomogeneous term. while their action on transition functions is

$\Omega_{n}(x)\rightarrow
g(x+\hat{e}_{n}L_{n})\Omega_{n}(x)g^{-1}(x)\;\leavevmode\nobreak\
,\leavevmode\nobreak\ n=1,2,3,4,\leavevmode\nobreak\ \forall
x\in{\mathbb{R}}^{4}.$ (90)

Gauge transformations $g(x)$ which leave the transition functions invariant
will be called “$\Omega$-periodic.” With our choice of gauge for the
transition functions, $\Omega$-periodic gauge transformations $g(x)$ are
periodic in $x_{1}$ and $x_{3}$ but not in $x_{2}$ and $x_{4}$. The only
constant $\Omega$-periodic transformations are the abelian ones,
$g=e^{i\alpha\tau^{3}}$.

For future use, we note that a fundamental-representation Wilson loop along a
unit-winding loop $C_{1}$ (winding once along $x_{n}$) is

$\displaystyle W(C_{1},x)=\,{\rm
tr}\,\left({\cal{P}}e^{i\int\limits_{x}^{x+\hat{e}_{n}L_{n}}A_{k}(x^{\prime})dx^{\prime}_{k}}\;\Omega_{n}(x)\right).$
(91)

Here, the insertion of $\Omega_{n}(x)$ ensures invariance under (90).

A particular field configuration obeying (89) is the constant field strength
Abelian background, the “fractional instanton” introduced by ’t Hooft, see
tHooft:1981sps ; tHooft:1981nnx ; vanBaal:1984ar :

$\displaystyle\bar{A}_{n}(x,z)=\bar{A}_{n}^{3}(x,z){\tau^{3}\over
2}:\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{A}_{1}^{3}$ $\displaystyle=$
$\displaystyle{2\pi x_{2}\over L_{1}L_{2}}+{z_{1}\over L_{1}},$ (92)
$\displaystyle\bar{A}_{2}^{3}$ $\displaystyle=$ $\displaystyle{z_{2}\over
L_{2}},$ $\displaystyle\bar{A}_{3}^{3}$ $\displaystyle=$ $\displaystyle{2\pi
x_{4}\over L_{3}L_{4}}+{z_{3}\over L_{3}},$ $\displaystyle\bar{A}_{4}^{3}$
$\displaystyle=$ $\displaystyle{z_{4}\over L_{4}}.$

Here, $z_{n}$ are constants whose significance as collective coordinates
associated with the instanton will be discussed at length later. The field
strength of the abelian background (92) is:

$\displaystyle F_{mn}^{(0)}$ $\displaystyle=$ $\displaystyle{\tau^{3}\over
2}\left(\begin{array}[]{cccc}0&-{2\pi\over L_{1}L_{2}}&0&0\cr{2\pi\over
L_{1}L_{2}}&0&0&0\cr 0&0&0&-{2\pi\over L_{3}L_{4}}\cr 0&0&{2\pi\over
L_{3}L_{4}}&0\end{array}\right)\leavevmode\nobreak\ .$ (97)

The abelian background (92,97) has the following properties:

  1. 1.

The field strength $F_{mn}^{(0)}$ from (97) can be used to explicitly verify
that the abelian background (92) has topological charge $1/2$. This can be
seen by recalling that the topological charge only depends on the transition
functions. Its fractional nature is owing to the nonzero twists
$n_{12}=n_{34}=1$.

  2. 2.

In addition, it also follows from (97), that for a “symmetric”
${\mathbb{T}}^{4}$—one where $L_{1}L_{2}=L_{3}L_{4}$—the background (92) is
self-dual and hence stable, i.e. it has minimal action for the given
topological charge. The action of the self-dual abelian solution is
$S_{0}={4\pi^{2}\over g^{2}}$, half that of the BPST instanton.

  3. 3.

For use below, it is convenient to introduce the variables

$(\hat{z}_{1},\hat{z}_{2},\hat{z}_{3},\hat{z}_{4})\equiv(z_{1}+{2\pi
x_{2}\over L_{2}},z_{2}-{2\pi x_{1}\over L_{1}},z_{3}+{2\pi x_{4}\over
L_{4}},z_{4}-{2\pi x_{3}\over L_{3}})$ (98)

The $\hat{z}$-variables are important since all gauge invariants
characterizing the nonabelian instanton background depend on $\hat{z}_{n}$
only. In the gauge we are using, the fact that the background depends on
$\hat{z}_{1},\hat{z}_{3}$ is already evident in (92). The appearance of the
combinations $\hat{z}_{2},\hat{z}_{4}$ follows from the action of translations
in $x_{1}$ and $x_{3}$: in order to preserve the transition functions, an
$x_{1},x_{3}$ translation is accompanied by a non-$\Omega$-periodic gauge
transformation which shifts $z_{2}$ and $z_{4}$. This ensures that all gauge
invariant quantities only depend on $\hat{z}_{n}$.303030For example, a
translation $x_{1}\rightarrow x_{1}+\epsilon_{1}$ is accompanied by
$g_{1}(\epsilon_{1},x_{2})=e^{i2\pi{\epsilon_{1}x_{2}\over
L_{1}L_{2}}{\tau^{3}\over 2}}$ ensuring that $\Omega_{2}(x_{1})$ is invariant:
as per (90), $\Omega_{2}(x_{1})$ transforms into
$g_{1}(x_{2}=L_{2})\Omega_{2}(x_{1}+\epsilon_{1})g_{1}^{-1}(x_{2}=0)=\Omega_{2}(x_{1})$.
At the same time, the $g_{1}$ action on $\bar{A}$ shifts $z_{2}\rightarrow
z_{2}-{2\pi\epsilon_{1}\over L_{1}}$, showing that the variable
$\hat{z}_{2}=z_{2}-{2\pi x_{1}\over L_{1}}$ is invariant under the combined
action of $g_{1}$ and translation $e^{\epsilon_{1}\partial_{1}}$. Our
nonabelian solution also exhibits this property: considering, for example, its
$W_{1}$ component of eqn. (A.1), we observe that it is invariant under the
combined action of translation of $x_{1}$, the shift of $z_{2}$ given above,
and a gauge transformation by $g_{1}$, which multiplies $W_{1}$ by a phase.
Gauge invariants built from the solution will then only depend on the
$\hat{z}_{n}$ combinations.

###  A.1 Constructing the self-dual fractional instanton for small $\Delta$

As explained in the main text, there are issues regarding the abelian solution
in the tuned ${\mathbb{T}}^{4}$ that have not yet been addressed in full.
Notably, they concern the lifting of the extra bosonic zero modes (found in
vanBaal:1984ar ) present in the $L_{1}L_{2}=L_{3}L_{4}$ limit. These issues,
as well as our desire to probe the more interesting asymmetric
${\mathbb{T}}^{4}$ limit313131This is because the asymmetric
${\mathbb{T}}^{4}$ connects to various semiclassical limits that have been
discussed in the literature, see the main text. prompt us to introduce a
“detuning” parameter $\Delta$, which we define as follows

$\Delta={L_{3}L_{4}-L_{1}L_{2}\over\sqrt{L_{1}L_{2}L_{3}L_{4}}}={L_{3}L_{4}-L_{1}L_{2}\over\sqrt{V}}\,.$
(99)

We always take $L_{3}L_{4}>L_{1}L_{2}$, i.e. $\Delta>0$. For small positive
$\Delta$, a nonabelian solution of the self-duality equations has been
constructed as a series expansion in powers of $\sqrt{\Delta}$
GarciaPerez:2000aiw , with the leading contribution being the abelian
background (92).

In this Section, we exhibit this solution, to the leading nontrivial order in
$\Delta$, using our notation and carefully including the dependence on
$z_{n}$. To begin, consider the classical background

$\displaystyle A_{n}(x,z)=(\bar{A}_{n}^{3}(x,z)+S_{n}(x,z)){\tau^{3}\over
2}+W_{n}(x,z)\tau^{+}+W_{n}^{*}(x,z)\tau^{-}\,,$ (100)

where $S,W$ are the deviations from the abelian background, to be determined
in terms of an expansion in $\sqrt{\Delta}$. Let us momentarily denote them by
$a_{n}\equiv S_{n}{\tau^{3}\over 2}+W_{n}\tau^{+}+W_{n}^{*}\tau^{-}$. The
construction of the self-dual solution on the asymmetric torus by means of an
expansion in $\Delta$ proceeds by imposing a self-duality condition on the
field strength of (100) and solving the resulting equations in a series
expansion in $\sqrt{\Delta}$ GarciaPerez:2000aiw . In order to solve for
$a_{n}$, it is subjected to the background-Lorentz gauge condition:

$\displaystyle D^{n}(\bar{A})a_{n}$ $\displaystyle=$
$\displaystyle\partial^{n}a_{n}+i[\bar{A}^{n},a_{n}]=0,\leavevmode\nobreak\
\text{or in components:}$ (101) $\displaystyle\leavevmode\nobreak\
\partial_{n}S^{n}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\leavevmode\nobreak\ (\partial_{n}+i\bar{A}_{n}^{3})W^{n}$
$\displaystyle=$ $\displaystyle 0.$

Further, the boundary conditions (89) imply that $S_{n}$ is periodic in all
$x_{n}$, while $W_{n}$ is periodic in $x_{1}$ and $x_{3}$, but not in $x_{2}$
and $x_{4}$. Explicitly the boundary conditions are

$\displaystyle S_{n}(x+\hat{e}_{k}L_{k})$ $\displaystyle=$ $\displaystyle
S_{n}(x),\forall k,$ (102) $\displaystyle W_{n}(x+\hat{e}_{1}L_{1})$
$\displaystyle=$ $\displaystyle W_{n}(x),$ $\displaystyle
W_{n}(x+\hat{e}_{2}L_{2})$ $\displaystyle=$ $\displaystyle e^{-i{2\pi
x_{1}\over L_{1}}}W_{n}(x),$ $\displaystyle W_{n}(x+\hat{e}_{3}L_{3})$
$\displaystyle=$ $\displaystyle W_{n}(x),$ $\displaystyle
W_{n}(x+\hat{e}_{4}L_{4})$ $\displaystyle=$ $\displaystyle e^{-i{2\pi
x_{3}\over L_{1}}}W_{n}(x).$

To proceed with the construction of the instanton on the twisted asymmetric
${\mathbb{T}}^{4}$, following GarciaPerez:2000aiw , we calculate the field
strength of (100), set its antiselfdual part to zero, and solve the equations
imposing self-duality in a series expansion in powers of $\sqrt{\Delta}$. To
write the subsequent equations, it is convenient to use a quaternion notation.
Thus, we introduce the matrices 323232The $n=1,2,3$ components of the four-
vector matrix $\sigma^{n}$ defined here should not be confused with
$\vec{\sigma}=\\{\sigma_{a},a=1,2,3\\}=(\sigma_{1},\sigma_{2},\sigma_{3})$,
the usual Pauli matrices. We use lower-case indices $a=1,2,3$ to denote the
usual Pauli matrices $\sigma_{a}$ and reserve $\tau^{a}$ for the $SU(2)$
generators.

$\displaystyle w$ $\displaystyle=$
$\displaystyle\sigma^{n}W_{n}\leavevmode\nobreak\ ,\leavevmode\nobreak\
\text{where}\leavevmode\nobreak\ \sigma^{n}=(i\vec{\sigma},1_{2\times
2}),\leavevmode\nobreak\
\bar{\sigma}^{n}=(\sigma^{n})^{\dagger}\leavevmode\nobreak\ ,$ (103)
$\displaystyle s$ $\displaystyle=$
$\displaystyle\sigma^{n}S_{n}\leavevmode\nobreak\ ,$ $\displaystyle w_{c}$
$\displaystyle=$ $\displaystyle Cw^{*}C\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{where}\leavevmode\nobreak\
C=\left(\begin{array}[]{cc}0&-i\cr i&0\end{array}\right)\leavevmode\nobreak\
.$ (106)

Here $\vec{\sigma}$ are the usual Pauli matrices, acting in spinor space, not
to be confused with the $\tau^{3},\tau^{\pm}$ group generators from earlier.
In terms of the matrices (103), the condition that the field strength of (100)
be self dual is $F_{mn}\bar{\sigma}^{m}\sigma^{n}=0$. This, using (101),
becomes

$\displaystyle
F_{mn}\bar{\sigma}^{m}\sigma^{n}=0\implies\bar{\sigma}^{n}\partial_{n}s$
$\displaystyle=$
$\displaystyle{2\pi{\Delta}\over\sqrt{V}}i\tau^{3}-i(w_{c}^{\dagger}w_{c}-w^{\dagger}w),$
(107) $\displaystyle\text{and}\leavevmode\nobreak\
\bar{\sigma}^{n}(\partial_{n}+i\bar{A}_{n}^{3})w$ $\displaystyle=$
$\displaystyle-{i\over 2}(s^{\dagger}w-w_{c}^{\dagger}s).$

The self-duality equations (107) admit a solution as a series expansion in the
${\mathbb{T}}^{4}$ detuning parameter ${\Delta}$ (99),

$\displaystyle w$ $\displaystyle=$
$\displaystyle\Delta^{1/2}\;w_{1}+\Delta^{3/2}\;w_{2}+\Delta^{5/2}\;w_{3}+\ldots,$
(108) $\displaystyle s$ $\displaystyle=$
$\displaystyle\Delta\;s_{1}+\Delta^{2}\;s_{2}+\Delta^{3}\;s_{3}+\ldots\leavevmode\nobreak\
.$

The difference in powers of $\sqrt{\Delta}$ in the series for $w$ and $s$
follows by the structure of the self-duality equations (107).

Here, we shall study only the leading-order solution. To exhibit it
explicitly, we substitute the expansions in (108) in (107) and keeping the
leading term in each equation, we find that $s_{1},w_{1}$ obey

$\displaystyle\bar{\sigma}^{n}\partial_{n}s_{1}$ $\displaystyle=$
$\displaystyle{2\pi\over\sqrt{V}}i\tau^{3}-i(w_{1c}^{\dagger}w_{1c}-w_{1}^{\dagger}w_{1})\,,$
(109) $\displaystyle\bar{\sigma}^{n}(\partial_{n}+i\bar{A}_{n}^{3})w_{1}$
$\displaystyle=$ $\displaystyle 0\,.$

We begin with the equation for $w_{1}$—the complex quaternion
$\sigma^{m}W_{m}$, obeying the boundary conditions (102). Periodicity in
$x_{1},x_{3}$ is obeyed by writing a Fourier series

$\displaystyle
w_{1}=\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}w_{1}^{n_{1},n_{3}}(x_{2},x_{4}),$ (110)

while the conditions

$\displaystyle w_{1}^{n_{1},n_{3}}(x_{2}+L_{2},x_{4})$ $\displaystyle=$
$\displaystyle w_{1}^{n_{1}+1,n_{3}}(x_{2},x_{4})\leavevmode\nobreak\ ,$ (111)
$\displaystyle w_{1}^{n_{1},n_{3}}(x_{2},x_{4}+L_{4})$ $\displaystyle=$
$\displaystyle w_{1}^{n_{1},n_{3}+1}(x_{2},x_{4})\leavevmode\nobreak\ ,$

ensure the rest of (102) and, by induction, imply that all $w^{n_{1},n_{3}}$
can be expressed via a single function,

$w_{1}^{n_{1},n_{3}}(x_{2},x_{4})=u_{1}(x_{2}-n_{1}L_{2},x_{4}-n_{3}L_{4}).$
(112)

Before we continue, we note that (112) together with (110) implies that the
norm of $w_{1}$ on ${\mathbb{T}}^{4}$ can be expressed as an
${\mathbb{R}}^{2}$-integral

$\displaystyle||w_{1}||$ $\displaystyle\equiv$
$\displaystyle\int_{T^{4}}d^{4}x\,{\rm
tr}\,w_{1}^{\dagger}w_{1}=\sum_{n_{1},n_{3}}\int\limits_{0}^{L_{2}}dx_{2}\int\limits_{0}^{L_{4}}dx_{4}\,{\rm
tr}\,(w_{1}^{n_{1},n_{3}})^{\dagger}w_{1}^{n_{1},n_{3}}$ (113)
$\displaystyle=$
$\displaystyle\int\limits_{-\infty}^{\infty}dx_{2}\int\limits_{-\infty}^{\infty}dx_{4}\,{\rm
tr}\,u_{1}^{\dagger}(x_{2},x_{4})u_{1}(x_{2},x_{4})\leavevmode\nobreak\ .$

Thus, finiteness of $||w_{1}||$ implies that the elements of the quaternion
$u_{1}(x_{2},x_{4})$ should be square-normalizable functions on the
${\mathbb{R}}^{2}$-plane spanned by $x_{2},x_{4}$. Any complete basis of such
functions on ${\mathbb{R}}^{2}$ can be used, but below, we shall argue that an
especially convenient basis is given by appropriately chosen simple harmonic
oscillator eigenfunctions.

Continuing with our self-duality equations (109), now the second equation in
(109) together with (112) implies that

$\displaystyle\left(\begin{array}[]{cc}(\partial_{4}+i{z_{4}\over
L_{4}})+{2\pi\over L_{3}L_{4}}[x_{4}-(n_{3}-{z_{3}\over
2\pi})L_{4}]&-(\partial_{2}+i{z_{2}\over L_{2}})+{2\pi\over
L_{1}L_{2}}[x_{2}-(n_{1}-{z_{1}\over 2\pi})L_{2}]\\\
(\partial_{2}+i{z_{2}\over L_{2}})+{2\pi\over
L_{1}L_{2}}[x_{2}-(n_{1}-{z_{1}\over 2\pi})L_{2}]&(\partial_{4}+i{z_{4}\over
L_{4}})-{2\pi\over L_{3}L_{4}}[x_{4}-(n_{3}-{z_{3}\over
2\pi})L_{4}]\end{array}\right)$ (116) $\displaystyle\times
u_{1}(x_{2}-n_{1}L_{2},x_{4}-n_{3}L_{4})=0\,.$

The holonomies in the partial derivatives can be absorbed by defining a new
quaternion $\tilde{u}_{1}(x_{2}-n_{1}L_{2},x_{4}-n_{3}L_{4})$ by means of the
redefinition

$u_{1}(x_{2}-n_{1}L_{2},x_{4}-n_{3}L_{4})=e^{-i{z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})-i{z_{4}\over
L_{4}}(x_{4}-n_{3}L_{4})}\tilde{u}_{1}(x_{2}-n_{1}L_{2},x_{4}-n_{3}L_{4}).$
(118)

Using this, equation (116) becomes

$\displaystyle\left(\begin{array}[]{cc}\partial_{4}+{2\pi\over
L_{3}L_{4}}[x_{4}-(n_{3}-{z_{3}\over 2\pi})L_{4}]&-\partial_{2}+{2\pi\over
L_{1}L_{2}}[x_{2}-(n_{1}-{z_{1}\over 2\pi})L_{2}]\\\ \partial_{2}+{2\pi\over
L_{1}L_{2}}[x_{2}-(n_{1}-{z_{1}\over 2\pi})L_{2}]&\partial_{4}-{2\pi\over
L_{3}L_{4}}[x_{4}-(n_{3}-{z_{3}\over 2\pi})L_{4}]\end{array}\right)$ (121)
$\displaystyle\times\tilde{u}_{1}(x_{2}-n_{1}L_{2},x_{4}-n_{3}L_{4})=0\,.$
(122)

As already alluded to, we now recognize that the first order differential
operators appearing above are the creation and annihilation operators of
simple harmonic oscillators (SHO) of frequencies $2\pi/(L_{1}L_{2})$ and
$2\pi/(L_{3}L_{4})$. Let $h_{p}^{ij}(x)$ be the normalized $p$-th state of the
SHO of frequency

$\omega_{ij}={2\pi\over L_{i}L_{j}}.$ (123)

The relations that we need in what follows are the orthonormality relations

$\displaystyle\int\limits_{-\infty}^{\infty}dxh^{ij}_{n}(x)h^{ij}_{m}(x)$
$\displaystyle=$ $\displaystyle\delta_{nm}\leavevmode\nobreak\ ,n,m=0,1,2...,$
(124)

as well as the raising and lowering operators which obey the usual equations

$\displaystyle{\sqrt{L_{i}L_{j}\over 4\pi}}\;\left[\partial_{x}+{2\pi\over
L_{i}L_{j}}x\right]\;h^{ij}_{n}(x)$ $\displaystyle=$
$\displaystyle\sqrt{n}\;h^{ij}_{n-1}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
A_{ij}|n\rangle_{ij}=\sqrt{n}|n-1\rangle_{ij},$ (125)
$\displaystyle\leavevmode\nobreak\ {\sqrt{L_{i}L_{j}\over
4\pi}}\;\left[-\partial_{x}+{2\pi\over L_{i}L_{j}}x\right]\;h^{ij}_{n}(x)$
$\displaystyle=$ $\displaystyle\sqrt{n+1}\;h^{ij}_{n+1}(x)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
A^{\dagger}_{ij}|n\rangle_{ij}=\sqrt{n+1}|n+1\rangle_{ij}\,,$

where we have indicated that the differential operators in the middle and
bottom line are the familiar lowering and raising operators $A_{ij}$,
$A_{ij}^{\dagger}$ of SHOs of frequencies $\omega_{ij}$ (123). In terms of
$A_{12},A_{12}^{\dagger},A_{34},A_{34}^{\dagger}$,333333With appropriately
shifted center, as in the argument of $\tilde{u}_{1}$ below, a fact which we
do not make explicit for brevity. we can rewrite (122) as

$\displaystyle\left(\begin{array}[]{cc}\sqrt{2\omega_{34}}A_{34}&\sqrt{2\omega_{12}}A_{12}^{\dagger}\\\
\sqrt{2\omega_{12}}A_{12}&-\sqrt{2\omega_{34}}A_{34}^{\dagger}\end{array}\right)\tilde{u}_{1}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2},x_{4}-(n_{2}-{z_{3}\over 2\pi})L_{2})=0\,.$ (128)

It is then clear that this equation is satisfied by $\tilde{u}_{1}$ given by
the product of ground state wave functions of the $\omega_{12}$ and
$\omega_{34}$ SHOs times a constant $2\times 2$ matrix $C_{1}$:

$\tilde{u}_{1}=h_{0}^{12}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2})\;h_{0}^{34}(x_{4}-(n_{3}-{z_{3}\over 2\pi})L_{4})\;C_{1},$ (129)

where $C_{1}$ obeys, as a consequence of (128)

$\left(\begin{array}[]{cc}0&\sqrt{2\omega_{12}}\\\
0&-\sqrt{2\omega_{34}}\end{array}\right)C_{1}=0\implies
C_{1}=\left(\begin{array}[]{cc}a&b\\\
0&0\end{array}\right)\leavevmode\nobreak\ ,$ (130)

with arbitrary complex numbers $a,b$.

B

efore continuing, let us make the following important remark:

  * •

W

e stress that (129, 130) is the unique solution and not simply a consistent
one. Because of the ${\mathbb{R}}^{2}$-normalizability (113), any solution can
be expressed in terms of $\omega_{12}$ and $\omega_{34}$ SHO eigenfunctions.
Explicitly, for the $ab$-th element of the quaternion $\tilde{u}_{1}$, we have
$\tilde{u}_{1\;ab}(x_{2},x_{4})=\sum\limits_{n,m=0}^{\infty}c_{ab}^{nm}h^{12}_{n}(x_{2})h^{34}_{m}(x_{4})$.
Plugging into (128) and using (124, 125) one finds a set of linear equations
for the constants $c_{ab}^{nm}$:

$\displaystyle\sqrt{\omega_{34}}\;c_{11}^{p,q+1}\sqrt{q+1}+\sqrt{\omega_{12}}\;c_{21}^{p-1,q}\sqrt{p}$
$\displaystyle=$ $\displaystyle 0,\leavevmode\nobreak\
\text{for}\leavevmode\nobreak\ p,q=0,1,2,3...,$
$\displaystyle\sqrt{\omega_{34}}\;c_{12}^{p,q+1}\sqrt{q+1}+\sqrt{\omega_{12}}\;c_{22}^{p-1,q}\sqrt{p}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sqrt{\omega_{12}}\;c_{11}^{p+1,q}\sqrt{p+1}-\sqrt{\omega_{34}}\;c_{21}^{p,q-1}\sqrt{q}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\sqrt{\omega_{12}}\;c_{12}^{p+1,q}\sqrt{q+1}-\sqrt{\omega_{34}}\;c_{22}^{p,q-1}\sqrt{q}$
$\displaystyle=$ $\displaystyle 0.$ (131)

One begins by noticing that $c_{11}^{00}$ and $c_{12}^{00}$ do not appear
above (this is easy to see by inspection recalling the range of $p,q$, which
ensure that terms containing the nonexistent coefficients, like
$c^{-1,*}_{**}$, are always multiplied by $0$). Then, recursively solving the
equations, one finds that their only solution is that all $c_{ij}^{nm}$ vanish
except for the two that do not appear in equations (• ‣ A.1), the undetermined
$c_{11}^{00}$ and $c_{12}^{00}$, called $a$ and $b$ in (129).

Let us flesh out the argument. We consider the first and third equations in (•
‣ A.1); the second and fourth can be considered similarly and are left as an
exercise. First notice that (• ‣ A.1) imply that $c_{11}^{0,p}=c_{11}^{p,0}=0$
for $p=1,2,3...$. Further, denote $\delta=\sqrt{\omega_{12}\over\omega_{34}}$
and shift the $p$’s and $q$’s in (• ‣ A.1) such that both equations involve
$c_{11}^{p,q}$ we find

$\displaystyle c_{11}^{p,q}\sqrt{q}+\delta c_{21}^{p-1,q-1}\sqrt{p}$
$\displaystyle=$ $\displaystyle 0,$ (132) $\displaystyle
c_{11}^{p,q}\sqrt{p}-{1\over\delta}c_{21}^{p-1,q-1}\sqrt{q}$ $\displaystyle=$
$\displaystyle 0.$

Next, we multiply the top equation by $\sqrt{p}$ and the bottom by $\sqrt{q}$,
subtract them, and find

$c_{21}^{p-1,q-1}(p\delta+{q\over\delta})=0.$ (133)

Recalling that the appropriate range for $p,q$ in (133) is $1,2,3...$, this
implies that all $c_{21}$’s vanish. Hence by (132), all $c_{11}^{p,q}$ with
$(p,q)\neq(0,0)$ vanish as well.

W

e next turn to the first of the self-duality equations in (109). Since $s_{1}$
is periodic, the r.h.s. of the equation has no constant Fourier mode. Thus,
consistency implies that the constant mode of the r.h.s vanishes as well; this
will be seen to fix the coefficients $a,b$ in terms of the volume of the torus
up to an overall phase. To determine them, let us collect everything—in
reverse order, eqns. (129), (122), (118), (112), and (110)—and exhibit the
full solution for $w_{1}$ we found so far:

$\displaystyle w_{1}=F(x,z)C_{1}=F(x,z)\left(\begin{array}[]{cc}a&b\\\
0&0\end{array}\right)\,,$ (136)

where

$\displaystyle F(x,z)$ $\displaystyle=$
$\displaystyle\sqrt{L_{2}L_{4}}\sum\limits_{n_{1},n_{3}=-\infty}^{\infty}e^{-i2\pi(n_{1}{x_{1}\over
L_{1}}+n_{3}{x_{3}\over L_{3}})}e^{-i{z_{2}\over
L_{2}}(x_{2}-n_{1}L_{2})-i{z_{4}\over L_{4}}(x_{4}-n_{3}L_{4})}$ (137)
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \times\leavevmode\nobreak\
\leavevmode\nobreak\ h_{0}^{12}(x_{2}-(n_{1}-{z_{1}\over
2\pi})L_{2})\;h_{0}^{34}(x_{4}-(n_{3}-{z_{3}\over
2\pi})L_{4})\leavevmode\nobreak\ ,$

with the normalization

$\int_{T^{4}}|F|^{2}=V,$ (138)

recalling (125). Note that $F$ is dimensionless and $a,b$ have dimension one.
In addition, under the reflection of all $x_{n}$ and all $z_{n}$, after
relabeling $n_{1},n_{3}\rightarrow-n_{1},-n_{3}$ and noting that $h_{0}^{ij}$
are even functions of their arguments, we find

$F(-x,-z)=F(x,z)\leavevmode\nobreak\ .$ (139)

With (136, 137), we have that the vanishing constant mode of $\partial s$ from
equation (109) implies that

$\displaystyle{2\pi\sqrt{V}}\tau^{3}$ $\displaystyle=$
$\displaystyle(C_{c}^{\dagger}C_{c}-C^{\dagger}C)\int\limits_{T^{4}}d^{4}x|F(x,z)|^{2}\leavevmode\nobreak\
,\text{or}$ $\displaystyle{2\pi\sqrt{V}}\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}|b|^{2}-|a|^{2}&-2a^{*}b\\\
-2ab^{*}&|a|^{2}-|b|^{2}\end{array}\right)V\,.$ (144)

We thus conclude that $a=0$ while $|b|^{2}={2\pi\over V^{1/2}}$. Thus, our
solution has the form

$\displaystyle w(x,z)$ $\displaystyle=$
$\displaystyle\sigma^{n}W_{n}=\sqrt{\Delta}w_{1}(x,z)=\sqrt{\Delta}F(x,z)e^{i\alpha}{(2\pi)^{1/2}\over
V^{1/4}}\left(\begin{array}[]{cc}0&1\\\
0&0\end{array}\right)+{\cal{O}}(\Delta)\,,$ (147) $\displaystyle s(x,z)$
$\displaystyle=$ $\displaystyle\sigma^{n}S_{n}={\cal{O}}(\Delta)\,,$ (148)

with the phase $\alpha$ due to the gauge freedom to rotate around the
$\tau^{3}$ isospin direction.

Before we continue, let us exhibit the self-dual solution we found, eqn.
(147), $w=\Delta^{1/2}w_{1}$ in terms of the 4-vector gauge potentials,
$W_{n},S_{n}$ of (100):

$\displaystyle S_{n}$ $\displaystyle=$ $\displaystyle{\cal{O}}(\Delta)\,,$
$\displaystyle W_{1}(x,z,\alpha)$ $\displaystyle=$ $\displaystyle-{i\over
2}\sqrt{\Delta}F(x,z)e^{i\alpha}{\sqrt{2\pi}\over
V^{1/4}}+{\cal{O}}(\Delta^{3/2})\equiv-\gamma
e^{i\alpha}F(x,z),\text{where}\leavevmode\nobreak\ \gamma\equiv{i\over
2}\sqrt{2\pi\Delta\over\sqrt{V}},$ $\displaystyle W_{2}(x,z,\alpha)$
$\displaystyle=$ $\displaystyle{1\over
2}\sqrt{\Delta}F(x,z)e^{i\alpha}{\sqrt{2\pi}\over
V^{1/4}}+{\cal{O}}(\Delta^{3/2})=iW_{1}+{\cal{O}}(\Delta^{3/2}),$
$\displaystyle W_{3}$ $\displaystyle=$ $\displaystyle{\cal{O}}(\Delta^{3/2}),$
$\displaystyle W_{4}$ $\displaystyle=$ $\displaystyle{\cal{O}}(\Delta^{3/2}).$
(149)

We note that the above implies that the only gauge-field zero modes in our
background are the one due to translations (corresponding to shifting $z_{n}$)
and constant gauge rotations around $\tau^{3}$ (corresponding to shifts of
$\alpha$). The former are physical and gauge invariant quantities
characterizing the solution will be seen to depend on $z_{n}$, while the
latter are gauge artifacts with no physical quantity exhibiting
$\alpha$-dependence.

For calculating the action density and gauge-invariant “electric” and
“magnetic” fields to leading nontrivial order, one also needs the expression
for $S_{1}$, which was not given in (A.1). Here we simply describe how this
can be found. We have, with $w_{1}$ from (147)

$\displaystyle w_{1}$ $\displaystyle=$ $\displaystyle
F\left(\begin{array}[]{cc}0&b\cr 0&0\end{array}\right)\leavevmode\nobreak\
,\leavevmode\nobreak\ b=e^{i\alpha}{\sqrt{2\pi}\over V^{1\over 4}}\implies
w_{1}^{\dagger}w_{1}=|F|^{2}\left(\begin{array}[]{cc}0&0\cr
0&|b|^{2}\end{array}\right)\,,$ (154) $\displaystyle w_{1c}$ $\displaystyle=$
$\displaystyle
F^{*}\left(\begin{array}[]{cc}0&0\cr-b^{*}&0\end{array}\right)\implies
w_{1c}^{\dagger}w_{1c}=|F|^{2}\left(\begin{array}[]{cc}|b|^{2}&0\cr
0&0\end{array}\right)\,.$ (159)

Hence, from the top equation in (109), we obtain the equation for $s_{1}$:

$\displaystyle\bar{\sigma}^{n}\partial_{n}s_{1}=i\tau^{3}{2\pi\over\sqrt{V}}\left(1-|F(x,z)|^{2}\right)\leavevmode\nobreak\
.$ (160)

The consistency of this equation (the absence of a constant Fourier mode of
the r.h.s.) discussed above is yet again seen to follow from (138). We also
note that equation (160) allows adding a constant to $s_{1}$; this is a
contribution to the constants $z_{n}$, already included in our background
(92). To solve (160), we multiply by $\sigma^{m}\partial_{m}$ to obtain

$\displaystyle\partial_{n}\partial^{n}s_{1}=-i\sigma^{m}\tau^{3}{2\pi\over\sqrt{V}}(F^{*}\partial_{m}F+F\partial_{m}F^{*})\leavevmode\nobreak\
.$ (161)

This equation is solved by expanding both $s_{1}$ and the r.h.s. in Fourier
modes in $x_{1}...x_{4}$ and equating the coefficients. This can be explicitly
done and a unique $s_{1}$ can be written down, but we shall not need the
explicit form here. As we mentioned in Section 1.3, understanding the higher
orders in the $\Delta$ expansion is an interesting problem for future studies.

###  A.2 The field strength tensor of the fractional instanton to order
$\Delta$

To find the gauge invariants of the solution to order $\Delta$, we need the
field strength of $s_{1}$, namely343434As stressed above, we shall not need
its explicit form, but only the relations it satisfies, which are important to
ensure self-duality.

$\displaystyle F^{s}_{nm}\equiv(\partial_{n}S_{m}-\partial_{m}S_{n})\,,$ (162)

where for brevity we use $S_{n}$ to denote the coefficient of the
${\cal{O}}(\Delta)$ part of $S_{n}$. The l.h.s. of (160) has the form

$\displaystyle\left(\begin{array}[]{cc}-i\partial_{3}+\partial_{4}&-i\partial_{1}-\partial_{2}\cr-i\partial_{1}+\partial_{2}&i\partial_{3}+\partial_{4}\end{array}\right)\left(\begin{array}[]{cc}iS_{3}+S_{4}&iS_{1}+S_{2}\cr
iS_{1}-S_{2}&-iS_{3}+S_{4}\end{array}\right)$ (167) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\partial_{n}S^{n}+i(F^{s}_{12}-F^{s}_{34})&-F^{s}_{13}-F^{s}_{24}+i(F^{s}_{23}-F^{s}_{14})\cr
F^{s}_{13}+F^{s}_{24}+i(F^{s}_{23}-F^{s}_{14})&\partial_{n}S^{n}-i(F^{s}_{12}-F^{s}_{34})\end{array}\right)\,.$
(170)

Comparing with the r.h.s. of (160), using $\partial_{n}S^{n}=0$, we conclude
that

$\displaystyle F^{s}_{12}-F^{s}_{34}$ $\displaystyle=$
$\displaystyle{2\pi\over\sqrt{V}}(1-|F(x,z)|^{2}),$ $\displaystyle
F^{s}_{13}+F^{s}_{24}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
F^{s}_{23}-F^{s}_{14}$ $\displaystyle=$ $\displaystyle 0.$ (171)

Note that these terms should be multiplied by $\Delta$. Also note that these
equations by themselves do not determine the individual $F_{12}^{s}$,
$F_{34}^{s}$ (etc.) but only their non-self-dual parts. To find the individual
$F_{mn}^{s}$, we need to solve (161) to first find $s_{1}$.

The field strength of the ${\cal{O}}(1)$ abelian background (92) is
$F_{mn}^{(0)}$, already given in (97). Combining this with the results for
$F_{mn}^{s}$ from (A.2), we find

$\displaystyle F_{mn}^{(0+1(s))}$ $\displaystyle=$
$\displaystyle{\tau^{3}\over 2}\left(\begin{array}[]{cccc}0&-{2\pi\over
L_{1}L_{2}}+\Delta F_{12}^{s}&\Delta F_{13}^{s}&\Delta F_{14}^{s}\cr{2\pi\over
L_{1}L_{2}}-\Delta F_{12}^{s}&0&\Delta F_{14}^{s}&-\Delta F_{13}\cr-\Delta
F_{13}^{s}&-\Delta F_{14}^{s}&0&-{2\pi\over L_{3}L_{4}}+\Delta
F_{34}^{s}\cr-\Delta F_{14}^{s}&\Delta F_{13}^{s}&{2\pi\over
L_{3}L_{4}}-\Delta F_{34}^{s}&0\end{array}\right)\,,$ (176)

where $F_{34}^{s}$ is understood to be expressed via $F_{12}^{s}$ through the
first equation in (A.2).

To study the complete field strength tensor in the $\tau^{3}$ direction to
order $\Delta$, we must add to (176) the ${\cal{O}}(\Delta)$ contribution to
the $\tau^{3}$ field strength coming from the commutator
$F_{mn}^{(1(w))}=i[W_{m}\tau^{+}+W_{m}^{*}\tau^{-},W_{n}\tau^{+}+W_{n}^{*}\tau^{-}]={\tau^{3}\over
2}2i(W_{m}W_{n}^{*}-W_{m}^{*}W_{n})$. For the background (A.1), this is only
nonzero for $F_{12}^{(1(w))}$. We find

$\displaystyle F_{12}^{(1(w))}=-F_{21}^{(1(w))}={\tau^{3}\over
2}{2\pi\Delta\over\sqrt{V}}|F(x,z)|^{2}\leavevmode\nobreak\
,\leavevmode\nobreak\ \text{all other}\;F_{mn}^{(1(w))}=0.$ (178)