${1\over 2}({dh\over d\text{x}_{3}})^{2}+V(h)=C$. The minimum action
corresponds to when $C=0$ therefore ${1\over 2}({dh\over
d\text{x}_{3}})^{2}=-V(h)$:

$\displaystyle S[h]$
$\displaystyle=\int^{+\infty}_{-\infty}d\text{x}_{3}\\{{1\over 2}({dh\over
d\text{x}_{3}})^{2}-V(h)\\}=2\int^{0}_{b\over 2}{d\text{x}_{3}\over
dh}dh\\{-2V(h)\\}=2m\int_{0}^{b\over 2}dhh\sqrt{1+{2\lambda h^{2}\over
m^{2}}}\leavevmode\nobreak\ .$ (101)

We expand the square root424242The Taylor series expansion of
$\sqrt{1+\text{x}}$ converges for $|\text{x}|<r=1$. Evaluating
$\text{x}={2\lambda h^{2}/m^{2}}$ for $h={b/2}$ with values of parameters from
below equation (78) gives $|\text{x}|={\pi^{2}/12}<1$ which lies within the
radius of convergence. As a reminder we mention that the condition
$|\text{x}|=|{2\lambda h^{2}/m^{2}}|<1$ is not strictly required in dYM since
the full potential is cosine which would allow for a wider range of these
parameters. and evaluate the integral term by term. We also multiply the
action by a negative sign to take into account the negative in the exponent.
We find:

$\displaystyle-S[h]$ $\displaystyle=\int_{0}^{b\over
2}dh\\{-2mh+-{2\lambda\over
m}h^{3}+\overset{\infty}{\underset{n=2}{\sum}}(-1)^{n}{2\lambda^{n}\over
m^{2n-1}}{(2n-3)!!\over n!}h^{2n+1}\\}$ $\displaystyle=-m{b^{2}\over
4}-{\lambda b^{4}\over
32m}+\overset{\infty}{\underset{n=2}{\sum}}{(-\lambda)^{n}b^{2n+2}\over
m^{2n-1}}{(2n-3)!!\over 2^{2n+2}(n+1)!}$ (102) $\displaystyle{=-{mb^{2}\over
4}\\{1+{1\over 2}{\lambda b^{2}\over
4m^{2}}-\overset{\infty}{\underset{n=2}{\sum}}{(-1)^{n}{(2n-3)!!\over(n+1)!}({\lambda
b^{2}\over 4m^{2}})^{n}}\\}}\leavevmode\nobreak\ .$

The order $\lambda$ and $\lambda^{2}$ terms in (4.1) match with the
coefficients of ${\hat{R}\hat{T}\over\beta}$ in (96) and (99) respectively.
This further demonstrates the validity of the perturbative evaluation of the
saddle point. The verification to higher orders in $\lambda$ ($n\geq 3$) can
be made if the corresponding diagrams are evaluated.

The higher order terms in (77) ($g^{6}_{1},g^{8}_{1},...$) and their cross
terms with each other can also be evaluated similarly.

Next we will compare the $SU(2)$ $k$-string result of the perturbative
evaluation of the saddle point to next to leading term, with the exact result
of $SU(2)$ $k$-string in Table 2 of Section 3.4. The exact $SU(2)$ saddle
point area law is given by (58), (61):

$\displaystyle{Z^{\eta}\over
Z^{0}}=\text{exp}(-T_{1}RT)=\text{exp}(-{2\bar{T}_{1}\over\sqrt{2}{\beta}}\hat{R}\hat{T})=\text{exp}(-{8\over{\beta}}\hat{R}\hat{T}),\
\hat{R},\hat{T}\rightarrow\infty,\ \beta\rightarrow 0\leavevmode\nobreak\ ,$
(103)

where $R={1\over m_{\gamma}}\hat{R}$, $T={1\over m_{\gamma}}\hat{T}$,
$\beta={m^{3}_{\gamma}/\tilde{\zeta}}$ and, from Table 2 of Section 3.4,
$\bar{T}_{1}={8\over\sqrt{2}}$.

Using (89) and (96) the perturbative saddle point method gives:

$\displaystyle{Z^{\eta}\over Z^{0}}=\text{exp}(-{1\over\beta}({mb^{2}\over
4}+{\lambda b^{4}\over
32m}+...)\hat{R}\hat{T})=\text{exp}(-{1\over\beta}(7.84+...\
)\hat{R}\hat{T}),\hat{R},\hat{T}\rightarrow\infty,\ \beta\rightarrow
0\leavevmode\nobreak\ .$ (104)

From the comment below equation (78), the values of $m=2$, $\lambda=-{8\over
4!}$ and $b=\sqrt{2}\pi$ have been replaced. This shows the convergence of the
$SU(2)$ perturbative saddle point method result to the exact value obtained by
a direct calculation of the saddle point.

###  4.2 Evaluation for SU(N)

Having shown how the method works for $SU(2)$, in this Section we will
evaluate the $k$-string tensions perturbatively to next to leading order for
$SU(N)$. We start from (recall (73)):

$Z^{\eta}=\int[D\sigma]\text{exp}(-{1\over\beta}\int_{{\rm
I\\!R}^{3}}d^{3}\hat{\text{x}}\\{{1\over
2}(\partial_{l}\sigma)^{2}-(\mu_{k})_{j}\partial_{l}\sigma_{j}\partial_{l}\eta+{1\over
2}(\partial_{l}\eta)^{2}{\mu_{k}}^{2}+{1\over
2}(\sigma_{j}-\sigma_{j+1})^{2}-{1\over
4!}(\sigma_{j}-\sigma_{j+1})^{4}+...\\})\leavevmode\nobreak\ .$ (105)

The mass term in (105) can be diagonalized. Let
$(\sigma_{j}-\sigma_{j+1})^{2}=\sigma^{T}A\sigma$ (for $j=1,...,N$) where $A$
is the following $N\times N$ matrix for $N\geq 3$:

$A_{ij}=2\ \text{for}\ i=j\ ,\ A_{ij}=-1\ \text{for}\ |i-j|=1\ ,\
A_{1N}=A_{N1}=-1\ ,\ A_{ij}=0\ \text{otherwise}\leavevmode\nobreak\ ,$ (106)

while for ${SU(2)}$:

$A_{11}=A_{22}=2\ \text{and}\ A_{12}=A_{21}=-2\leavevmode\nobreak\ .$ (107)

The matrix $A$ is symmetric and can be diagonalized by an orthogonal
transformation $D$, explicitly $D^{T}D=I$, $A=D\Lambda D^{T}$. $D=(v_{1}\
v_{2}\ ...\ v_{N})$ with $v_{q}$ the eigenvectors of $A$. This gives
$(\sigma_{j}-\sigma_{j+1})^{2}=g^{T}\Lambda g$ with $g=D^{T}\sigma$. $A$ has
an eigenvalue434343The diagonalization matrix $D$ has the effect of an
${\mathbb{Z}}_{N}$ Fourier transform and the eigenvalues of $A$ are
$\Lambda_{q}=4\sin^{2}{\pi q\over N}$, $q=1,...,N-1$ and $\Lambda_{N}=0$. As
discussed in Cherman:2016jtu , this is the spectrum of a latticized emergent
dimension of $N$ sites. of zero corresponding to the eigenvector
$v^{T}_{N}\equiv({1\over\sqrt{N}},...,{1\over\sqrt{N}})$. The corresponding
field $g_{N}=(\sigma_{1}+...+\sigma_{N})/\sqrt{N}$ would be a massless
component which decouples from the rest of the fields and hence can be
neglected. We will now express the higher order interaction terms in a form
convenient for a perturbative expansion. For this define the matrix $B$ as
follows:

$B_{ij}=1\ \text{for}\ i=j\ ,\ B_{ij}=-1\ \text{for}\ j=i+1\ ,\ B_{ij}=0\
\text{otherwise}\leavevmode\nobreak\ .$ (108)

Defining $h_{q}\equiv\sigma_{q}-\sigma_{q+1}$ for $q=1,...,N-1$ we have
$BDg=B\sigma=\big{(}\begin{array}[]{c}h\\\ \sigma_{N}\end{array}\big{)}$. Also
define the top left $(N-1)\times(N-1)$ block of the matrix $BD$ as
$K\equiv(BD)_{N-1\times N-1}$. Since the first $N-1$ elements of the last
column of $BD$ are zero444444Because $D=(v_{1},...,v_{N})$, where $v_{q}$,
$q=1,...N-1$ are the eigenvectors of $A$ with eigenvalues
$\Lambda_{q}=4\sin^{2}{\pi q\over N}$ and
$v_{N}={1\over\sqrt{N}}(1,1,1...,1)^{T}$ is the zero eigenvector. For use
below, the other $N-1$ eigenvectors, for brevity shown for odd $N$ only, with
components $v_{q}^{l}$ are: $v_{q<{N\over 2}}^{l}=\sqrt{2\over N}\sin{2\pi
ql\over N}$ and $v_{{N\over 2}<q<N}^{l}=\sqrt{2\over N}\cos{2\pi ql\over N}$.
this gives: $K_{pq}g_{q}=h_{p}\equiv\sigma_{p}-\sigma_{p+1}$ for
$p=1,...,N-1$. Using the previous notations and definitions, (105) can now be
rewritten as:

$\begin{split}Z^{\eta}=\int[Dg]\text{exp}(&-{1\over\beta}\int_{{\rm
I\\!R}^{3}}d^{3}{\text{x}}\\{{1\over 2}(\partial_{l}g_{N})^{2}+{1\over
2}(\partial_{l}g_{q})^{2}+{1\over
2}\Lambda_{q}g^{2}_{q}-(\mu_{k})_{j}D_{jq}\partial_{l}g_{q}\partial_{l}\eta\\\
&+{1\over 2}(\partial_{l}\eta)^{2}{\mu_{k}}^{2}-{1\over
4!}\underset{p}{\sum}(K_{pq}g_{q})^{4}-{1\over
4!}(\underset{p}{\sum}K_{pq}g_{q})^{4}+...\\})\leavevmode\nobreak\
,\end{split}$ (109)

where a summation over $p,q$ and $l$ is implicit. Note that
$(\mu_{k})_{j}D_{jN}=0$, hence the massless mode $g_{N}$ completely decouples
from the rest of the modes and interactions. Integrating by parts the Wilson
loop term and rescaling $g_{q}\rightarrow\sqrt{\beta}g_{q}$, we cast it,
similar to (81), into a form appropriate for a perturbative evaluation of the
saddle point:

$\displaystyle Z^{\eta}=\int[Dg]\text{exp}($ $\displaystyle-\int_{{\rm
I\\!R}^{3}}d^{3}{\text{x}}\\{{1\over 2}(\partial_{l}g_{N})^{2}+{1\over
2}(\partial_{l}g_{q})^{2}+{1\over 2}\Lambda_{q}g^{2}_{q}+{1\over
2}({b_{q}\over 2\pi})^{2}(\partial_{l}\eta)^{2}$ (110)
$\displaystyle+\beta{\lambda\over
8}\underset{p}{\sum}(K_{pq}g_{q})^{4}+\beta{\lambda\over
8}(\underset{p}{\sum}K_{pq}g_{q})^{4}+...\\})\text{exp}(+{b_{q}\over\sqrt{\beta}}\int_{A}d\text{x}_{1}d\text{x}_{2}\partial_{3}g_{q})\leavevmode\nobreak\
.$

Here, we defined $b_{q}=2\pi(\mu_{k})_{j}D_{jq}$ and $\lambda=-{8\over 4!}$.
To evaluate the Wilson loop exponent using the quadratic terms, we follow
steps similar to the ones leading from (82) to (87). We obtain the analogue of
(87) for $SU(N)$:

${\\{Z^{\eta}_{g^{4}}\\}_{\lambda=0}\over\\{Z^{\eta}_{g^{4}}\\}_{b_{q}=\lambda=0}}=\text{exp}(-{1\over\beta}\\{{\Lambda_{q}b_{q}^{2}\over
2}\underset{\text{A
A}}{\iint^{\prime}}P_{q}(\text{x}-\text{x}^{\prime})+{b_{q}^{2}\over
2}\underset{b(\text{A})\
b(\text{A})}{\iint^{\prime}}d\text{x}^{l}d\text{x}^{\prime
k}\delta^{lk}(P_{q}(\text{x}-\text{x}^{\prime})-{1\over
4\pi|\text{x}-\text{x}^{\prime}|})\\})\leavevmode\nobreak\ ,$ (111)

where
$P_{q}(\text{x}-\text{x}^{\prime})={\text{exp}(-\sqrt{\Lambda_{q}}|\text{x}-\text{x}^{\prime}|)/(4\pi|\text{x}-\text{x}^{\prime}|})$
and with an implicit summation over $q$. In the limit of a large Wilson loop
area ($\hat{R},\hat{T}\rightarrow\infty$), eq. (111), following a similar
calculation as in (4.1), reduces to:

${\\{Z^{\eta}_{g^{4}}\\}_{\lambda=0}\over\\{Z^{\eta}_{g^{4}}\\}_{b_{q}=\lambda=0}}=\text{exp}(-{1\over\beta}\\{{\sqrt{\Lambda_{q}}b^{2}_{q}\over
4}\hat{R}\hat{T}+{b_{q}^{2}\over 2}\underset{b(\text{A})\
b(\text{A})}{\iint^{\prime}}d\text{x}^{l}d\text{x}^{\prime
k}\delta^{lk}(P_{q}(\text{x}-\text{x}^{\prime})-{1\over
4\pi|\text{x}-\text{x}^{\prime}|})\\})\leavevmode\nobreak\ ,$ (112)

Using the explicit form of the eigenvectors given in Footnote 44, we can
analytically show that confining strings have finite tension in the large-N
limit, despite the vanishing mass gap.454545Note that the string tension
remains fixed at large-N, despite the vanishing mass gap, as there is a number
of dual photons of nonzero mass ($\sim m_{\gamma}$) whose flux is confined, as
well as a number of dual photons approaching zero mass ($\sim m_{\gamma}/N$)
whose flux spreads out. Thus the finite tension confining string in the
gapless abelian large-N limit is a rather fuzzy object. We defer a further
study until the large-N corrections, discussed in Cherman:2016jtu  for sYM,
are better understood in the dYM case. In particular, for $k=1$ strings we
find the infinite-N limit
$a_{1}=\lim\limits_{N\rightarrow\infty}\sum\limits_{q=1}^{N-1}{1\over
4}\sqrt{\Lambda_{q}}b^{2}_{q}=4\pi$, consistent with the results from Table 3.
For further comments on the large-N limit, see Sections 5.1.2 and 5.2.
Appendix E discusses the large-N limit of the string tensions in product
representations and gives more details on the analytic calculations using the
leading-order saddle point method of this Section.

Next, we evaluate the leading corrections to this result for $SU(N)$ and
compare the values obtained with the numerical results in Table 1. The
integrals we need to do are the generalization of (94) from the $SU(2)$
calculation:

$\displaystyle\text{I}_{\lambda}\equiv\big{\langle}-\int
d^{3}\text{x}\\{\beta{\lambda\over
8}\underset{p}{\sum}(K_{pq}g_{q})^{4}+\beta{\lambda\over
8}(\underset{p}{\sum}K_{pq}g_{q})^{4}\\}{1\over
4!}({b_{q}\over\sqrt{\beta}}\int_{A}d^{2}\text{y}\partial_{\text{y}_{3}}g_{q})^{4}\big{\rangle}_{0,C}$
(113) $\displaystyle=-{1\over\beta}{4!\over 4!}{\lambda\over
8}P_{q_{1}q_{2}q_{3}q_{4}}\\{\prod^{4}_{i=1}K_{pq_{i}}b_{q_{i}}+\prod^{4}_{i=1}K_{p_{i}q_{i}}b_{q_{i}}\\}\leavevmode\nobreak\
,$

and a summation over $p$, $q_{i}$, $p_{i}$ ($i=1,2,3,4$) from $1,...,N-1$ is
implicit. The quantities $P_{q_{1}q_{2}q_{3}q_{4}}$ are given by:

$P_{q_{1}q_{2}q_{3}q_{4}}\equiv\int
d^{3}\text{x}\prod^{4}_{i=1}\int_{A}d^{2}\text{y}^{i}\partial_{\text{y}^{i}_{3}}P_{q_{i}}(\text{x}-\text{y}^{i})\leavevmode\nobreak\
.$ (114)

In the limit that $a\rightarrow\infty$ ($a=\hat{R}=\hat{T}$) using (4.1) we
have:

$P_{q_{1}q_{2}q_{3}q_{4}}\overset{a\rightarrow\infty}{=}{a^{2}\over
16}\bar{P}_{q_{1}q_{2}q_{3}q_{4}}\ \ \text{with}\ \
\bar{P}_{q_{1}q_{2}q_{3}q_{4}}=\int^{+\infty}_{-\infty}d\text{x}_{3}\prod^{4}_{i=1}\text{E}(\sqrt{\Lambda_{q_{i}}}\text{x}_{3})\leavevmode\nobreak\
,$ (115)

hence $\text{I}_{\lambda}$ becomes:

$\text{I}_{\lambda}\overset{a\rightarrow\infty}{=}-{1\over\beta}{\lambda\over
128}\bar{P}_{q_{1}q_{2}q_{3}q_{4}}\\{\prod^{4}_{i=1}K_{pq_{i}}b_{q_{i}}+\prod^{4}_{i=1}K_{p_{i}q_{i}}b_{q_{i}}\\}\hat{R}\hat{T}\leavevmode\nobreak\
,$ (116)

where $K_{pq}=B_{pj}D_{jq}$ and $b_{q}=2\pi(\mu_{k})_{j}D_{jq}$ for
$q,p=1,...,N-1$. The leading area law term in (112) and its leading correction
in (116) need to be evaluated numerically with mathematical software. The
results are summarized in Appendix D; results for $k=1$ strings for
$SU(3)-SU(10)$ were already shown in Table 3; the inclusion of the leading
order correction brings the numerical value closer to the numerical or
analytic (for $SU(3)$) value.

##  5 $\mathbf{N}$-ality dependence and large-$\mathbf{N}$ behaviour of
$\mathbf{k}$-strings in dYM

In this Section we give a discussion on two main questions regarding the
properties of $k$-string tensions: their $N$-ality dependence and large-$N$
behaviour in dYM theory. The $N$-ality of an irreducible representation of a
gauge group $SU(N)$ refers to the number of boxes in the Young tableaux of the
representation mod $N$ 09  or the charge of the representation under the
action of the center element $\text{exp}(-i{2\pi\over N})\text{I}$ of the
gauge group. It is believed that asymptotically the string tensions in a gauge
theory depend only on the $N$-ality of that representation, see 04 . This is
due to the screening effect by gluons. A cloud of gluons would transform any
charge in a representation with $N$-ality $k$ to its $k$-antisymmetric
representation which carries the stable lowest energy k-string among different
representations with the same $N$-ality $k$.

We will argue that this is also true in dYM theory and show that the
asymptotic string tensions will only depend on the $N$-ality $k$ of the
representation $k$. The screening by gluons, in the framework of dYM theory,
is due to the pair production of $W$-bosons, an effect (in principle)
calculable using weak coupling semiclassical methods.

We discuss qualitatively the role of the unbroken ${\mathbb{Z}}_{N}$ center
symmetry in dYM for the confining string properties and contrast them to those
in another theory with abelian confinement—Seiberg-Witten theory. We also
derive an approximate analytic formula for k-string ratios in dYM theory for
$N\sim 10$ and smaller and have a comparison with known scaling laws of
k-string ratios.

In regards to their large-$N$ behaviour we show that dYM $k$-string ratios
favour even power corrections similar to the sine law scaling and derive the
leading terms in the $1/N$ expansion of $k$-string ratios in dYM theory. At
the end we will argue that at large $N$ $k$-strings are not necessarily free
in gauge theories; in other words, $T$${}_{\text{k}}$ can remain smaller than
$kT$${}_{\text{1}}$ in the large-$N$ limit.

###  5.1 $N$-ality dependence

####  5.1.1 Asymptotic string tensions depend only on the $N$-ality of the
representation

The expectation value of the Wilson loop for charges in a representation $r$
with $N$-ality $k$ of $SU(N)$ evaluated using the low energy effective theory
in dYM theory in the limit of $\hat{R},\hat{T}\rightarrow\infty$ and
$\beta(={m^{3}_{\gamma}\over\tilde{\zeta}})\rightarrow 0$ using (54) and (61)
is given by:

$\langle
W_{r}(R,T)\rangle=\sum^{d(r)}_{i=1}\text{exp}(-T^{i}_{r}RT)=\sum^{d(r)}_{i=1}\text{exp}(-{2\bar{T}^{i}_{r}\over\sqrt{2}\beta}\hat{R}\hat{T})\leavevmode\nobreak\
,$ (117)

where $d(r)$ refers to the dimension of the representation $r$ and
$\bar{T}^{i}_{r}$ is given by a similar expression as (61) but with $\mu_{k}$
replaced by the weight $\mu^{i}_{r}$ of representation $r$ with $N$-ality $k$:

$\displaystyle{\bar{T}^{i}_{r}}=\underset{f(\text{z})}{\text{min}}\int_{0}^{+\infty}d\text{z}\\{({\partial
f\over\partial\text{z}})^{2}+\underset{j}{\sum}[1-\text{cos}(f_{j}-f_{j+1})]\\},\
\ \ \ f(+\infty)=0,f(0)=\pi\mu_{r}^{i}\leavevmode\nobreak\ .$ (118)

Expression (117) is the sum of exponential of area laws. The leading
exponential in the limit of large $\hat{T}$ and $\hat{R}$ would give the
string tension for charges in representation $r$ with $N$-ality $k$. Any
representation of $SU(N)$ with $N$-ality $k$ contains the fundamental weight
$\mu_{k}$ as one of its weights (Appendix C.1). Therefore in order to show
that string tensions would only depend on the $N$-ality of the representation
$r$ of the group $SU(N)$ we have to show that the lowest string tension action
in (118) corresponds to boundary conditions dictated by the fundamental weight
$\mu_{k}$ among all the weights $\mu_{r}^{i}$ ($i=1,...,d(r)$) of the
representation $r$. Any weight of a representation of $SU(N)$ can be obtained
from the highest weight by lowering with the simple roots 09  and as noted
above any representation with $N$-ality $k$ contains $\mu_{k}$ as one of its
weights. Therefore any weight of a representation $r$ with $N$-ality $k$ can
be obtained from the fundamental weight $\mu_{k}$ by adding or subtracting the
simple roots.

We will now qualitatively (but convincingly) argue that adding or subtracting
any simple root from $\mu_{k}$ would result in boundary conditions that would
give a value for the minimum of the action (118) which is equal
to464646Degenerate string tensions will occur when the corresponding weights
are related by the unbroken ${\mathbb{Z}}_{N}$ center symmetry. For example,
for the fundamental representation all weights have the same string tension,
see Section 5.1.2. For higher $N$-ality representations, the dim($r$) weights
fall into distinct ${\mathbb{Z}}_{N}$ orbits, each of which has degenerate
string tensions. or more than the value obtained by boundary conditions of
$\mu_{k}$.

The saddle point solutions $f_{j}$ of (118) for $\mu_{r}^{i}=\mu_{k}$ start
from $\pi(\mu_{k})_{j}$ at $z=0$ and decrease or increase monotonically to
zero at $z=+\infty$. From the form of $\mu_{k}$ given in (59), one sees that
there are two discontinuities, as a function of $j$, in the boundary
conditions for $f_{j}$. These occur between $j=N$ and $j=1$, since
$(\mu_{k})_{N}=-k/N$ and $(\mu_{k})_{1}=1-k/N$, and between $j=k$ and $j=k+1$,
as $(\mu_{k})_{k}=1-k/N$ and $(\mu_{k})_{k+1}=-k/N$. These two discontinuities
in the boundary conditions make the corresponding terms in the cosine
potential $1-\text{cos}(f_{N}-f_{1})$ and $1-\text{cos}(f_{k}-f_{k+1})$ to
start from 2 at $z=0$ and reach $0$ at $z=+\infty$ (this is in contrast to all
the other terms, which start from $0$ at $z=0$ and reach $0$ again at
$z=\infty$). Thus, for boundary conditions given by $\mu_{k}$ we would
summarize the behaviour of $f_{j}$’s as follows. For the kinetic term in (118)
we would have $k$ functions $f_{1},...,f_{k}$ that start from $\pi(1-k/N)$ at
$z=0$ and reach $0$ at $z=+\infty$ and $N-k$ functions $f_{k+1},...,f_{N}$
that start from $-\pi k/N$ at $z=0$ and reach $0$ at $z=+\infty$. For the
potential term, since only the difference between the $f_{j}$’s enters the
cosine, the $1-\text{cos}(f_{N}-f_{1})$ and $1-\text{cos}(f_{k}-f_{k+1})$
terms start at 2 at $z=0$ and reach 0 at $z=+\infty$, while the rest of the
terms start from 0 at $z=0$ and reach 0 at $z=+\infty$.

Now let us ask how this picture would change if simple roots are added or
subtracted from $\mu_{k}$. The picture of the kinetic term will either remain
the same (k functions start from $\pi(1-k/N)$ and $N-k$ functions starting
from $-\pi k/N$) or it would become worse (and thus increase the value of
action) in a way that the boundaries values at $z=0$ become higher and result
in an increase in the kinetic term (since it is the square of the derivative
of the functions). The same is true for the cosine term—any addition or
subtraction of the simple roots from the weight $\mu_{k}$ would either not
change the picture of the cosine term or it would make it worse (increase the
value of action) in a way that we would have more than 2 discontinuities in
the boundaries that would result in more than two terms of the potential term
having to start from 2 at $z=0$, or we would still have two discontinuities
but the boundary conditions would have become larger and the cosine terms
corresponding to these discontinuities would oscillate between 2 and zero more
than once. Both our numerical results and the simple variational ansatz of
Section 5.1.3 confirm this picture.

####  5.1.2 Comparing different abelian confinements: strings in dYM vs.
softly-broken Seiberg-Witten theory

The two most-studied examples where confinement of quarks becomes analytically
calculable within quantum field theory are softly-broken Seiberg-Witten theory
on ${\mathbb{R}}^{4}$ and QCD(adj) with massive or massless adjoint fermions
on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$. This paper is devoted to the
study of dYM theory, which belongs to the second class, QCD(adj) with massive
adjoint fermions. Semiclassical calculability in dYM is achieved, as mentioned
many times, by taking the $NL\Lambda\ll 1$ limit.

In both dYM and Seiberg-Witten theory confinement is “abelian:”474747This
should not be taken to mean that the nonabelian nature of the theory is not
relevant: on the contrary, it is crucial in both examples. the confining
strings form in a regime where $W$-bosons are not relevant and the dynamics of
confinement is described by a weakly-coupled abelian gauge theory. In Seiberg-
Witten theory, this is the dual magnetic gauge theory on ${\mathbb{R}}^{4}$,
while in dYM it is the long-distance theory on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$—the theory of the dual photons
discussed at length in earlier Sections. In both cases, the confining dynamics
involves magnetically charged—and thus nonperturbative from the point of view
of the electric gauge theory—objects: the magnetic monopoles or dyons in
Seiberg-Witten theory condense to break the magnetic gauge symmetry, while in
dYM, the proliferation of monopole-instantons in the vacuum (which should not
really be called “condensation,” the title of Unsal:2007jx  nothwithstanding)
leads to the expulsion of electric flux.484848There are hints that the two
confinement mechanisms are related, see Poppitz:2011wy . We shall see, in the
next Section, that the physics of confinement in dYM has a flavor very similar
to the picture of the QCD vacuum underlying the MIT Bag Model.

Here, we want to stress two aspects in which dYM confinement is distinct from
Seiberg-Witten theory that have not been much discussed in the literature:

  1. 1.

The presence of a global unbroken ${\mathbb{Z}}_{N}$ (zero-form494949In the
terminology of Gaiotto:2014kfa .) center symmetry in dYM vs. the fact that the
Weyl group in Seiberg-Witten theory is broken 15 . The unbroken
${\mathbb{Z}}_{N}$ symmetry has implications for the “meson” and “baryon”
spectra of the theory, as we explain further in this Section.

  2. 2.

The abelian large-$N$ behaviour: confining string tensions remain finite in
dYM in the large-$N$, fixed $\Lambda NL\ll 1$ limit. This is different from
their behaviour in the analogous limit of Seiberg-Witten theory, where the
string tensions vanish along with the mass gap 15 . For further discussion,
see Section 5.2.

Here we concentrate on the first point above: the unbroken ${\mathbb{Z}}_{N}$
center symmetry in dYM. In the long-distance theory, in the $N$-dimensional
basis of dual photons we are using, this symmetry appears as a clock symmetry,
taking $\sigma_{i}\rightarrow\sigma_{i+1}$, with $N+1\equiv 1$. In gauge-
variant terms, the action of the ${\mathbb{Z}}_{N}$ center symmetry resembles
that of an unbroken cyclic subgroup of the Weyl symmetry of $SU(N)$, as can be
seen by noting that it cyclically interchanges the $N$ monopole-instantons
associated with the simple and affine root of the Lie algebra.505050In terms
independent of the choice of basis vectors of the root lattice, the
${\mathbb{Z}}_{N}$ center acts on the dual photons ${\sigma}$ as the ordered
product of Weyl reflections with respect to all simple roots, see
Anber:2015wha . On the other hand, in Seiberg-Witten theory, the Weyl group is
spontaneously broken, as pointed out long ago 15 .

The different global symmetry realization has interesting implications for the
nature of confining strings in the two theories. To illustrate the differences
it suffices to consider the confinement of fundamental quarks in $SU(3)$. In
dYM theory, there are degenerate “mesons” composed of quarks (introduced as
static sources) of the three different colors, of weights $\nu_{1}=\mu_{1}$,
$\nu_{2}=\nu_{1}-\alpha_{1}$, and $\nu_{3}=\nu_{2}-\alpha_{2}$, respectively.
These mesons are confined by distinct flux tubes related by the
${\mathbb{Z}}_{N}$ global symmetry action (the action of ${\mathbb{Z}}_{N}$ on
the weights of the $SU(N)$ fundamental representation is to cyclically permute
them). Furthermore, the fluxes carried by these three strings add up to zero,
so one can form a “baryon,” where the “baryon vertex” is a junction of three
strings (a domain wall junction), as illustrated on Figure 2.

Figure 2: Strings between static quarks of different colors (denoted by color
circles) in $SU(3)$ dYM theory. Left panel: $Q_{i}\overline{Q}_{i}$ mesons in
$SU(3)$ dYM are degenerate, due to the unbroken $Z_{3}$ center symmetry. There
are three flux tubes carrying fluxes $\nu_{i}$ ($i=1,2,3$), one for
fundamental quarks of each weight (color). Right panel: A “baryon vertex” in
dYM is a 3-domain wall junction, which exists due to the vanishing total flux
$\nu_{1}+\nu_{2}+\nu_{3}=0$. Similar structures persist for arbitrary number
of colors in dYM theory.

In contrast, in $SU(3)$ Seiberg-Witten theory, there are two $U(1)$ magnetic
gauge groups broken by the monopole condensate, giving rise to two Abrikosov-
Nielsen-Olesen (ANO) vortices. The flux of one vortex is proportional to
$\mu_{1}$ and confines quarks in the highest weight of the fundamental
representation. The other flux tube carries electric flux proportional to
$\mu_{2}$ (the second fundamental weight of $SU(3)$) and confines quarks in
the highest weight of the two-index antisymmetric representation (anti-quarks,
for $SU(3)$). There is no third flux tube. The picture of “mesons” in $SU(3)$
Seiberg-Witten theory that results is shown on Figure 3: the lowest and
highest weights of the fundamental quarks are confined by the two ANO flux
tubes, while the middle-weight quark is confined by two flux tubes: one of
flux $\mu_{2}$ and an anti-flux tube of flux $\mu_{1}$. The lack of a third
flux tube becomes especially noticeable when baryons are considered: baryons
in Seiberg-Witten theory are “linear molecules” only, as shown on Figure 3.
This difference persists and becomes more pronounced for higher rank $SU(N)$
gauge groups.515151See 16  for a description of confining strings in softly-
broken Seiberg-Witten theory within its $M$-theory embedding.

Figure 3: Strings in $SU(3)$ Seiberg-Witten theory. Left panel:
$Q_{i}\overline{Q}_{i}$ mesons for different color quarks are non degenerate,
due to existence of only two ANO flux tubes (denoted by lines with a single or
double arrow) carrying electric fluxes $\mu_{1}$ and $\mu_{2}$ (notice that
$\nu_{2}=\mu_{2}-\mu_{1}$), respectively. Right panel: Only linear baryons
exist in Seiberg-Witten theory. Similar pictures hold for any number of
colors.

As the $SU(3)$ example illustrates, the different symmetry realizations in dYM
and Seiberg-Witten theory have implications for the spectrum of mesons and
baryons. We shall not pursue this further here, but only note that in dYM one
can add dynamical massive quarks and the meson, baryon (as well as glueball)
spectra can be studied within weakly-coupled field theory, revealing many
unusual and surprising features discussed in Aitken:2017ayq .

####  5.1.3 An approximate form of k-string ratios and the MIT Bag Model

Here, we shall derive a naive analytic upper bound for the half $k$-string
tensions in Table 1 by approximating the integral in (61) in a simple manner.
We shall arrive at a simple $k$-string tension scaling law, which is in good
agreement with the available data, as described further below. We shall also
elaborate on the similarity between confinement in dYM and the MIT Bag Model
of the Yang-Mills vacuum.

We begin by repeating the dimensionless half $k$-string tension action (recall
e.g. eq. (118)):

$\displaystyle{\bar{T_{k}}}=\underset{f(\text{z}^{\prime})}{\text{min}}\int_{0}^{+\infty}d\text{z}^{\prime}\\{({\partial\vec{f}\over\partial\text{z}^{\prime}})^{2}+\sum\limits_{j=1}^{N}[1-\text{cos}(f_{j}-f_{j+1})]\\},\leavevmode\nobreak\
\leavevmode\nobreak\
\vec{f}(+\infty)=0,\vec{f}(0)=\pi\vec{\mu}_{k}\leavevmode\nobreak\
.\leavevmode\nobreak\ $ (119)

Here $\vec{f}$ represents the $N$-dimensional vector of dual photon fields
(whose components are summed explicitly in the second term; we omit the arrows
in what follows) and the boundary conditions at the origin and at infinity are
the ones appropriate for static sources in the highest weight of the $k$-index
antisymmetric tensor representation.

A simple variational ansatz for the half domain wall extremizing (119) can be
obtained by approximating the first term in the action as a linear function
connecting the boundary value $\pi\mu_{k}$ at $z=0$ to zero at a finite
positive $z=J$. The second term is approximated by simply taking its value at
$z=0$ (i.e. with $f=\pi\mu_{k}$) multiplied by $J$; in other words, the fields
$f_{i}$ are taken in the vacuum (where the potential term in (119) vanishes)
outside a region of width $J$ which represents the thickness of the flux tube
in our variational ansatz. As the form of $\mu_{k}$ and the potential term
imply, for $f=\pi\mu_{k}$ only two terms in the sum of $N$ cosines contribute
a factor of $2$ each, giving rise to second term in (120), while the remaining
$N-2$ terms do not contribute.525252As discussed in Section 5.1.1, one of the
qualitative reasons why charges $\mu_{k}$ are confined by strings of the
lowest tension (for every representation) is that adding or subtracting any
root from $\mu_{k}$ leads to higher “vacuum energy” cost. Collecting
everything, using the explicit form of the fundamental weight $\mu_{k}$ from
(59), we obtain the string tension as a function of the one variational
parameter $J$, the flux tube thickness:

$\bar{T_{k}}^{naive}(x)=J\\{({\pi{N-k\over N}\over J})^{2}k+({\pi{k\over
N}\over J})^{2}(N-k)\\}+4J={\beta_{k}\over J}+4J\leavevmode\nobreak\ ,$ (120)

where the parameter

$\beta_{k}\equiv(\pi{N-k\over N})^{2}k+(\pi{k\over
N})^{2}(N-k)=\pi^{2}{(N-k)k\over N},$ (121)

is proportional to the quadratic Casimir of the $k$-index antisymmetric
tensor. Extremizing (120) with respect to $J$ gives
$J_{k,\text{ext}}={\sqrt{\beta_{k}}\over 2}$. The value of the string tension
at the extremum point is:

$\bar{T}_{k}^{\text{naive}}=4\pi\sqrt{(N-k)k\over N}\leavevmode\nobreak\ .$
(122)

Although the relation (122) is only a naive upper bound estimate for the
$k$-strings in Table 1, its ratio with the fundamental ($k=1$) $k$-string
gives a good fit to the ratios of $k$-strings of Table 1:

${\bar{T}_{k}^{\text{naive}}\over\bar{T}_{1}^{\text{naive}}}=\sqrt{(N-k)k\over
N-1}\leavevmode\nobreak\ .$ (123)

The relation (123) is, in fact, known as the “square root of the Casimir”
scaling law for $k$-string ratios. It was first seen to arise in the MIT Bag
Model of the QCD vacuum a long time ago 13 .535353Ref. 13  studied a rotating
string solution, but a simpler static one exists, see discussion below and
ref. Hasenfratz:1977dt , which also contains a review of the physical picture
underlying the MIT Bag Model of the Yang-Mills vacuum. As far as we are aware,
dYM theory is the only known example where this “square root of Casimir”
k-string scaling has been seen to arise within a controlled approximation in
quantum field theory.

We shall now discuss the physics behind (120) and (123) and will argue that
the similarity of strings in dYM to those in the MIT Bag Model is not an
accident. The first term on the r.h.s. of (120) represents the gradient energy
of the $\sigma$-field. Recall that the duality relation (38) maps spatial
gradients of the dual photon field to electric fields in the perpendicular
direction (i.e. to electric flux going from the quark to the antiquark, which
are here taken at infinite separation). Thus, the $\beta_{k}\over J$ term
represents the electric field energy cost (per unit length) for a flux tube of
thickness $J$. The coefficient $\beta_{k}$, the total electric flux, is
determined by the sources—quarks in the $k$-index antisymmetric tensor
representation—and is proportional to the quadratic Casimir of that
representation, as in the classical MIT Bag Model of the confining
string.545454The classical chromoelectric flux of static sources in a given
representation is proportional to the quadratic Casimir, see Section 3.3 in
Hasenfratz:1977dt . Also note that the “square root of Casimir” scaling is
obtained in the Bag Model without surface tension and that introducing
additional Bag Model parameters, e.g. bag surface tension, modifies the
scaling with the Casimir of the representation. Naturally, in order to
minimize its energy, the electric flux tube wants to expand, i.e. maximize
$J$—in a perturbative vacuum, the chromoelectric field would relax to the
dipole field of the quarks. The second term on the r.h.s. of (120), equal to
$4J$, represents the energy cost per unit length to “expelling the vacuum” and
replacing it with electric flux in a region of width $J$. This term represents
the “volume energy cost,” proportional to the bag constant parameter of the
MIT Bag Model. In dYM, the vacuum is a monopole-antimonopole medium which
abhors electric flux and wants to minimize $J$; the “bag constant” in dYM is
not a model parameter, but is determined by the fugacity of monopole-
instantons, ultimately fixed by the underlying gauge theory. The compromise
between the two contributions to the energy results in $k$-strings of width
$J_{k,\text{ext}}=\sqrt{\beta}_{k}/2$ and tensions given in (122).

As we already alluded to, the agreement between the dYM and MIT Bag Model
$k$-string tensions is not accidental. In the MIT Bag Model, the major
assumption is that the chromoelectric fields within the confines of the
(presumably small) bag can be treated classically, owing to asymptotic
freedom. The “bag constant" of the YM vacuum, characterizing its abhorrence of
electric flux, is introduced as a model parameter. In dYM, both the classical
treatment of the Cartan electric fields and the expulsion of electric flux are
dynamical features arising from the judiciously chosen deformation of YM
theory and are justified in the $N\Lambda L\ll 1$ limit.

Finally, we note that while the physical picture in dYM is similar to that in
the bag model, the “square root of Casimir” scaling of $k$-string tensions
discussed here is not exact in dYM, as it results from a simple variational
estimate. It is only an upper bound on the string tensions in dYM, see the
following Section and, in particular, Figure 4.

####  5.1.4 Comparison with known scaling laws

It is known that the asymptotic string tensions depend only on the $N$-ality
$k$ of the representation of the confined charges, hence they are often
referred to as the $k$-strings. Different models of confinement make different
predictions for the ratios of $k$-string tensions. The main ones are the sine
law and Casimir scaling. We also include the square root of Casimir scaling in
the list below, due to its similarity with the $k$-string ratios in dYM theory
for N $\sim$ 10 and smaller:

$\displaystyle\text{Sine law}:{T_{k}\over T_{1}}={\text{sin}(\pi{k\over
N})\over{\text{sin}({\pi\over N})}},$ $\displaystyle\text{Casimir
scaling}:{T_{k}\over T_{1}}={k(N-k)\over N-1},$ (124)
$\displaystyle\text{Square root of Casimir scaling}:{T_{k}\over
T_{1}}=\sqrt{k(N-k)\over N-1}\leavevmode\nobreak\ .$

In field theory calculations, usually the corresponding $k$-string tension is
calculated to leading order in a small parameter expansion. It has to be noted
that the above relations correspond to the leading order result in that
expansion and, in each case, are subject to corrections.

Figure 4: Comparison of $SU(10)$ $k$-string ratios of (122) with dYM
$k$-string ratios, labeled by “dYM”, to other $k$-string tension laws. The
Sine law labeled by “sin”, the Casimir scaling by “cas”, and scaling with the
Square root of the Casimir scaling by “sqrtcas”. From the known theoretical
models predicting different scalings of $k$-string tensions, the ones in dYM
are closest to the MIT Bag Model “square root of Casimir” $k$-string tension
law. There is a clear physical reason behind this similarity, explained in
Section 5.1.3.

The Sine law is found in Seiberg-Witten theory 15 , in MQCD 16 , in three-
dimensional SU(N) gauge theories with massless Dirac or Majorana fermions 61 ,
and in some AdS/CFT-inspired models 17 . Casimir scaling of string tensions
refers to the relation between string tensions
$T_{r}/T_{F}=C_{2}(r)/C_{2}(F)$, where $C_{2}(r)$ and $C_{2}(F)$ are the
quadratic Casimir of representation $r$ and the fundamental representation,
respectively ($T_{r}$ denotes the string tension for charges in representation
$r$). This relation can be derived from the “dimensional reduction” form of
the Yang-Mills vacuum wave functional 18 , from the stochastic vacuum picture
19 , and from certain supersymmetric dual models 20 .

$SU(3)$ lattice simulations have shown scaling with the Casimir of the
representation $C_{2}(r)$ with a good accuracy 21 : it holds at intermediate
distances ($\lessapprox$ 1 fm) but at larger distances (asymptotically) gluons
screen the charges down to the representation of the same non-zero $N$-ality
with the lowest dimensionality which carries the most stable lowest string
tension—then $C_{2}(r)$ is replaced by the Casimir of the $k$-antisymmetric
representation which leads to the Casimir scaling relation shown in (5.1.4);
notice however, that for $N=3$ $T_{1}=T_{2}$. Lattice studies of
$3$-dimensional YM theory seem to also favor Casimir scaling of $k$-string
tensions ratios for gauge groups up to $SU(8)$ Bringoltz:2008nd , while
studies of $4$-dimensional YM theory (for similar number of colors) appear to
favor scaling in-between the sine and Casimir laws, see Lucini:2012gg  for
references and discussion.

The various $k$-string ratios shown in (5.1.4) are compared with dYM
$k$-string ratios for $SU(10)$ in Figure 4. It is clear from the figure that
the square root of Casimir scaling shows most similarity with the dYM
$k$-string ratios. This scaling arises in the MIT Bag Model of QCD 13  and the
reasons for the similarity was discussed in Section 5.1.3.

###  5.2 Large-N behaviour

One feature of the abelian large-N limit in dYM was already mentioned: in the
$N\rightarrow\infty$, fixed-$NL\Lambda\ll 1$ double scaling limit, the mass
gap vanishes, but the string tensions stay finite. This large-N behaviour is
quite different from a similar abelian large-N limit of Seiberg-Witten theory,
where both the string tensions and mass gap vanish 15 . Furthermore, as
observed in Cherman:2016jtu , in the above double-scaling limit on
${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, where the size of the dimension
$L\rightarrow 0$ and the number of colors $N\rightarrow\infty$, with
$NL$-fixed, in both super Yang-Mills and dYM, the infrared theory can be
viewed as a theory “living” in an emergent latticized dimension, in a manner
reminiscent of T-duality in string theory. This is a behaviour not quite
expected of quantum field theory and clearly deserves a better understanding.

The results of this paper show the nonvanishing of the string tension in dYM
at large N. In the remainder of this Section, we study the leading large-N
corrections to k-string ratios and their large-N behaviour, for a range of N
that includes exponentially large values, but does not strictly extent to
infinity.

The reason for this restriction, already mentioned in Section 1.1 of the
Introduction, is that our analysis has neglected the fact that at large values
of N, the virtual effects of the W-bosons become important, as there is a
large number of them. In particular, W-boson loops induce mixing between the
Cartan algebra photons (and hence between dual photons), which were not
incorporated in our effective Lagrangian. Similar to the discussion of ref.
Cherman:2016jtu  for sYM (using the calculations of refs. 08 ; Anber:2014sda
), we estimate that these mixing terms become important when $N$ becomes
comparable to $N^{*}=2\pi e^{+{24\pi^{2}\over(11-4n_{f})(Ng^{2})}}$. This
exponentially large value of $N^{*}$ is the one that applies to massless
adjoint QCD, and uses the computations in vito . The corresponding calculation
for dYM (adjoint QCD with massive adjoint flavors) has not yet been performed,
but we expect the appearance of a similar exponentially large $N^{*}$.
Studying the role of these corrections in dYM is an interesting task for
future work, which will allow to further study the intriguing features of the
abelian large-N limit.

####  5.2.1 Leading large-N terms

In this Section, we derive the leading large-N corrections to k-string ratios
in dYM theory for T2/T1 and T3/T1. We will show that the k-string ratios in
dYM theory favour even power corrections in $1\over N$. For this we add
noise555555Noise of order $\epsilon$ refers to a random fluctuation of order
$\epsilon$ imposed on the data. The fluctuation can be a Gaussian, uniform,
etc., distribution of width $\epsilon$ centred on the data point. We have used
a uniform distribution. of order $0.0005$, the typical value of error of dYM
k-string ratios565656We consider half of the upper bound estimate of the error
in Table 1 ($-0.006/2=-0.003$) as the value of error for k-strings. Hence for
k-string ratios as a typical example we get:
$T_{2}/T_{1}=8.0006_{-0.003}/6.8583_{-0.003}\approx
1.1666^{+0.0005}_{-0.0004}$. The reader has to be reminded that an error of
$-0.003$ is still a high confidence interval for the true value of k-strings.
to the exact k-string ratios of the Casimir scaling and sine law, whose
scaling behaviour is known, and analyze them along with the k-string ratios in
dYM theory. From Figures 6 and 6 it can clearly be seen that the coefficient
of the linear correction term in dYM k-string ratios similar to the sine law
is suppressed (whereas for the Casimir scaling law it is of same order)
compared to the constant or the coefficient of the second order term therefore
it can be concluded that dYM $k$-string ratios similar to the sine law
disfavour a linear correction term and favour even power corrections.

To find the leading term and leading correction term, we add noise of order
$0.0005$ to the exact k-string ratios of the sine law for $\text{SU}(5\leq
N\leq 10)$ to generate data with errors of order of the errors of the dYM
data. Next we generate n = 1000 noised data for dYM and the sine law data with
noise and make even power polynomial fits:
$c_{0}+c_{2}\text{x}^{2}+...+c_{p}\text{x}^{p}$ for $p=2k,k\geq 0$, with
$\text{x}=1/N$. The average and standard deviation of $c_{0}$ and $c_{2}$ give
estimates for the values of these coefficients and their errors. We increase
$p$ and make higher order polynomial fits until consistent results are
reached. Tables 4 and 5 summarize the values obtained by this analysis. It can
be seen that consistent results are obtained for $p=6$ and $p=8$ polynomial
fits. In fact the values of the $p=6$ column for the noised sine law data are
in agreement with the exact coefficients of the $1\over N$ expansion of the
sine law k-string ratios as can be seen from (5.2.1). This is not limited to
the sine law. Any other function with even power corrections shows a similar
behaviour and the results for $c_{0}$ and $c_{2}$ coefficients for an even
polynomial fit with $p=6$ would agree with the true values of the coefficients
in its $1/N$ expansion (e.g. doing the same analysis for a cosine(x) function
with $\text{x}={1\over N}$). So assuming dYM k-string ratios have only even
power corrections, the values of the coefficients in the $p=6$ column would be
in agreement with the true values in dYM theory.

The following relations summarize the leading large $N$ corrections in sine,
Casimir, square root of Casimir and dYM k-string ratios:

$\displaystyle\text{Sine}:{\text{sin}(k{\pi\over N})/\text{sin}({\pi\over
N})}=k+(k/6-k^{3}/6)\pi^{2}({1\over N})^{2}+...\leavevmode\nobreak\ ,$
$\displaystyle\text{Casimir}:{k(N-k)/(N-1)}=k+(k-k^{2})({1\over
N})+...\leavevmode\nobreak\ ,$ $\displaystyle\text{Sqrt of
Casimir}:\sqrt{k(N-k)/(N-1)}=k^{1\over 2}+{1\over 2}(k^{1\over 2}-k^{3\over
2})({1\over N})+...\leavevmode\nobreak\ ,$ (125)
$\displaystyle\text{dYM}:\text{T}_{2}/\text{T}_{1}=1.347\pm 0.001+(-2.7\pm
0.2)({1\over N})^{2}+...\leavevmode\nobreak\ ,$
$\displaystyle\phantom{{}=\text{abC}}\text{T}_{3}/\text{T}_{1}=1.570\pm
0.001+(-7.5\pm 0.2)({1\over N})^{2}+...\leavevmode\nobreak\ .$

Figure 5: Polynomial fits for $T_{2}/T_{1}$ $k$-string ratios in dYM and
noised ($\sim$ 0.0005) data in sine, Casimir for $\text{SU}(6\leq\text{N}\leq
10)$. Sine: $2.007-0.150\text{x}-9.013\text{x}^{2}$, Casimir:
$1.995-1.873\text{x}-2.981\text{x}^{2}$, dYM:
$1.345+0.035\text{x}-2.917\text{x}^{2}$.

Figure 6: Polynomial fits for $T_{3}/T_{1}$ $k$-string ratios in dYM and
noised ($\sim$ 0.0005) data in sine, Casimir for $\text{SU}(6\leq\text{N}\leq
10)$. Sine: $3.105-2.222\text{x}-26.447\text{x}^{2}$, Casimir:
$2.980-5.552\text{x}-9.178\text{x}^{2}$, dYM:
$1.551+0.390\text{x}-9.734\text{x}^{2}$.

| p=2 | p=4 | p=6 | p=8
---|---|---|---|---
$\text{c}_{0}$(sin) | 1.9962 $\pm$ 0.0001 | 2.001 $\pm$ 0.0004 | 2.001 $\pm$ 0.001 | 1.998 $\pm$ 0.005
$\text{c}_{2}$(sin) | -9.458 $\pm$ 0.006 | -9.91 $\pm$ 0.03 | -9.9 $\pm$ 0.2 | -9 $\pm$ 1
$\text{c}_{0}$(dYM) | 1.3482 $\pm$ 0.0001 | 1.3465 $\pm$ 0.0004 | 1.347 $\pm$ 0.001 | 1.347 $\pm$ 0.005
$\text{c}_{2}$(dYM) | -2.822 $\pm$ 0.006 | -2.65 $\pm$ 0.03 | -2.7 $\pm$ 0.2 | -3 $\pm$ 1
Table 4: $c_{0}$ and $c_{2}$ for even power polynomial fits of order p for $\text{T}_{2}/\text{T}_{1}$ for noised ($\sim$ 0.0005) sine law data and dYM | p=2 | p=4 | p=6 | p=8
---|---|---|---|---
$\text{c}_{0}$(sin) | 2.9397 $\pm$ 0.0001 | 2.9984 $\pm$ 0.0004 | 3.000 $\pm$ 0.001 | 2.999 $\pm$ 0.005
$\text{c}_{2}$(sin) | -33.334 $\pm$ 0.006 | -39.17 $\pm$ 0.03 | -39.4 $\pm$ 0.2 | -39 $\pm$ 1
$\text{c}_{0}$(dYM) | 1.5815 $\pm$ 0.0001 | 1.5682 $\pm$ 0.0004 | 1.570 $\pm$ 0.001 | 1.569 $\pm$ 0.005
$\text{c}_{2}$(dYM) | -8.594 $\pm$ 0.006 | -7.27 $\pm$ 0.03 | -7.5 $\pm$ 0.2 | -7 $\pm$ 1
Table 5: $c_{0}$ and $c_{2}$ for even power polynomial fits of order p for
$\text{T}_{3}/\text{T}_{1}$ for noised ($\sim$ 0.0005) sine law data and dYM

As a short summary of this Section, we argued that k-strings in dYM are not
free at large N, i.e. $T_{k}/T_{1}\neq k$, and leading corrections to
$T_{k}/T_{1}$ are of order $1/N^{2}$. In the next Section, we discuss some
theoretical questions behind these findings.

####  5.2.2 Comments on free k-strings and large-N factorization

An often-discussed expected behaviour of $k$-strings at large $N$ is that they
become free, meaning that the string tension with $N$-ality $k$ becomes $k$
times the tension of the fundamental $k=1$-string at large $N$ 14 ; 10 . From
the previous Section, in particular (5.2.1), it can be clearly seen that the
k-string tensions in dYM theory show a different behaviour:
$\underset{N\rightarrow\infty}{\text{lim}}\text{T}_{2}=(1.347\pm
0.001)\text{T}_{1}<2\text{T}_{1}$ and
$\underset{N\rightarrow\infty}{\text{lim}}\text{T}_{3}=(1.570\pm
0.001)\text{T}_{1}<3\text{T}_{1}$. The usual line of reasoning that leads to
the conclusion that k-strings become free at large $N$ is based on large-$N$
factorization and assumes the commutativity of the large-$N$ and large
Euclidean time $T$ limits. We will show that factorization and commutativity
of limits should be treated more carefully.575757For another discussion on the
non-commutativity of the large-$N$ and large-$T$ limits refer to 62 .

W

e first briefly review the usual arguments that lead to free k-strings at
large $N$:
A correlator of two gauge invariant operators A and B can always be written as
a factorized expectation value plus a connected expectation value. In the
lattice strong coupling expansion and in perturbation theory in gauge theories
it is known that the leading term in the large $N$ limit is the factorized one
11 . Assuming a normalization $\langle\text{AB}\rangle\sim O(1)$ we have:

$\displaystyle\langle\text{A}\text{B}\rangle=\langle\text{A}\rangle\langle\text{B}\rangle+\langle\text{A}\text{B}\rangle_{C},\
\ \ \ \langle\text{A}\rangle\langle\text{B}\rangle\sim O(1),\ \ \ \
\langle\text{A}\text{B}\rangle_{C}\sim O({1\over N^{2}})\leavevmode\nobreak\
.$ (126)

In particular, we will apply this formula to the expectation value of a Wilson
loop in the product representation:

$\displaystyle\langle
W_{\square\otimes\square}\rangle\equiv\langle\text{tr}(U_{\square\otimes\square}...U_{\square\otimes\square})\rangle=\langle\text{tr}\\{(U_{\square}...U_{\square})\otimes(U_{\square}...U_{\square})\\}\rangle=\langle
W_{\square}W_{\square}\rangle$ $\displaystyle\Longrightarrow\langle
W_{\square\otimes\square}\rangle=\langle W_{\square}W_{\square}\rangle=\langle
W_{\square}\rangle\langle W_{\square}\rangle+\langle
W_{\square}W_{\square}\rangle_{C}\leavevmode\nobreak\ .$ (127)

A subscript of a “square” (as in $W_{\square}$) refers to the fundamental
representation. The product of the link matrices $U$ is being taken along a
rectangular Wilson loop $R\times T$. To find the k-string tensions we take the
large $T$ and $R$ limit and consider the leading exponential on the right hand
side of (5.2.2). To consider the properties of the k-strings at large $N$ we
also take the large $N$ limit. If the large $T$ and $R$ and large $N$ limits
commute, then we can reverse the order of limits. Taking the large $N$ limit
first makes the connected term vanish, then taking the large $T$ and $R$ limit
we would find:

$\langle W_{\square\otimes\square}\rangle\sim\langle W_{\square}\rangle\langle
W_{\square}\rangle,\leavevmode\nobreak\ \langle
W_{\square\otimes\square}\rangle\sim\text{exp}(-T_{2}RT),\leavevmode\nobreak\
\langle W_{\square}\rangle\sim\text{exp}(-T_{1}RT)\leavevmode\nobreak\
\leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
T_{2}=2T_{1},$

i.e. the result that the $k=2$-string tension is twice the fundamental string
tension.

The line of reasoning represented above leads to the result that $k$ strings
are “non-interacting” and would be correct if the large $T$ and $R$ and large
$N$ limits commuted, which is not always true, as we discuss at length below
(see 62  for a discussion in a similar framework and Witten:1978qu  for a
reminder that large-distance and large-$N$ limits’ non-commutativity has a
long history). An important difference between the large area limit and
large-$N$ limit relevant to their non-commutativity is the fact that the large
area limit is taken in the same quantum field theory where as the large-$N$
limit is taken in different quantum field theories.

To study the general properties of field theories at large $N$, e.g. large $N$
factorization, one takes the large $N$ limit first but to study the asymptotic
k-string tensions at large $N$, due to the non-commutativity of the large area
and large N limits, one should not take the large $N$ limit first. For any
$SU(N)$ theory, the proper way to find asymptotic $k$-string tensions at large
$N$ is to first solve for the $k$-string tensions at fixed $N$, this is done
by taking the large area $(RT)$ limit and considering the leading exponential
in this limit. Then, once the asymptotic $k$-string tensions are determined
for each $N$ from the coefficient of the area term of the leading exponential,
the large-$N$ limit of $k$-string tensions can be taken.

The limits cannot be taken in reverse order as the leading exponential in the
large area ($RT$) limit, which gives the $k$-string tension for the given
value of $N$, can be suppressed in the large-$N$ limit compared to
exponentials sub-leading in the large area limit. Let us first illustrate this
important point in a toy example, similar to the way the non-commutativity of
limits is realized in dYM. As the discussion of dYM is somewhat lengthy and
slightly technical585858See further below the discussion of this Section
(between eqs. (129) and (131)) as well as the explicit calculations in
Appendix E. we prefer to first illustrate the result by the following example.
Consider the function

$g(A,N)=\text{exp}(-T_{N}A)+{1\over N^{p}}\leavevmode\nobreak\
\text{exp}(-T^{\prime}_{N}A)\leavevmode\nobreak\ ,\leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ p>0,$ (128)

where $A$ stands in for the area of the Wilson loop. Let the large-$N$ limits
of $T_{N}$ and $T^{\prime}_{N}$,
$\underset{N\rightarrow\infty}{\text{lim}}T^{\prime}_{N}=T^{\prime}$,
$\underset{N\rightarrow\infty}{\text{lim}}T_{N}=T$, be such that
$T^{\prime}<T$. For any large but fixed $N$ the leading term in the large-$A$
limit is $g(A,N)\sim\text{exp}(-T^{\prime}A)$ and for any large but fixed $A$
the leading term in the large-$N$ limit is $g(A,N)\sim\text{exp}(-TA)$.
Therefore, if one is interested in the leading exponential in the large-$A$
limit, one should not take the large-$N$ limit first, as this will make the
second term in (128), which is leading in the large area limit, vanish. One
would then find $g(A,N)\sim\text{exp}(-TA)$, which is an incorrect result for
the leading exponential in the large-$A$ limit. A similar behaviour happens in
dYM as we discuss in detail further below, see discussion after (129).

In what follows, we shall see that in the regime of parameters studied in this
work, in particular in the framework of a bounded large-$N$ (see the comments
in the beginning of Section 5.2), the leading exponential in the large $T$ and
$R$ limits, which determines the k-string tensions, comes from the connected
term, although it can be shown that for fixed $R$ and $T$ this term will be
sub-leading in $N$ compared to the factorized term, similar to the toy example
of eqn. (128).

First, we argue how this can be seen more explicitly from the results of
Section 5.1.1. There, it was argued that the lowest string tension action in
the product representation of $N$-ality $2$ corresponds to boundary conditions
on the dual photon fields determined by the fundamental weight $\mu_{2}$
($\bar{T}^{i}_{\square\otimes\square}$ in relation (118) for
$\mu^{i}_{\square\otimes\square}=\mu_{2}$) and for a fundamental
representation of unit $N$-ality it corresponds to $\mu_{1}$
($\bar{T}^{i}_{\square}$ in relation (118) for $\mu^{i}_{\square}=\mu_{1}$).
Hence the leading exponential of the factorized term in the limit of large
$\hat{T}$ and $\hat{R}$ is
$\text{exp}(-2\times{2\bar{T}_{1}\over\sqrt{2}\beta}\hat{R}\hat{T})$ and the
leading exponential for the Wilson loop in the product representation is
$\text{exp}(-{2\bar{T}_{2}\over\sqrt{2}\beta}\hat{R}\hat{T})$. For large
values of $N$ from equation (5.2.1) we have
$\underset{N\rightarrow\infty}{\text{lim}}2\bar{\text{T}}_{2}=(1.347\pm
0.001)2\bar{\text{T}}_{1}<2\times 2\bar{\text{T}}_{1}$. Clearly, the
factorized term can never produce this leading exponential which should,
therefore, come from the connected term.

This result quoted above can also be obtained without referring to numerics,
via the perturbative saddle point method developed in Section 4, as shown in
Appendix E.

Next, we wish to verify the large $N$ factorization result in dYM and directly
argue that, for a Wilson loop in the product representation the factorized
term is leading and the connected term is sub-leading for large $N$. The
discussion that we begin now becomes more transparent and explicit after
reviewing the calculations of Appendix E.

Consider the expectation value of a Wilson loop in the product representation
${\square\otimes\square}$. Based on (117), for fixed but large $R$ and $T$ we
have:

$\langle
W_{{\square\otimes\square}}(R,T)\rangle\sim\sum^{d({\square\otimes\square})}_{h=1}\text{exp}(-T^{h}_{{\square\otimes\square}}RT)\leavevmode\nobreak\
,$ (129)

where $d({\square\otimes\square})=N^{2}$ refers to the dimension of the
product representation. In words, the expectation value of the Wilson loop in
the product representation is given, in the abelianized dYM theory, by a sum
of decaying exponentials, one for each weight $h$ of the product
representation, with string tension $T^{h}_{{\square\otimes\square}}$
corresponding to each weight.

On the other hand, for a Wilson loop in the fundamental representation
$\square$ we have:

$\langle
W_{\square}(R,T)\rangle\sim\sum^{d(\square)}_{i=1}\text{exp}(-T^{i}_{\square}RT)=N\text{exp}(-T_{1}RT)\leavevmode\nobreak\
.$ (130)

Similar to (129), this is a sum of decaying exponentials, one for every weight
of the fundamental representation, with the only simplification ocurring
because of the unbroken ${\mathbb{Z}}_{N}$ center symmetry, ensuring that the
string tensions for all weights of the fundamental representation have the
same value $T_{1}$.

Now, let us study (129) in more detail. Our considerations from this point to
eqn. (131) are more qualitative than quantitatively rigorous (although, as
already mentioned, they can be justified in the leading order perturbative
evaluation of the saddle point, see Appendix E). They carry similar flavor to
our argument of Section 5.1.1 that strings sourced by quarks with charges in
the highest weight of the $k$-index antisymmetric representation have the
smallest string tension. However, we find the considerations below quite
suggestive and intuitive, supporting the large-$N$ vs. large-$RT$ limit
subtlety.

The weights of the product representation are given by the sum of the weights
of the fundamental representation in (161):
$\mu^{h}_{{\square\otimes\square}}\equiv\mu^{(ij)}_{{\square\otimes\square}}=\mu^{i}_{\square}+\mu^{j}_{\square}$
for $1\leq h\leq N^{2}$ and $1\leq i,j\leq N$. These weights enter the
boundary conditions of the string tension action (55). In what follows, we
show that for large $N$ and for $|i-j|\gg 1$ and $|i-j|\ll N$ the string
tensions of the product representation become approximately equal to two times
the string tension of the fundamental representation at large $N$, i.e.
$T^{h}_{\square\otimes\square}\equiv T^{(ij)}_{\square\otimes\square}\approx
2T_{1}$. As there are $O(N^{2})$ such tensions, it will be concluded, after
considering eq. (131), that $\langle
W_{{\square\otimes\square}}(R,T)\rangle-\langle W_{\square}(R,T)\rangle\langle
W_{\square}(R,T)\rangle<O(N^{2})$ and therefore the connected term would be
sub-leading in $N$.

Consider the fundamental string tension $T^{q}_{\square}$ given by (55) with
$r=\square$ and $q$ denoting one of the weights of the fundamental
representation. The weight $\mu_{\square}^{q}$ is given in (161). Recall, from
Section 3.2, that the dual photon configuration extremizing the action of a
given string interpolates between a value at the origin given by
$\pi\mu^{q}_{\square}$ and zero at infinity. Since the $p$-th component of
$\mu_{\square}^{q}$ is $(\mu_{\square}^{q})_{p}=-{1\over N}+\delta^{pq}$, for
large values of $N$ all the components of $\sigma_{a}$, $1\leq a\leq N$ at
$z=0$ approach zero except for the $q$’th component which approaches $\pi$.
The fact that one component, namely $\sigma_{q}$, differs in its boundary
conditions from the rest would result in a non-zero action for
$T^{q}_{\square}$, otherwise if all components had the same boundary
conditions (e.g. $-\pi/N$) at $z=0$ they would be linear functions
interpolating between $-\pi/N$ at $z=0$ and zero at $z=J$ and when $J$ is
taken to infinity would result in a zero action. This suggests that the main
contribution to $T^{q}_{\square}$ would come from the components of
$\sigma_{a}$ near595959Note that due to the $Z_{N}$ symmetry of (55) the $N$
components of $\sigma_{a}$, $1\leq a\leq N$ can be considered similar to $N$
points on a circle corresponding to angles $\theta={2\pi a/N}$. The components
near the $q$’th component are defined as the points (components) close to the
$q$’th point on this circle. the $q$’th (also, see Appendix E). Conversely,
the components farther away from the $q$’th component would approach a linear
configuration, similar to the case when all boundary conditions were the same,
in order to minimize the action as much as possible and will have negligible
effect on the value of the string tension action $T^{q}_{\square}$, with their
contribution being suppressed by a power of $1/N$.

A similar picture is true for $T^{(ij)}_{\square\otimes\square}$ with
$\mu^{h}_{\square\otimes\square}=\mu^{(ij)}_{{\square\otimes\square}}$.606060In
components, the weights of the product representation are
$(\mu^{(ij)}_{\square\otimes\square})_{p}=\delta^{i}_{p}+\delta^{j}_{p}-2/N$.
The ${\mathbb{Z}}_{N}$ symmetry acts as
$\mu^{(ij)}_{\square\otimes\square}\rightarrow\mu^{(i+1({\rm mod}N),j+1({\rm
mod}N))}_{\square\otimes\square}$, i.e. the $N^{2}$ weights of the product
representation fall into $N$ ${\mathbb{Z}}_{N}$ orbits. Due to the $Z_{N}$
symmetry of the action without loss of generality we can take
$i=[(N-\Delta_{ij}+1)/2]$ and $j=[(N+\Delta_{ij}+1)/2]$ with
$\Delta_{ij}=|i-j|\neq 0$; the square brackets refer to the integer part. For
large $N$, all components $(\mu^{(ij)}_{\square\otimes\square})_{p}$ approach
zero, except for the $i$-th and $j$-th components, which approach $1$.

Consider now the calculation of the string tension
$T^{(ij)}_{\square\otimes\square}$ for large $N$ with $|i-j|\gg 1$ and
$|i-j|\ll N$. The components $\sigma_{a}$, $1\leq a\leq N$ interpolate between
$\pi(\mu^{(ij)}_{\square\otimes\square})_{a}$ at the origin and zero at
infinity and therefore for large values of $N$ all the components would
approach zero at $z=0$ except for $\sigma_{i}$ and $\sigma_{j}$ which approach
$\pi$. Similar to the picture above for $T^{q}_{\square}$ it can be seen that
the components of the dual photon fields $\sigma_{a}$ near the $\sigma_{i}$
and $\sigma_{j}$ components would make the main contribution to the string
tension $T^{(ij)}_{\square\otimes\square}$. The components farther away from
the $i$’th and $j$’th components would approach a linear configuration,
similar to the case when all boundary conditions are the same, in order to
minimize the action as much as possible and will have negligible effect on the
values of the string tension action $T^{(ij)}_{\square\otimes\square}$, with
their contribution being suppressed by a power of $1/N$. Next, we divide the
string tension action of $T^{(ij)}_{\square\otimes\square}$ into two parts,
one part associated with the components $\sigma_{a}$ for $1\leq a\leq N/2$ and
one for $N/2<a\leq N$; at large-$N$ and $|i-j|\gg 1$, $|i-j|\ll N$, these
actions become independent of each other. In each part, the components closer
to the $i$’th and $j$’th components of $\sigma_{a}$, which are relevant to the
string tension value and their boundary conditions are similar to the
components near the $q$’th component of $\sigma_{a}$ for $T^{q}_{\square}$ for
$SU([N/2])$.616161In this regard, notice that the components of
$\mu^{(ij)}_{\square\otimes\square}$ with $i\neq j$ can also be written as
$-1/(N/2)$ or $1-1/(N/2)$, similar to the components of $\mu^{i}_{\square}$
for $SU([N/2])$ (when $N$ is odd the difference would be clearly negligible).
For large $N$, these string tensions approach—as per our numerical results of
Table 3 or from the analytic study, recall paragraph after (112)—a nonzero
value $T_{1}$ with their differences suppressed by a power of $1/N$. From this
observation, we conclude that for large $N$ and for $|i-j|\gg 1$ and $|i-j|\ll
N$, $T^{(ij)}_{\square\otimes\square}\approx 2T_{1}$. Clearly, there are
$O(N^{2})$ such string tensions at large $N$.

On the other hand, the highest weight of the antisymmetric two index
representation, $\mu_{2}$, see (59), which was argued and numerically found to
give rise to the smallest $N$-ality two string tension, $T_{2}<2T_{1}$, is
obtained from $\mu^{(ij)}_{{\square\otimes\square}}$, by taking $i=j+1$
(mod$N$). There are $O(N)$ such string tensions, including the
${\mathbb{Z}}_{N}$-center orbit of the highest weight of the antisymmetric
two-index representation.

We now combine the results of the previous two paragraphs to conclude,
recalling (129), that at large $N$

$\langle
W_{{\square\otimes\square}}(R,T)\rangle\sim\sum^{d({\square\otimes\square})}_{h=1}\text{exp}(-T^{h}_{{\square\otimes\square}}RT)\leavevmode\nobreak\
\approx O(N^{2})\;e^{-2T_{1}RT}+O(N)\;e^{-T_{2}RT}.$ (131)

The first term in the last expression above represents the contribution of the
$O(N^{2})$ string tensions of weights $\mu^{(ij)}_{{\square\otimes\square}}$
with $|i-j|\gg 1$ and $|i-j|\ll N$. The second term is the contribution of the
$O(N)$ $k=2$ strings in the ${\mathbb{Z}}_{N}$ orbit of the highest weight of
the two-index antisymmetric representation. We now note that eqn. (131)
exactly mirrors the situation described and discussed earlier in eqn. (128),
showing the subtlety of taking the large-$N$ vs. large-$RT$ limit. See also
Appendix E, where (131) is recovered using the leading-order perturbative
saddle point, evaluated analytically for large-$N$.

The discussion in this Subsection demonstrates that in the framework of a
bounded large $N$ studied in this work (recall the preamble626262At larger
values of $N$, as mentioned in the preamble of Section 5.2, the virtual
effects of the W-bosons become important which has not been taken into account
in this work. We speculate, based on preliminary results, that with taking
these effects into account the same picture, i.e. large $N$ factorization and
interacting k-strings, persists at large $N$. of Section 5.2) large $N$
factorization would not necessarily imply free k-strings and the leading
exponential in the large area limit can come from the connected term of the
correlator of two Wilson loops in the fundamental representation that is sub-
leading in $N$ compared to the factorized term. We remind the reader that
although the connected term is sub-leading in $N$ it would still contribute to
the k-string tensions at large $N$—since, as already stressed at the beginning
of this Section, to find asymptotic k-string tensions, the large area limit
must be taken first and the leading exponential in this limit should be
considered. Thus, no matter how large $N$ is, the connected term, which
contains the leading exponential in the large area limit, would contribute to
the k-string tensions at large $N$.

####  5.2.3 A comment on “holonomy-decorated” Wilson loops

Here, we want to make a point which gives additional justification of our
emphasis to study $k$-strings of minimal tensions, corresponding to quark
sources of a particular weight (e.g. $\mu^{(ij)}_{{\square\otimes\square}}$,
with $i=j+1$ (mod$N$) for $k=2$). So far, we only considered gauge invariant
Wilson loop operators without insertions of the Higgs field (holonomy). In the
small-$L$ abelianized regime of dYM theory, one can isolate the contribution
of individual components of the fundamental quarks by inserting powers of the
holonomy inside the trace defining the Wilson loop. This gives rise to Wilson
loops in ${\mathbb{R}}^{3}$ “decorated” by loops winding around the
${\mathbb{S}}^{1}$, similar to the construction of Cherman:2016jtu ;
Aitken:2017ayq . The construction of these loops shows that the abelian
strings of different tensions (due to quarks of a single weight) in product
representations are physical, i.e. they are created by gauge invariant
operators. We now define a “decorated” Wilson loop as follows. The fundamental
representation holonomy around the ${\mathbb{S}}_{1}$ is

$\Omega(x)_{F}={\cal{P}}e^{i\oint\limits_{{\mathbb{S}}_{1}}A^{a}_{4}(x,x_{4})t^{a}_{F}dx_{4}}\leavevmode\nobreak\
,\ \ \ \ \ \ a=1,...,N-1.$ (132)

The gauge invariant Wilson loop projecting on a single component of a quark
field can then be written as

$W^{k}_{F}={\rm tr}_{F}{\cal P}\left[{1\over
N}\sum\limits_{p=1}^{N}\omega_{N}^{-(N-k)p}(\Omega(x)_{F})^{p}\right]\;e^{i\int\limits_{\text{x}}^{\text{x}}A_{\mu}d\text{x}^{\mu}}\leavevmode\nobreak\
,$ (133)

where $\omega_{N}=e^{i{2\pi\over N}}$ and the integral $\int\limits_{x}^{x}$
is taken along a large $RT$ contour in ${\mathbb{R}}^{3}$, broken up at the
point $x$ where the Higgs field is inserted. In the center symmetric vacuum at
weak coupling $\Omega$ can be replaced by its vacuum expectation value,
$\langle\Omega\rangle$, given (for brevity, shown below only for odd $N$ and
recalling (2.2.1)), by

$\langle\Omega\rangle={\rm
diag}(\omega_{N}^{N-1},\omega_{N}^{N-2},...,\omega_{N},1)\leavevmode\nobreak\
.$ (134)

Hence, the holonomy insertion and discrete Fourier transform in (133)project
(the term in square brackets inside the trace in (133)) the Wilson loop to an
abelian component corresponding to a source given by the $k$-th component
(weight) of the fundamental quark (in the ordering of eigenvalues as in
(134)). Using (133), one can construct sources of various weights in product
representations.

##  Appendix A Derivation of $\mathbf{W}$-boson spectrum

Consider two analogs of off-diagonal $SU(2)$ generators in $SU(N)$, namely
$T_{(kl)}^{1}$ and $T_{(kl)}^{2}$ (analogs of $\tau^{1}/2$ and $\tau^{2}/2$ in
$SU(2)$ respectively), $1\leq k,l\leq N$, where $k\neq l$, refer to the row
and column of the non-zero components of these generators. We will work out
the quadratic mass terms associated with their corresponding gauge fields
$A_{i}^{1}T_{(kl)}^{1}$ and $A_{i}^{2}T_{(kl)}^{2}$. The mass term comes from
the $F^{2}_{i4}$ term in (8) with
$F_{i4}=\partial_{i}A_{4}-\partial_{4}A_{i}-i[A_{i},A_{4}]$:

$\displaystyle\\{{1\over
2g^{2}}\text{tr}F^{2}_{i4}(x)\\}_{\text{quad},A_{i}^{1,2}}$ $\displaystyle=$
$\displaystyle{1\over
2g^{2}}\\{\text{tr}(\partial_{4}A^{1,2}_{i})^{2}-\text{tr}([A^{1,2}_{i},A^{vev}_{4}])^{2}$
$\displaystyle+2i\text{tr}(\partial_{4}A^{1,2}_{i}[A^{1,2}_{i},A^{vev}_{4}])\\}\leavevmode\nobreak\
,$

with $A_{i}^{1,2}=A_{i}^{1}T_{(kl)}^{1}+A_{i}^{2}T_{(kl)}^{2}$. Noting that
$[A^{1,2}_{i},A^{vev}_{4}]={2\pi|l-k|\over
NL}i(A_{i}^{2}T_{kl}^{1}-A_{i}^{1}T_{kl}^{2})$, expanding each component in
its Fourier modes using (11), and integrating over the compact $\text{x}_{4}$
direction we have:

$\displaystyle\int^{L}_{0}d\text{x}_{4}\\{{1\over
2g^{2}}\text{tr}F^{2}_{i4}(x)\\}_{quad,A_{i}^{1,2}}$ $\displaystyle={L\over
2g^{2}}\overset{+\infty}{\underset{m=0}{\sum}}\\{({2\pi m\over
L})^{2}(A_{i,m}^{1}A_{i,m}^{1\dagger}+A_{i,m}^{2}A_{i,m}^{2\dagger})$
$\displaystyle+({2\pi|l-k|\over
NL})^{2}(A_{i,m}^{1}A_{i,m}^{1\dagger}+A_{i,m}^{2}A_{i,m}^{2\dagger})$ (136)
$\displaystyle+2i{2\pi|l-k|\over NL}{2\pi m\over
L}(A^{1\dagger}_{i,m}A^{2}_{i,m}-A^{2\dagger}_{i,m}A^{1}_{i,m})\\}\leavevmode\nobreak\
.$

Expanding in real and imaginary parts of Fourier components we have:

$\displaystyle\int^{L}_{0}d\text{x}_{4}\\{{1\over
2g^{2}}\text{tr}F^{2}_{i4}(x)\\}_{quad,A_{i}^{1,2}}$ $\displaystyle={{L\over
2g^{2}}\overset{+\infty}{\underset{m=0}{\sum}}\\{({2\pi m\over
L})^{2}[(A_{i,m1}^{1})^{2}+(A_{i,m2}^{1})^{2}+(A_{i,m1}^{2})^{2}+(A_{i,m2}^{2})^{2}]}$
$\displaystyle{+4{2\pi|l-k|\over NL}{2\pi m\over
L}(A^{1}_{i,m2}A^{2}_{i,m1}-A^{2}_{i,m2}A^{1}_{i,m1})}$ (137)
$\displaystyle{+({2\pi|l-k|\over
NL})^{2}[(A_{i,m1}^{1})^{2}+(A_{i,m2}^{1})^{2}+(A_{i,m1}^{2})^{2}+(A_{i,m2}^{2})^{2}]\\}}\leavevmode\nobreak\
,$

with $A_{i,02}^{1}=A_{i,02}^{2}=0$. The above mass terms can be diagonalized
by defining the following fields:

$\begin{split}\bar{A}^{1}_{i,m}\equiv(A^{1}_{i,m1}+A^{2}_{i,m2})/\sqrt{2},\bar{A}^{2}_{i,m}\equiv(A^{1}_{i,m2}+A^{2}_{i,m1})/\sqrt{2}\\\
\bar{A}^{3}_{i,m}\equiv(A^{1}_{i,m2}-A^{2}_{i,m1})/\sqrt{2},\bar{A}^{4}_{i,m}\equiv(A^{1}_{i,m1}-A^{2}_{i,m2})/\sqrt{2}\leavevmode\nobreak\
,\end{split}$ (138)

leading to the quadratic Lagrangian for the off-diagonal components:

$\begin{split}\int^{L}_{0}d\text{x}_{4}\\{{1\over
2g^{2}}\text{tr}F^{2}_{i4}(x)\\}_{quad,A_{i}^{1,2}}=&{L\over
2g^{2}}\overset{+\infty}{\underset{m=0}{\sum}}\\{({2\pi m\over
L}-{2\pi|l-k|\over
NL})^{2}{[(\bar{A}^{1}_{i,m})^{2}+(\bar{A}^{3}_{i,m})^{2}]}\\\ &+({2\pi m\over
L}+{2\pi|l-k|\over
NL})^{2}{[(\bar{A}^{2}_{i,m})^{2}+(\bar{A}^{4}_{i,m})^{2}]}\\}\leavevmode\nobreak\
.\end{split}$ (139)

Relation (139) shows that there are W-bosons
${W^{\pm}_{1}=(\bar{A}^{1}_{i,m}\pm i\bar{A}^{3}_{i,m})/\sqrt{2}}$ and
${W^{\pm}_{2}=(\bar{A}^{4}_{i,m}\pm i\bar{A}^{2}_{i,m})/\sqrt{2}}$ with masses
$|{2\pi m\over L}-{2\pi|l-k|\over NL}|$ and $|{2\pi m\over L}+{2\pi|l-k|\over
NL}|$ respectively for $m=0,1,2,...$ and $1\leq l<k\leq N$.

##  Appendix B Error analysis

###  B.1 Truncation error

In this Section we will discuss relation (70). Consider (67) at
$m\rightarrow\infty$ and at its minimum solution:

$\bar{T}^{m,J}_{k,\text{min}}=\bar{T}^{m,J}_{k1,\text{min}}+\bar{T}^{m,J}_{k2,\text{min}}\leavevmode\nobreak\
.$ (140)

The only explicit dependence on $J$ in (140) is through $\Delta z={J/m}$.
Extracting this explicit dependence and suppressing the indices we have:

$H^{J}={H_{1}\over J}+JH_{2},\ \ \ \bar{T}^{m,J}_{k,\text{min}}\equiv H^{J},\
\ \ \bar{T}^{m,J}_{k1,\text{min}}\equiv{H_{1}\over J},\ \ \
\bar{T}^{m,J}_{k2,\text{min}}\equiv{JH_{2}}\leavevmode\nobreak\ .$ (141)

Taking the derivative of $H^{J}$ with respect to $J$ gives:

${dH^{J}\over dJ}=-{H_{1}\over J^{2}}+H_{2}+{1\over J}{\partial
H_{1}\over\partial f_{jh}}{df_{jh}\over dJ}+J{\partial H_{2}\over\partial
f_{jh}}{df_{jh}\over dJ}\leavevmode\nobreak\ .$ (142)

Since the partial derivatives at the minimum solution vanish we have:

${dH^{J}\over dJ}={1\over J}(-{H_{1}\over J}+JH_{2})\leavevmode\nobreak\ .$
(143)

For $J_{1}<J_{2}$ it can be shown that $H^{J_{2}}<H^{J_{1}}$. The minimum
solution of $H^{J_{i}}$ is a path $P_{i}$ that connects the boundary point
$\pi\mu_{k}$ at $\text{z}=0$ to $0$ at $\text{z}=J_{i}$ for $i=1,2$ ( Section
3.2). If we extend path $P_{1}$ on the z-axis from $\text{z}=J_{1}$ to
$\text{z}=J_{2}$ we would obtain a path $\tilde{P_{1}}$ that connects the
boundary point $\pi\mu_{k}$ at $\text{z}=0$ to $0$ at $\text{z}=J_{2}$. But
the value of the action of the paths $P_{1}$ and $\tilde{P_{1}}$ is the same
since the portion of the path $\tilde{P_{1}}$ that is on the z-axis gives zero
action. On the other hand the action of $\tilde{P_{1}}$ should be higher than
the action of $P_{2}$ since $P_{2}$ is the minimizing path of $H^{J_{2}}$
hence $H^{J_{2}}<H^{J_{1}}$. Due to this ${dH^{J}/dJ<0}$. From (143) this
gives ${H_{1}/J}>JH_{2}$ and hence
$\bar{T}^{m,J}_{k1,\text{min}}>\bar{T}^{m,J}_{k2,\text{min}}$ for all
$0<J<\infty$. Also we should have
$\underset{{J\rightarrow\infty}}{\lim}{dH^{J}/dJ}=0$ since $H^{J}$ is a
decreasing function of $J$ and it is bounded from below
($\underset{{J\rightarrow\infty}}{\lim}{H^{J}}=\bar{T_{k}}$). This shows that
$\bar{T}^{m,\infty}_{k1,\text{min}}=\bar{T}^{m,\infty}_{k2,\text{min}}$. Also
in the limit of $J\rightarrow 0$ the relations $\underset{{J\rightarrow
0}}{\lim}\bar{T}^{m,J}_{k1,\text{min}}=\infty$ and $\underset{{J\rightarrow
0}}{\lim}\bar{T}^{m,J}_{k2,\text{min}}=0$ can be easily verified from (61).
This can also be seen from relation (141), $H_{1}$ and $H_{2}$ are finite
quantities hence the previous limits follow. Summarizing the previous
relations derived we have:

$\displaystyle\bar{T}^{m,J}_{k1,\text{min}}>\bar{T}^{m,J}_{k2,\text{min}},\ \
\bar{T}^{m,J}_{k,\text{min}}>\bar{T}^{m,\infty}_{k,\text{min}},\ \ \ \ \ \ \ \
\ \ \ 0\leq J<\infty$ (144)
$\displaystyle\bar{T}^{m,0}_{k1,\text{min}}=\infty,\ \ \ \ \ \ \ \
\bar{T}^{m,0}_{k2,\text{min}}=0,\ \ \ \ \ \ \ \
\bar{T}^{m,\infty}_{k1,\text{min}}=\bar{T}^{m,\infty}_{k2,\text{min}}={1\over
2}\bar{T}^{m,\infty}_{k,\text{min}}\leavevmode\nobreak\ .$

From (144) it can be seen that $\bar{T}^{m,J}_{k2,min}$ starts from $0$ at
$J=0$ and approaches ${1\over 2}\bar{T}^{m,\infty}_{k,\text{min}}$ at
$J=\infty$. We conjecture that for $0<J<\infty$,
$\bar{T}^{m,J}_{k2,\text{min}}<{1\over
2}\bar{T}^{m,\infty}_{k,\text{min}}$636363 This behaviour has been verified in
the numerical simulations up to $J=14$. It has also been verified for cases
when an analytic solution is possible. For example expanding the cosine term
and keeping only the quadratic term. For this case it would be possible to
solve the saddle point analytically for a finite boundary condition at
$\text{z}=J$ and see that $\bar{T}^{\infty,J}_{k2,min}$ starts from $0$ at
$\text{z}=0$ and increases monotonically to ${1\over
2}\bar{T}^{\infty,\infty}_{k,\text{min}}$ at $\text{z}=\infty$.. Also from
(144) we have ${1\over 2}\bar{T}^{m,\infty}_{k,\text{min}}<{1\over
2}\bar{T}^{m,J}_{k,\text{min}}={1\over
2}(\bar{T}^{m,J}_{k1,\text{min}}+\bar{T}^{m,J}_{k2,\text{min}})<\bar{T}^{m,J}_{k1,\text{min}}$
so we obtain the following inequalities:

$\bar{T}^{m,J}_{k2,\text{min}}<{1\over
2}\bar{T}^{m,\infty}_{k,\text{min}}<{1\over
2}\bar{T}^{m,J}_{k,\text{min}}<\bar{T}^{m,J}_{k1,\text{min}}\leavevmode\nobreak\
.$ (145)

From (145) relation (70) easily follows.

###  B.2 Sample error calculation

We have shown data for half $k$-string tensions of $SU(10)$, k = 5 in Tables 6
and 7 to perform a sample error calculation.

m | n | $<T_{k1}>$ | $\sigma_{k1}\over\sqrt{R}$ | $<T_{k2}>$ | $\sigma_{k2}\over\sqrt{R}$ | $<T_{k}>$ | $\sigma_{k}\over\sqrt{R}$
---|---|---|---|---|---|---|---
100 | 18 | 5.82434259 | 6.9E-05 | 5.82600776 | 6.2E-07 | 11.6503503 | 6.2E-07
100 | 19 | 5.82423417 | 1.4E-05 | 5.82610277 | 1.4E-05 | 11.6503369 | 1E-07
100 | 20 | 5.82420397 | 3.9E-06 | 5.82612949 | 3.9E-06 | 11.6503335 | 1.3E-08
200 | 20 | 5.82500582 | 1.20E-04 | 5.82415592 | 1.20E-04 | 11.6491617 | 6.90E-07
200 | 21 | 5.82486451 | 5.00E-05 | 5.82428283 | 5.00E-05 | 11.6491473 | 1.70E-07
200 | 22 | 5.82482467 | 2.30E-05 | 5.82431903 | 2.30E-05 | 11.6491437 | 4.00E-08
Table 6: $SU(10)$, $k=5$ sample data with error in the mean ($\sigma\over\sqrt{R}$) m | n | $<T_{k1}>$ | $\Delta_{n}$ | $<T_{k2}>$ | $\Delta_{n}$ | $<T_{k}>$ | $\Delta_{n}$
---|---|---|---|---|---|---|---
100 | 18 | 5.82434259 | - | 5.82600776 | - | 11.6503503 | -
100 | 19 | 5.82423417 | -(1E-04) | 5.82610277 | 1E-04 | 11.6503369 | -(1E-05)
100 | 20 | 5.82420397 | -(3E-05) | 5.82612949 | 3E-05 | 11.6503335 | -(3E-06)
200 | 20 | 5.82500582 | - | 5.82415592 | - | 11.6491617 | -
200 | 21 | 5.82486451 | -(1E-04) | 5.82428283 | 1E-04 | 11.6491473 | -1E-05
200 | 22 | 5.82482467 | -(4E-05) | 5.82431903 | 4E-05 | 11.6491437 | -(4E-06)
Table 7: $SU(10)$, $k=5$ sample data with $\Delta_{n}=<X_{n}>-<X_{n-1}>$.

Sample minimization error calculation:
Sample calculation for $SU(10)$, $k=5$ and $m=100$:
Min. Error for $<T_{k2}>=|\Delta_{20}|+{\sigma_{k}\over\sqrt{R}}$ = 3E-05 +
3.9E-06 $\sim$ 3E-05

From Tables 6 and 7 and the sample calculation above it is clear that the
minimization errors are of order $10^{-5}$ and less so they can be safely
neglected in comparison to the discretization and truncation errors.

Sample discretization error calculation:
The difference between $<T_{k1}>$, $<T_{k2}>$ and $<T_{k}>$ for $m=100$ and
$m=200$ is:

$\begin{split}&<T_{k}>_{200}-<T_{k}>_{100}=11.6491437-11.6503335=-0.0011898\\\
&<T_{k1}>_{200}-<T_{k1}>_{100}=5.82482467-5.82420397=0.0006207\\\
&<T_{k2}>_{200}-<T_{k2}>_{100}=5.82431903-5.82612949=-0.00181046\leavevmode\nobreak\
.\end{split}$ (146)

Hence we predict that the continuum value of the string tensions of $SU(10)$,
$k=5$ for $J=14.0$ would be:

$\begin{split}&<T_{k}>=11.6491_{-0.001}\\\ &<T_{k1}>=5.8248^{+0.0006}\\\
&<T_{k2}>=5.8243_{-0.0018}\leavevmode\nobreak\ .\end{split}$ (147)

Sample truncation error calculation:

Relation (70) gives an upper bound estimate for the truncation error. Based on
(147):

$\begin{split}|<T_{k1}>-<T_{k2}>|\lessapprox
5.8248+0.0006-(5.8243-0.0018)=0.0029\\\ \Longrightarrow\text{Trunc.}\ E.\
\lessapprox 2\times 0.0029=0.0058\leavevmode\nobreak\ .\end{split}$ (148)

Adding the truncation and discretization error and neglecting the minimization
error we have:

$\text{Total Error}=\text{Trunc. E.}+\text{Dis. E.}=0.0058+0.001=0.0068\approx
0.007\leavevmode\nobreak\ .$ (149)

Hence we predict the value of the half string tension for $SU(10)$ and $k=5$
is: $11.6491_{-0.007}$. The errors obtained by this method for different half
$k$-string tensions varied from 0.005 to 0.007 therefore we have considered
the average as an upper bound estimate for the value of the error for all half
$k$-string tensions and included it in Table 1. It has to be noted that upper
bound estimates for errors always overestimate the true value of the errors as
can be seen from Table 2.

##  Appendix C Derivation of group theory results

###  C.1 Any representation of ${SU(N)}$ with ${N}$-ality ${1\leq k\leq N-1}$
contains the fundamental weight ${\mu_{k}}$ as one of its weights

The simple roots $\alpha_{i}$ and fundamental weights $\mu_{k}$ of $SU(N)$ are
given by the following relations:

$\begin{split}&\alpha_{i}=(0,..,0,\overset{\text{i-th}}{\widehat{1}},-1,0,...,0),\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i\leq N-1\\\ &\mu_{k}=({N-k\over
N},...,\overset{\text{k-th}}{\widehat{{N-k\over N}}},{-k\over N},...,{-k\over
N}),\ \ \ \ \ \ 1\leq k\leq N-1\end{split}$ (150)

An arbitrary representation of $SU(N)$ with $N$-ality $1\leq k\leq N-1$ can be
represented by its highest weight $w_{k}$:

$\displaystyle w_{k}=h_{i}\mu_{i}\ \text{with}\ h\equiv
h_{1}+2h_{2}+...+(N-1)h_{N-1}=mN+k,\ m,h_{i}\in\mathbb{Z},\ m,h_{i}\geq
0\leavevmode\nobreak\ ,$ (151)

where ${h_{i}\geq 0}$ are the $N-1$ Dynkin indices of the representation,
which determine $k$, its $N$-ality, by the mod($N$) relation given above. The
proof involves two steps. First we will prove the following lemma:
Lemma $w_{k}=\mu_{k}+a_{i}\alpha_{i}$ for $a_{i}\in\mathbb{Z}\ \text{and}\
a_{i}\geq 0$

###### Proof.

It can be easily seen that:

$\mu_{k}=k\mu_{1}-\beta_{k}\ \ \text{with}\ \
\beta_{k}=(k-1)\alpha_{1}+(k-2)\alpha_{2}+...+\alpha_{k-1},\ \ \beta_{1}=0$
(152)

Hence $w_{k}$ can be written as:

$w_{k}=(h_{1}+2h_{2}+...+(N-1)h_{N-1})\mu_{1}-(h_{2}\beta_{2}+...+h_{N-1}\beta_{N-1})\\\
$ (153)

With knowing $N\mu_{1}=(N-1)\alpha_{1}+(N-2)\alpha_{2}+...+\alpha_{N-1}$ and
(152) we have:

$\begin{split}h\mu_{1}=(mN+k)\mu_{1}=&\
\mu_{k}+(m(N-1)+k-1)\alpha_{1}+...+(m(N-(k-1))+1)\alpha_{k-1}\\\
&+m(N-k)\alpha_{k}+...+m\alpha_{N-1}\end{split}$ (154)

Therefore using (152) and (154), $w_{k}$ in (153) can be written as:

$w_{k}=\mu_{k}+b_{i}\alpha_{i}\ \ \ \ \text{with}\ \ \ \ b_{i}\in\mathbb{Z}$
(155)

We need to show that $b_{i}$ is greater than or equal to zero. Lets assume the
contrary. First lets assume $b_{k}<0$:

$\begin{split}&b_{k}<0\Longrightarrow b_{k}\leq-1\\\ &\eqref{eq:B6}\ \&\
\eqref{eq:B2}\Longrightarrow\alpha_{k}\cdot
w_{k}=2b_{k}+1-b_{k+1}-b_{k-1}=h_{k}\geq 0,\\\ &b_{k}\leq-1\ \&\
2b_{k}+1-b_{k+1}-b_{k-1}\geq 0\Longrightarrow b_{k+1}\leq b_{k}\ \text{or}\
b_{k-1}\leq b_{k}\leavevmode\nobreak\ .\end{split}$ (156)

Lets assume $b_{k-1}\leq b_{k}$:

$\begin{split}&\eqref{eq:B6}\ \&\
\eqref{eq:B2}\Longrightarrow\alpha_{k-1}\cdot
w_{k}=2b_{k-1}-b_{k}-b_{k-2}=h_{k-1}\geq 0,\\\ &b_{k-1}\leq b_{k}\ \&\
2b_{k-1}-b_{k}-b_{k-2}\geq 0\Longrightarrow b_{k-2}\leq
b_{k-1}\leavevmode\nobreak\ .\end{split}$ (157)

Similarly, it can be concluded that $0>b_{k}\geq b_{k-1}\geq b_{k-2}\geq
b_{k-3}\geq...\geq b_{2}\geq b_{1}$. Here we will clearly have a contradiction
since we have:

$\begin{split}&\eqref{eq:B6}\ \&\ \eqref{eq:B2}\Longrightarrow\alpha_{1}\cdot
w_{k}=2b_{1}-b_{2}=h_{1}\geq 0\\\ &\text{But if}\ b_{1}\leq
b_{2}<0\Longrightarrow 2b_{1}-b_{2}<0\leavevmode\nobreak\ .\end{split}$ (158)

Similarly, a contradiction occurs if it is assumed that $b_{k+1}\leq b_{k}$.
Now, if any other $b_{i}<0$ for $i\neq k$, similarly it can be argued that
either $b_{i+1}\leq b_{i}$ or $b_{i-1}\leq b_{i}$ and concluded that
$2b_{1}-b_{2}<0$ or $2b_{N-1}-b_{N-2}<0$ or $b_{k}\leq b_{i}<0$, which would
lead to contradictions similar to above. ∎

The next step of the proof is to show that given a highest weight $w_{k}$ of a
representation with $N$-ality $k$, it is always possible to lower with the
simple roots to obtain $\mu_{k}$. Given a weight $\mu$ of a representation of
$SU(N)$, the master formula in 09  can be applied:

${2\mu\cdot\alpha_{i}\over\alpha^{2}_{i}}=\mu\cdot\alpha_{i}=-(p_{i}-q_{i})$
(159)

Where $p_{i}\in\mathbb{Z}\ \text{and}\ p_{i}\geq 0$ is the number of times
which we can raise $\mu$ with $\alpha_{i}$ and $q_{i}\in\mathbb{Z}\
\text{and}\ q_{i}\geq 0$ is the number of times which we can lower $\mu$ with
$\alpha_{i}$. Based on the above Lemma, we have
$w_{k}=\mu_{k}+a_{i}\alpha_{i}$. If $a_{i}=0$ for all $i$ then the
representation contains $\mu_{k}$ as one of its weights but if at least one is
greater than zero then we will show that for some $\alpha_{i}$ which
$a_{i}>0$, $w_{k}\cdot\alpha_{i}>0$ which would imply that $q_{i}>0$ and hence
$w_{k}$ can be lowered with some $\alpha_{i}$ which $a_{i}>0$. Let
$a_{j}=\text{Max}\\{a_{i}|1\leq i\leq N-1\\}$ for some $1\leq j\leq N-1$.
Since we assumed at least one $a_{i}$ is greater than zero then $a_{j}>0$. If
$j=1$, $j=N-1$ or $j=k$ then $w_{k}\cdot\alpha_{j}=2a_{1}-a_{2}>0$,
$w_{k}\cdot\alpha_{j}=2a_{N-1}-a_{N-2}>0$ or
$w_{k}\cdot\alpha_{j}=2a_{k}+1-a_{k-1}-a_{k+1}>0$ respectively, since we
assumed that $a_{j}>0$ and is the maximum among others. Otherwise if
$a_{j}>a_{j+1}$ or $a_{j}>a_{j-1}$ then
$w_{k}\cdot\alpha_{j}=2a_{j}-a_{j+1}-a_{j-1}>0$. But if
$a_{j}=a_{j+1}=a_{j-1}$ then $w_{k}\cdot\alpha_{j}=0$. In this case $a_{j-1}$
and $a_{j+1}$ are greater than zero and both maximum among other $a_{i}$.
Hence we can repeat what we did for $a_{j}$ for $a_{j+1}$ or $a_{j-1}$ for a
number of steps until $j+r$ in $a_{j+r}$ for an $r\neq 0$ becomes $j+r=1$,
$j+r=N-1$ or $j+r=k$ or we would have $a_{j+r}>a_{j+r+1}$ or
$a_{j+r}>a_{j+r-1}$ which in that case $w_{k}$ can be lowered with
$\alpha_{j+r}$. So we proved that if at least one $a_{i}$ is greater than zero
then $w_{k}$ can always be lowered with some $\alpha_{h}$ which $a_{h}>0$.
Hence we continue this process until all the $\alpha_{i}$ are removed from the
highest weight $w_{k}=\mu_{k}+a_{i}\alpha_{i}$ and we reach $\mu_{k}$. This
shows that any representation of $SU(N)$ with $N$-ality $k$ contains $\mu_{k}$
as one of its weights.

###  C.2 Derivation of (49)

To derive (49), we will work out the steps for the contribution of one
monopole with magnetic charge $q^{1}_{m}$. Consider the long distance
behaviour of (49) for generators in the fundamental representation of the
gauge group, located at $R\in{\mathbb{R}}^{3}$:

$\begin{split}\\{\text{tr}\ \text{exp}(i\oint_{R\times
T}A^{c}_{m}t_{F}^{c}d\text{x}^{m})\\}_{\text{1 mon.}}&=\text{tr}\
\text{exp}(i\int_{S(R\times
T)}\epsilon_{anm}\partial_{n}A^{c}_{m}t_{F}^{c}d\text{S}^{a})\\\ &=\text{tr}\
\text{exp}(i\int_{S(R\times T)}d\text{x}_{1}d\text{x}_{2}\;{\text{R}_{3}\over
2|R-\text{x}|^{3}}\;Q^{1})\leavevmode\nobreak\ .\end{split}$ (160)

Here $c=1,...,N-1$ labels the abelian generators of $SU(N)$,
$Q^{1}=\text{diag}(1,-1,0,...,0)$, and we substituted the magnetic field of a
monopole, (19) converted to Cartesian coordinates. Next, we first write
$Q^{1}$ as a linear combination of the abelian generators of the fundamental
representation
$Q^{1}=\bar{V}_{i}t^{i}_{F}=\text{diag}(\bar{V}\cdot\bar{\mu}^{1}_{F},...,\bar{V}\cdot\bar{\mu}^{N}_{F})$
for $i=1,...,N-1$; here $\bar{\mu}^{j}_{F}$ for $j=1,...,N$ are the
$(N-1)$-dimensional weight vectors of the fundamental representation of
$SU(N)$ and $\bar{V}$ is an $(N-1)$-dimensional vector. In order to transform
this to an arbitrary representation $r$ we replace $t^{c}_{F}$ by its
corresponding generator in the representation $r$ and write
$Q^{1}_{r}{\equiv}\bar{V}_{i}t^{i}_{r}=\text{diag}(\bar{V}\cdot\bar{\mu}^{1}_{r},...,\bar{V}\cdot\bar{\mu}^{d(r)}_{r})$.
In order to write this in the $N$-dimensional form of weights used in this
work (150), we note that the weight vectors of the fundamental representation
of $SU(N)$ in their $N$-dimensional form are:

$\mu^{j}_{F}=(-{1\over N},...,-{1\over
N},\overset{\text{j-th}}{\widehat{1-{1\over N}}},-{1\over N},...,-{1\over
N}),\ \ \ \ \ \ \ \ \ \ j=1,...,N\leavevmode\nobreak\ ,$ (161)

which can be easily verified by lowering the fundamental weight $\mu_{1}$ in
(150) with the simple roots. From (161), the $N$-dimensional form of
$\bar{V}$, named $V$, can be determined by requiring:
$Q^{1}=\text{diag}(\bar{V}\cdot\bar{\mu}^{1}_{F},...,\bar{V}\cdot\bar{\mu}^{N}_{F})=\text{diag}(V\cdot\mu^{1}_{F},...,V\cdot\mu^{N}_{F})$,
which gives $V=q^{1}_{m}=(1,-1,0,...,0)$. Hence $Q^{1}_{r}$, using the
$N$-dimensional form of weights, becomes
$Q^{1}_{r}=\text{diag}(V\cdot\mu^{1}_{r},...,V\cdot\mu^{d(r)}_{r})=\text{diag}(q^{1}_{m}\cdot\mu^{1}_{r},...,q^{1}_{m}\cdot\mu^{d(r)}_{r})$.
Therefore (160), for generators in an arbitrary representation $r$, becomes:

$\begin{split}\\{\text{tr}\ \text{exp}(i\oint_{R\times
T}A^{c}_{m}t_{r}^{c}d\text{x}^{m})\\}_{\text{1 mon.}}&=\text{tr}\
\text{exp}(i\int_{S(R\times
T)}\epsilon_{anm}\partial_{n}A^{c}_{m}t_{r}^{c}d\text{S}^{a})\\\
&=\overset{d(r)}{\underset{j=1}{\sum}}\ \text{exp}(i\int_{S(R\times
T)}d\text{x}_{1}d\text{x}_{2}\;\mu^{j}_{r}\cdot q^{1}_{m}\;{\text{R}_{3}\over
2|R-\text{x}|^{3}})\leavevmode\nobreak\ .\end{split}$ (162)

##  Appendix D Perturbative saddle point $\mathbf{k}$-strings: leading order
$\mathbf{+}$ leading correction

The following tables 8 \- 16 compare the values of $k$-strings obtained from a
perturbative saddle point calculation to their numerical values in Table 1.
The ”Leading” and ”Leading Corr.” column give values for the coefficient of
$-{RT\over\beta}$ in (112) and (116) respectively. If $T$ represents a half
$k$-string value in Table 1 and $T^{\prime}$ its corresponding one in the
”Num. value” column, they are related by:

$T^{\prime}=\text{ROUND}(\bar{T},3)\ \ \ \text{with}\ \ \
\bar{T}\equiv\text{ROUNDDOWN}(T,3)\times 2/\sqrt{2}$ (163)

We multiply the half $k$-strings by $2$ to obtain the full $k$-string then we
divide it by $\sqrt{2}$ to normalize them similar to the perturbative saddle
point method $k$-strings as in (103) and (104). There is a high chance that
the numerical half $k$-strings in Table 1 will match the exact half
$k$-strings rounded to the third decimal if they are rounded down to the 3rd
decimal. This is due to the fact that the true value of half $k$-strings
always lies below the values obtained in Table 1 and the true value of the
error is of order $0.001$ or less as can be seen from Table 2.

$SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
2 | 9.870 | -2.029 | 7.841 | 8.000 | 0.159
3 | 11.396 | -2.343 | 9.053 | 9.238 | 0.185
4 | 11.913 | -2.396 | 9.517 | 9.699 | 0.182
5 | 12.150 | -2.410 | 9.740 | 9.919 | 0.179
6 | 12.277 | -2.415 | 9.862 | 10.041 | 0.179
7 | 12.355 | -2.417 | 9.938 | 10.114 | 0.176
8 | 12.405 | -2.418 | 9.987 | 10.163 | 0.176
9 | 12.439 | -2.418 | 10.021 | 10.196 | 0.175
10 | 12.463 | -2.418 | 10.045 | 10.221 | 0.176
Table 8: Comparison of $N$-ality $1$ $k$-strings for $SU(2\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
3 | 11.396 | -2.343 | 9.053 | 9.238 | 0.185
4 | 13.958 | -2.870 | 11.088 | 11.314 | 0.226
5 | 15.018 | -2.99 | 12.028 | 12.247 | 0.219
6 | 15.568 | -3.029 | 12.539 | 12.751 | 0.212
7 | 15.891 | -3.045 | 12.846 | 13.055 | 0.209
8 | 16.097 | -3.052 | 13.045 | 13.253 | 0.208
9 | 16.237 | -3.056 | 13.181 | 13.388 | 0.207
10 | 16.337 | -3.058 | 13.279 | 13.485 | 0.206
Table 9: Comparison of $N$-ality $2$ $k$-strings for $SU(3\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
4 | 11.913 | -2.396 | 9.517 | 9.699 | 0.182
5 | 15.018 | -2.990 | 12.028 | 12.247 | 0.219
6 | 16.449 | -3.142 | 13.307 | 13.511 | 0.204
7 | 17.248 | -3.197 | 14.051 | 14.247 | 0.196
8 | 17.746 | -3.222 | 14.524 | 14.715 | 0.191
9 | 18.078 | -3.234 | 14.844 | 15.032 | 0.188
10 | 18.311 | -3.241 | 15.070 | 15.257 | 0.187
Table 10: Comparison of $N$-ality $3$ $k$-strings for $SU(4\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
5 | 12.15 | -2.410 | 9.740 | 9.919 | 0.179
6 | 15.568 | -3.029 | 12.539 | 12.751 | 0.212
7 | 17.248 | -3.197 | 14.051 | 14.247 | 0.196
8 | 18.237 | -3.262 | 14.975 | 15.159 | 0.184
9 | 18.876 | -3.292 | 15.584 | 15.761 | 0.177
10 | 19.317 | -3.307 | 16.010 | 16.183 | 0.173
Table 11: Comparison of $N$-ality $4$ $k$-strings for $SU(5\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
6 | 12.277 | -2.415 | 9.862 | 10.041 | 0.179
7 | 15.891 | -3.045 | 12.846 | 13.055 | 0.209
8 | 17.746 | -3.222 | 14.524 | 14.715 | 0.191
9 | 18.876 | -3.292 | 15.584 | 15.761 | 0.177
10 | 19.629 | -3.326 | 16.303 | 16.474 | 0.171
Table 12: Comparison of $N$-ality $5$ $k$-strings for $SU(6\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
7 | 12.355 | -2.417 | 9.930 | 10.114 | 0.176
8 | 16.097 | -3.052 | 13.045 | 13.253 | 0.208
9 | 18.078 | -3.234 | 14.844 | 15.032 | 0.188
10 | 19.317 | -3.307 | 16.010 | 16.183 | 0.173
Table 13: Comparison of $N$-ality $6$ $k$-strings for $SU(7\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
8 | 12.405 | -2.418 | 9.987 | 10.163 | 0.176
9 | 16.237 | -3.056 | 13.181 | 13.388 | 0.207
10 | 18.311 | -3.241 | 15.070 | 15.257 | 0.187
Table 14: Comparison of $N$-ality $7$ $k$-strings for $SU(8\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
9 | 12.439 | -2.418 | 10.021 | 10.196 | 0.175
10 | 16.337 | -3.058 | 13.279 | 13.485 | 0.206
Table 15: Comparison of $N$-ality $8$ $k$-strings for $SU(9\leq N\leq 10)$ $SU(N)$ | Leading | Leading Corr. | Sum(Lead.+Lead. Corr.) | Num. value | Num. value - Sum
---|---|---|---|---|---
10 | 12.463 | -2.418 | 10.045 | 10.221 | 0.176
Table 16: Comparison of $N$-ality $9$ $k$-strings for $SU(10)$

.

##  Appendix E Large-$N$ limit of string tensions for product representations:
a saddle point leading-order perturbative evaluation

Our starting point is (112). Recall that this equation gives the contribution
to the expectation value of the Wilson loop of quarks of charges (weight)
$\mu$, evaluated to leading order using the perturbative saddle point method.
For convenience, we now reproduce the area-law part of that equation
($\hat{R},\hat{T}\rightarrow\infty,\beta\rightarrow 0$):

${\\{Z^{\eta}_{g^{4}}\\}_{\lambda=0}\over\\{Z^{\eta}_{g^{4}}\\}_{b_{q}=\lambda=0}}=\text{exp}(-{1\over\beta}\\{{1\over
4}{\sum\limits_{q=1}^{N-1}\sqrt{\Lambda_{q}}b^{2}_{q}}\hat{R}\hat{T}\\})\leavevmode\nobreak\
.$ (164)

Here, $b_{q}\equiv 2\pi(\mu)_{j}D_{jq}$ for a representation of weight $\mu$;
$(\mu)_{j}$ denotes the $j$-th component, $j=1,...N$, of the weight vector.
Recall also that $\Lambda_{q}=4\sin^{2}{\pi q\over N}$ is the dimensionless
mass of the $q$-th dual photon and that the components of the matrix $D_{jq}$,
$1\leq j\leq N$, are $D_{jq}=\sqrt{2\over N}\sin{2\pi qj\over N}$, for $1\leq
q<{N\over 2}$ and $D_{jq}=\sqrt{2\over N}\cos{2\pi qj\over N}$, for ${N\over
2}<q<{N}$; for brevity, we only give the values for odd $N$. The $q=N$
component of $D_{jq}$ does not contribute to (164).

The main difference compared to the discussion in the main text is that we now
consider also weights corresponding to product representations, for
concreteness the ${\square\otimes\square}$ representation. Recall, from (131),
that the expectation value of the Wilson loop in the product representation is
given, in the abelianized regime of this paper, by a sum of exponentials, one
for each weight of the product representation:

$\langle
W_{{\square\otimes\square}}(R,T)\rangle\sim\sum^{d({\square\otimes\square})}_{h=1}\text{exp}(-T^{h}_{{\square\otimes\square}}RT)=\sum^{d({\square\otimes\square})}_{h=1}\text{exp}(-{1\over\beta}\hat{T}^{h}_{{\square\otimes\square}}\hat{R}\hat{T})\leavevmode\nobreak\
.$ (165)

Where we have also written it in its dimensionless form (recall the relations
$R=\hat{R}/m_{\gamma}$, $T=\hat{T}/m_{\gamma}$,
$\beta=m_{\gamma}^{3}/\tilde{\zeta}$ from the comment below (103)). Comparing
with (164) $\hat{T}^{h}_{{\square\otimes\square}}$ to leading order (l.o.) is
given by:

$\hat{T}^{h}_{{\square\otimes\square},\text{l.o.}}={1\over
4}{\sum\limits_{q=1}^{N-1}\sqrt{\Lambda_{q}}b^{2}_{q}}\leavevmode\nobreak\
,\leavevmode\nobreak\ \leavevmode\nobreak\ b_{q}\equiv
2\pi\sum\limits_{j=1}^{N}(\mu^{h})_{j}D_{jq}\leavevmode\nobreak\ ,$ (166)

with $\mu^{h}$—the $h$-th weight of the ${\square\otimes\square}$
representation.

The goal of this Appendix is to evaluate (166) for all weights of the
${\square\otimes\square}$ product representation, to leading order in the
analytic perturbative saddle point method and in the large-$N$ limit. We shall
see that the leading-order analytic considerations support the findings
discussed qualitatively after around Eqns. (128) and (131) of the main text of
the behaviour of the product-representation Wilson loop at large $N$.

We begin by noting that the weights of the ${\square\otimes\square}$
representation are labeled by two integers $a,b=1,...N$ (there are $N^{2}$
weights) and are given by

$(\mu^{h})_{j}\rightarrow(\mu^{ab})_{j}=\delta^{a}_{j}+\delta^{b}_{j}-{2\over
N}\approx\delta^{a}_{j}+\delta^{b}_{j}.$ (167)

The last equality is valid for sufficiently large $N$. From (166), recalling
that we consider odd-$N$, we find an explicit expression for the tension of
strings sourced by quarks with weight $\mu^{ab}$, (167), of the product
representation at leading order:646464To obtain (168), we noted that for large
and odd $N$, for $1\leq q<{N\over 2}$:  $\begin{split}{b_{q}\over
2\pi}=\sum\limits_{j=1}^{N}(\mu^{h})_{j}D_{jq}=\sqrt{{2\over N}}\\{\sin{2\pi
aq\over N}+\sin{2\pi bq\over N}-\sum\limits_{j=1}^{N}{2\over N}\sin{2\pi
jq\over N}\\}\approx\sqrt{{2\over N}}\\{\sin{2\pi aq\over N}+\sin{2\pi bq\over
N}+0\\},\end{split}$ (168) and used the fact that the last term in the first
line of (168) for large $N$ can be approximated by
$\approx-{1\over\pi}\int_{0}^{2\pi}dy\sin qy=0$. Also, a similar expression
can be written for ${N\over 2}<q<N$.

$\displaystyle\hat{T}^{ab}_{{\square\otimes\square},\text{l.o.}}=4\pi\left[\sum\limits_{q=1}^{\left[{N\over
2}\right]}{\pi\over N}\sin{\pi q\over N}\left(\sin{2\pi aq\over N}+\sin{2\pi
bq\over N}\right)^{2}+\sum\limits_{q=\left[{N\over 2}\right]+1}^{N-1}{\pi\over
N}\sin{\pi q\over N}\left(\cos{2\pi aq\over N}+\cos{2\pi bq\over
N}\right)^{2}\right]$ (169)

The sum in (169) can be evaluated exactly for arbitrary $N$, but to illustrate
our point it suffices to consider i.) the results of a numerical evaluation
and ii.) the evaluation of (169) at infinite $N$ by replacing the sum by an
integral.

We begin with a discussion of the numerical results for the $N^{2}$ product
representation string tensions shown on Figure 7. The $N^{2}$ string tensions
for $N=21$ are evaluated numerically using (169). As the plot shows, most of
the $N^{2}$ string tensions are of order $2T_{1,\text{l.o.}}$, while $2N$ of
them are approximately equal to the minimal value $T_{2,\text{l.o.}}$, and $N$
are equal to approximately $4T_{1,\text{l.o.}}$. Clearly, this is conforming
to the discussion in the main text, Section 5.2.2..

Figure 7: The product ${\square\otimes\square}$-representation string tensions
$\hat{T}^{ab}_{{\square\otimes\square}}$, evaluated for $N=21$. They take
values between $1.32$ and $3.99$ times the fundamental string tension. As the
plot shows, $42=O(N)$ string tensions take the minimal value (these correspond
to $a=b\pm 1$(mod $N$)), $21=O(N)$ tensions approximately equal four times the
fundamental string tension (corresponding to $a=b$) and the rest of the string
tensions ($378=O(N^{2})$) are slightly less than twice the fundamental string
tension

We can also evaluate (169) in the infinite-$N$ limit by replacing the sum by
an integral, for $a,b$ fixed, i.e.,

$\displaystyle{\hat{T}^{ab}_{{\square\otimes\square},\text{l.o.}}\over
4\pi}=\int\limits_{0}^{{\pi\over 2}}dx\sin
x\left(\sin{2xa}+\sin{2bx}\right)^{2}+\int\limits_{{\pi\over 2}}^{\pi}dx\sin
x\left(\cos{2ax}+\cos{2bx}\right)^{2}\leavevmode\nobreak\ .$ (170)

Thus, the product representation string tensions, normalized to the
fundamental string tension (equal to $4\pi$ in the leading saddle point
approximation), becomes in the large-N limit

${\hat{T}^{ab}_{{\square\otimes\square},\text{l.o.}}\over 4\pi}=2-{2\over
4(a-b)^{2}-1}\leavevmode\nobreak\ .$ (171)

Due to the $Z_{N}$ symmetry of the string tension action we expect to obtain
the same tensions for $|a-b|=n$ and $|a-b|=N-n$ for $1\leq n\leq[N/2]$. This
symmetry is lost in (171) due to the infinite $N$ limit, therefore it is best
to use this relation for $|a-b|\leq[N/2]$ only at large N and for
$|a-b|>[N/2]$ make the replacement $|a-b|\rightarrow N-|a-b|$ in (171). In the
limit $|a-b|\gg 1$, this relation approaches the value of $2$, while for
$|a-b|=1$, we obtain the value ${4\over 3}\approx 1.33$; the value of $4$ for
$a=b$ is also obtained. This distribution of the $N^{2}$ string tensions in
the infinite-$N$ limit is consistent with the numerical result shown for
$N=21$ and with the general discussion of Section 5.2.2.

At the end, we also acknowledge an additional subtlety one might be worried
about. The calculation that led to eqn. (169)—see (112) as well as eqns.
(81–89) which directly lead to it—assumes that $RT$ is larger than the inverse
mass squared of all dual photons, including the lightest one. Thus, strictly
speaking one expects (169) to pertain to the order of limits we advocated for
here: infinite area at fixed N, followed by $N\rightarrow\infty$ which is the
proper order of limits necessary for calculating k-strings at large $N$.

However for the discussion of large $N$ factorization in gauge theories the
large $N$ limit is taken first. Now, if $RT$ is smaller than the mass of some
dual photons, the area law due to these photons should be replaced with a
perimeter law contribution. This remark is relevant because if the large-N
limit is taken first, the masses of some dual photons vanish—recall that their
masses are scale as $\sqrt{\Lambda_{q}}=2\sin{\pi q\over N}$—and these dual
photons do not lead to an area law. To take this into account, consider the
integral (170) and omit contributions of dual photons of (dimensionless) mass
$2\sin{\pi q\over N}=2\sin x<{1\over\sqrt{\hat{R}\hat{T}}}$, as they do not
give rise to area law. Thus, the region of integration in (170), instead of
$(0,{\pi\over 2})$ and $({\pi\over 2},\pi)$, should be replaced by,
respectively, $(\epsilon,{\pi\over 2})$ and $({\pi\over 2},\pi-\epsilon)$,
with $\epsilon\sim{1\over\sqrt{\hat{R}\hat{T}}}$. In the further large
$\hat{R}\hat{T}$ limit, we have that $\epsilon\rightarrow 0$, showing that the
contributions to the string tension of dual photons of mass vanishing at
large-$N$ is negligible. Thus, we expect that if the order of limits is taken
as described now ($N$ to infinity first, large area next), the factorization
result analyzed above in terms of string tensions is recovered.

The discussion of large-$N$ factorization above and in the 2nd half of Section
5.2.2 was carried out in terms of string tensions, since its more explicit and
intuitive and allows for a qualitative analysis of large-$N$ factorization in
terms of the full saddle point as was done in Section 5.2.2. For this
analysis, the large area limit had to be taken to isolate the area law
contribution and find the string tensions, as done in the previous paragraph.
However one can show the large-$N$ factorization result in a more general and
abstract setting without the need to refer to any large area limits or
expressions for string tensions. Consider eq. (111), which is a general
expression for the saddle point at leading order, without reference to any
large area limit:

${\\{Z^{\eta}_{g^{4}}\\}_{\lambda=0}\over\\{Z^{\eta}_{g^{4}}\\}_{b_{q}=\lambda=0}}=\text{exp}(-{1\over\beta}\\{{\Lambda_{q}b^{2}_{q}\over
2}\underset{\text{A}\
\text{A}}{\iint}d^{2}\text{x}d^{2}\text{x}^{\prime}P_{q}(\text{x}-\text{x}^{\prime})+{b_{q}^{2}\over
2}\underset{b(\text{A})\
b(\text{A})}{\iint^{\prime}}d\text{x}^{l}d\text{x}^{\prime
k}\delta^{lk}(P_{q}(\text{x}-\text{x}^{\prime})-{1\over
4\pi|\text{x}-\text{x}^{\prime}|})\\})\leavevmode\nobreak\ ,$ (172)

Using (168) and noting that the integrals in (172) are finite quantities and a
function of $\hat{R},\hat{T}$ and $\sqrt{\Lambda_{q}}=2\sin{\pi q\over N}$
with $x\equiv{\pi q\over N}$, the large $N$ limit of the leading saddle point
(s.p.) result in (172) reduces to:

$\text{s.p.}_{\square\otimes\square,\text{l.o.}}=\int^{\pi/2}_{0}dx(\sin{2ax}+\sin{2bx})^{2}F_{\hat{R},\hat{T}}(\sin
x)+\int^{\pi}_{\pi/2}dx(\cos{2ax}+\cos{2bx})^{2}F_{\hat{R},\hat{T}}(\sin
x)\leavevmode\nobreak\ ,$ (173)

where $F_{\hat{R},\hat{T}}(\sin{\pi q\over N})$ is given by:

$F_{\hat{R},\hat{T}}(\sin{\pi q\over N})\equiv
4\pi{\Lambda_{q}}\underset{\text{A}\
\text{A}}{\iint}d^{2}\text{x}d^{2}\text{x}^{\prime}P_{q}(\text{x}-\text{x}^{\prime})+4\pi\underset{b(\text{A})\
b(\text{A})}{\iint^{\prime}}d\text{x}^{l}d\text{x}^{\prime
k}\delta^{lk}(P_{q}(\text{x}-\text{x}^{\prime})-{1\over
4\pi|\text{x}-\text{x}^{\prime}|})$ (174)

The expression corresponding to (173) in the fundamental representation of
$SU(N)$ is:

$\text{s.p.}_{\square,\text{l.o.}}=\int^{\pi/2}_{0}dx(\sin{2ax})^{2}F_{\hat{R},\hat{T}}(\sin
x)+\int^{\pi}_{\pi/2}dx(\cos{2ax})^{2}F_{\hat{R},\hat{T}}(\sin x)$ (175)

Making the change of variable $x\rightarrow\pi-x$ in the second integral of
(173) and (175), they can be simplified to:

$\displaystyle\text{s.p.}_{\square\otimes\square,\text{l.o.}}=2\int^{\pi/2}_{0}dxF_{\hat{R},\hat{T}}(\sin
x)+2\int^{\pi/2}_{0}dx\cos(2ax-2bx)\;F_{\hat{R},\hat{T}}(\sin
x)\leavevmode\nobreak\ ,$ (176)
$\displaystyle\text{s.p.}_{\square,\text{l.o.}}=\int^{\pi/2}_{0}dxF_{\hat{R},\hat{T}}(\sin
x)\leavevmode\nobreak\ .$ (177)

When $|a-b|\gg 1$ (and $|a-b|\ll N$ if the discrete form of (173) is
considered for a finite but large $N$), the second integral in (176), due to
the rapid oscillations of $\cos(2ax-2bx)$, is near zero therefore $O(N^{2})$
weights of the product representation give approximately twice the value of
the fundamental representation string tension in the leading saddle point
approximation from (177), which has the same value for all weights of the
fundamental representation. Therefore relations (176) and (177) clearly show
the large $N$ factorization result in dYM theory at leading order of the
saddle point without any reference to a large area limit.

Although the calculations in this Appendix were done at the leading order
saddle point level the same ideas and methods can be applied to show large $N$
factorization regarding the corrections (as in (113)) to these leading order
saddle point results.

###### Acknowledgements.

We thank Mohamed Anber and Aleksey Cherman for comments on the manuscript and
Adi Armoni for many enlightening discussions of k-strings. We acknowledge
support by a NSERC Discovery Grant. The computations for this paper were
performed on the gpc supercomputer at the SciNet HPC Consortium. SciNet is
funded by: the Canada Foundation for Innovation under the auspices of Compute
Canada; the Government of Ontario; Ontario Research Fund - Research
Excellence; and the University of Toronto.

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