Confinement on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ and Double-String Collapse

In this paper, we study in detail the double-string confinement mechanism, in the framework of the abelian EFT valid at $LN\Lambda\ll 1$, also accounting for the leading effect of virtual $W$-boson loops. This EFT is described in some detail in Section 2.

For the reader familiar with the ${\mathbb{R}}^{3}$ Polyakov model, but less familiar with confinement in circle-compactified theories at small $L$, we review, in Section 3, the main differences between Polyakov confinement and confinement in SYM on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, for gauge group $SU(2)$. This discussion does not suffer from too much Lie-algebraic notation needed to study $SU(N)$, but is sufficient to stress the main peculiarities of small-$L$ confinement in SYM theory, which persist to arbitrary values of $N$.

For general $SU(N)$ gauge groups, the double-string confinement mechanism is explained in Sections 4.1, 4.2. The taxonomy of the double-string configurations of Sections 4.2.1-4.2.4 captures most qualitative aspects of the physics for any number of colors $N$ and for quark sources of any $N$-ality $k$. It is found, via numerical simulations, to fail only for $N\geq 5$, $k=1$, when DWs exhibit attraction and the double string collapses.

Our numerical simulations were performed for $SU(N)$ gauge theories with $N\leq 10$. The numerical computations of the two-dimensional classical equations of motion with quark sources (see Section 5) can become quite demanding, especially when sources of all $N$-alities for up to $N=10$ are considered. Including the effect of virtual $W$-bosons demands even more resources due to the broadening of the double strings. Our computations for the confining string configurations were primarily performed on the Niagara cluster at the SciNet HPC Consortium niagara ; scinet . Each quark source configuration was run on a single hyperthreaded processor core, allowing us to run 80 quark sources simultaneously on a single compute node. In total, our simulations utilized approximately 15 core-years of compute time on Niagara.

Our main results are as follows:

1. 1.

   We first argue, in Section 4.2, that for static quark sources of $N$-ality $k$, the lowest tension confining strings are double-string configurations composed of two of the BPS 1-walls of SYM theory. These BPS 1-walls carry electric fluxes that add up to match the flux of the quark source $\vec{w}_{k}$, the highest weight of the $k$-index antisymmetric representation.

   We also show that quarks of $N$-ality $k$ with charges different from $\vec{w}_{k}$ are confined by double-string configurations made of two BPS $p$-walls, with⁴⁴4$p=0$ denotes non-BPS walls. $0\leq p\leq k$. These metastable configurations are expected to decay to the lowest tension ones by the pair production of $W$-bosons, an effect that cannot be seen by the small-$L$ abelian EFT.

2. 2.

   Numerically (see Section 6) the above qualitative expectations are found to hold for $N\leq 4$ for all $k$. However, for $N\geq 5$ double strings form only for $k>1$, i.e. for all but the fundamental representation quarks. Fundamental quark sources for $N\geq 5$ were found to be confined by a single-string configuration, formed upon a collapse of the two BPS 1-walls into a non-BPS bound state. See Section 6.1.

   The collapse of the double string for $k=1$ is the most unexpected result of our findings and, at present, we have no simple qualitative explanation for it. To make sure it is not a numerical artifact, we perform many checks on this collapse. One includes simulating the largest possible distance between the sources, making sure that the collapse is not a finite separation effect (see Fig. 9). Most importantly, we show that the fact that this collapse occurs only for $k=1$, $N\geq 5$ can already be seen by studying the interactions of parallel BPS DWs with appropriate fluxes. These one-dimensional configurations are studied in great detail in Section 7, confirming that the collapse occurs only for BPS 1-walls whose fluxes add to that of a fundamental quark.

3. 3.

   The results of the two items above imply that the $N$-ality dependence of the string tensions is rather weak. Qualitatively, this is because the $k$-strings of lowest tension with $k>1$ are all double strings made of BPS $1$-walls carrying appropriate fluxes. Hence, one expects that the $k$-string tension is approximately twice the tension of the BPS $1$-wall, for all $k>1$. This expectation is borne out in our numerical simulations, see Fig. 1 for $SU(10)$, as well as Section 6.2. The observed slight increase of the string tension with $k$ seen in the data is consistent with the increase of the length of the strings due to the larger transverse separation for larger $N$-ality. On the other hand, the fact that for $k=1$ the tension is significantly lower is due to the double-string collapse and reflects the binding energy of the non-BPS bound state.

   The $N$-ality dependence found here does not conform to the usual large-$N$ argument that $T(k,N)$ should approach $kT(1,N)$. The difference between the abelian and nonabelian large-$N$ has been emphasized many times Douglas:1995nw ; Poppitz:2011wy ; Cherman:2016jtu ; Poppitz:2017ivi : the confining strings discussed here do not become free in the $N\rightarrow\infty$, $LN$-fixed abelian large-$N$ limit.

4. 4.

   We find that, whenever confinement is due to a double string, the separation between the strings increases as a logarithm of the distance between the sources. This qualitatively matches with a simple model of exponential DW repulsion proposed earlier for an $SU(2)$ theory Anber:2015kea . The fit of the parameters to the logarithmic growth is in Sec. 6.3.

5. 5.

   We probe larger values of $L$ by including the effect of virtual $W$-boson (and superpartner) loops on double-string confinement for $N\leq 9$ in Sec. 6.4. These effects become more important as the ${\mathbb{S}}^{1}$ size increases. The main lesson is that, while still in the semiclassical regime, these effects do not change the qualitative picture of double-string confinement for all $k$ and $N\leq 9$: no separation of collapsed $k=1$ strings for $N>4$ occurs, nor do the $N\leq 4$ $k=1$ strings collapse. The quantitative changes due to increasing $L$ are the larger transverse separation of the double string and the change of the $k$-string tension ratio, see Appendix C.0.3 and C.0.4. Upon increasing $L$ within the abelianized regime, the range of variation of $f(N,k)$ is seen to be even smaller than that shown in Fig. 1 (see Fig. 15).

   The expectation for the string-tension ratio on ${\mathbb{R}}^{4}$, based on large-$N$ arguments and the M-theory embedding of SYM, is that $f(N,k)$ is smilar to the sine-law and Casimir-scaling curves on Fig. 1. The behavior we find indicates that the evolution of the string-tension ratio towards ${\mathbb{R}}^{4}$ may not be monotonic. It is also possible that at the finite values of the coupling (see eqns. (52, 55)) we chose to study, the unknown higher-order corrections to the Kähler potential qualitatively affect the string-tension ratios.

In this paper, we studied the details of double-string confinement in the semiclassical regime on ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$, in the framework of SYM. While originally found independently, double-string confinement in semiclassical circle-compactified theories is, in fact, required by anomaly inflow of the mixed anomalies involving the spontaneously broken discrete 0-form (e.g. chiral or CP) symmetry and the 1-form center symmetry Cox:2019aji . This confinement mechanism transcends SYM and also holds in other theories that have similar higher-form anomalies. Two examples are deformed Yang-Mills theory and QCD with adjoint fermions, and more can be found in Anber:2017pak ; Anber:2019nfu . It would be of interest to study double-string confinement in more detail in those theories as well. We note that QCD with adjoint fermions stands out among the many theories, since it has gapless modes in the bulk.

The question of the behavior of confining strings as $L$ is increased beyond the $\Lambda NL<1$ regime is, of course, of great interest. Lattice studies of adiabatic continuity in SYM, as in Bergner:2018unx , and in other theories should be able to shed light on the transition between the abelian confinement studied in this paper and the full-fledged non-abelian confinement in the ${\mathbb{R}}^{4}$ theory.

There are several other features of circle-compactified theories, their domain walls, and their relation to confinement that would be nice to understand better. All our studies here, and in virtually all papers on the subject, are done in the framework of the 3D EFT of the lightest fields, the Kaluza-Klein zero modes of the 4D fields. The small-$L$ theory, however, is weakly coupled at all scales and it should be possible—although it may be complicated, see Anber:2013xfa for related work and the recent remarks in Unsal:2020yeh —to study more interesting dynamical questions that necessitate the incorportation of the Kaluza-Klein modes in the description. One such question would be to understand dynamically how quarks “become anyons” on the domain walls. This aspect of anomaly inflow on domain walls, see e.g. Hsin:2018vcg , is not possible to see in our 3D EFT. Further, the pair production of “$W$-bosons,” which we allude to many times and which makes the string tensions $N$-ality-only dependent, is also outside of the realm of validity of the 3D EFT. Naturally, incorporating the Kaluza-Klein modes in the dynamics might shed further light on the large-$L$ behavior.

We hope to return to these interesting questions in the future.

At small $LN\Lambda\ll 1$, the bosonic part of the long distance theory is described by the weakly-coupled ${\mathbb{R}}^{1,2}$ Lagrangian:

$$L_{\rm EFT}=M\;\partial_{\mu}{x}^{a}K_{ab}\;\partial^{\mu}{\bar{x}}^{b}-M\;{m^{2}\over 4}\;{\partial W({\bm{x}})\over\partial x^{a}}\;K^{ab}\;{\partial\bar{W}(\bar{\bm{x}}))\over\partial\bar{x}^{b}}~{}.$$

(3)

Here, $W(\bm{x})$ is the holomorphic superpotential:

$$W({\bm{x}})=\sum_{A=1}^{N}e^{\bm{\alpha}_{A}\cdot{\bm{x}}}~{},$$

(4)

$\bm{\alpha}_{1},\ldots,\bm{\alpha}_{N-1}$ are the simple roots, and $\bm{\alpha}_{N}=-\sum\limits_{A=1}^{N-1}\bm{\alpha}_{A}$ is the affine (or lowest) root of the $SU(N)$ algebra.¹²¹²12Roots are normalized to have square length $2$; roots and coroots for $SU(N)$ are identified, and $\bm{\alpha}_{A}\cdot\bm{w}_{B}=\delta_{AB}$, $A,B=1,\ldots N-1$. The Kähler metric and its inverse are denoted by $K_{ab}$ and $K^{ab}$ in (3).¹³¹³13Because our Kähler metric (5,7) is real, symmetric, and constant, we do not distinguish between holomorphic and antiholomoprhic indices.

As opposed to the superpotential $W$, whose form can be determined by holomorphy and symmetries, the Kähler potential is under control only at small $L$. In fact, only one term in its weak-coupling expansion at $\Lambda NL\ll 1$ has been computed. The expression for the inverse Kähler metric is

$\displaystyle K^{ab}$

$\displaystyle=$

$\displaystyle\delta^{ab}+{3g^{2}N\over 16\pi^{2}}\;k^{ab}~{}.$

(5)

where the one-loop correction is $k^{ab}$. Physically, the matrix $k^{ab}$ determining the inverse Kähler metric represents the mixing of the electric Cartan photons due to the non-Cartan “$W$-boson” and superpartner loops. The form of the inverse Kähler metric $K^{ab}$ (5) is justified in the semiclassical $LN\Lambda\ll 1$ limit and the leading correction $k^{ab}$ was computed in various ways in Poppitz:2012sw ; Anber:2014sda ; Anber:2014lba . The Lagrangian (3) in terms of the dual photon fields is obtained after a linear-chiral superfield duality transformation, which takes into account the one-loop mixing. The duality is performed in an expansion in powers of $g^{2}$ and is described in detail in Anber:2014lba .¹⁴¹⁴14For use below, we only note that when the Cartan-photon mixing is accounted for, the photon-dual photon duality relation (2) is replaced by $\displaystyle K^{ab}F^{b}_{\mu\nu}={g^{2}\over 4\pi L}\epsilon_{\mu\nu\lambda}\partial^{\lambda}\sigma^{a}.$ (6) The one-loop correction to the (inverse) Kähler metric is currently known at the center symmetric point:

$\displaystyle k^{ab}$

$\displaystyle=$

$\displaystyle{1\over N}\sum\limits_{1\leq A<B\leq N}(\vec{\beta}^{AB})^{a}(\vec{\beta}^{AB})^{b}\left(\psi({B-A\over N})+\psi(1-{B-A\over N})\right).$

(7)

Here $\psi$ is the logarithmic derivative of the gamma function and $\vec{\beta}^{AB}$ denote the positive roots of $SU(N)$. These are $N-1$ dimensional vectors in the root lattice (whose components are denoted $(\vec{\beta}^{AB})^{a}$, $a=1,...,N-1$); recall that the simple roots $\vec{\alpha}_{A}=\vec{\beta}^{A,A+1}$, $A=1,...N-1$ are a subset of the positive roots. We note that (7) is a purely group theoretic factor; its implications will be discussed after we introduce the relevant scales.

The scales appearing in the long-distance theory (3) are determined by the microscopic dynamics of the underlying 4D SYM theory on ${\mathbb{S}}^{1}$ of size $L$. We begin with the strong coupling scale $\Lambda$ of the $SU(N)$ theory

$\displaystyle\left({\Lambda L\over 4\pi}\right)^{3}={16\pi^{2}\over 3Ng^{2}}\;e^{-{8\pi^{2}\over Ng^{2}}}~{}.$

(8)

Here, and everywhere in this paper, $g$ denotes the 4D $SU(N)$ gauge coupling taken at the scale $4\pi\over L$ (of order the mass of the heaviest $W$-boson). The mass of the lightest $W$-boson is $m_{W}={2\pi\over LN}$ and the condition for validity of our EFT, usually stated as $\Lambda LN\ll 1$, can more precisely be phrased as ${\Lambda}\ll m_{W}={2\pi\over LN}$.

All scales appearing in the long-distance theory (3) can be expressed in terms of $\Lambda$ and $L$ (or $\Lambda$ and $m_{W}$). The scale $M$ appearing in the kinetic term is

$\displaystyle M={g^{2}\over 16\pi^{2}L}\;\sim{m_{W}\over|3\log{\Lambda L\over 4\pi}|}(1+\ldots)~{},$

(9)

while the nonperturbative scale $m\ll M$, which determines the strength of the instanton effects generating the superpotential $W$ (see Poppitz:2012nz for the $SU(N)$ normalizations) is:

$\displaystyle m$

$\displaystyle=$

$\displaystyle 6\sqrt{2}\;\left({4\pi\over LN}\right)\left({16\pi^{2}\over g^{2}N}\right)\left({\Lambda LN\over 4\pi}\right)^{3}\;\sim m_{W}\left({\Lambda LN\over 4\pi}\right)^{3}\left(|3\log{\Lambda L\over 4\pi}|+\ldots\right)~{}.$

(10)

In both (9) and (10), we used (8) to express $g^{2}N$ through the strong coupling scale $\Lambda$ and $L$, to leading accuracy at $g^{2}N\ll 1$ (as indicated by the dots).

The one-loop mixing matrix $k^{ab}$ can be diagonalized using a discrete Fourier transform (as done in unpublished work with A. Cherman), but for our purposes a numerical evaluation will suffice. As already noted, $k^{ab}$ from (7) is purely group-theoretic in nature and it is easy to see that it has only negative eigenvalues, the smallest of which is about $-6$, for the range of $N$ we study. This means that, upon increasing $g^{2}N$, the eigenvalues of $K^{ab}$ decrease away from unity (their vanishing would signal a strong coupling singularity where our EFT breaks down). As (8) shows, at fixed strong coupling scale $\Lambda$, an increase of $g^{2}N$ is tantamount to increasing $L$. Thus, the decrease of the $K^{ab}$ eigenvalues shows that a strong coupling regime is approached, as expected, in the large-$L$ limit. Conversely, a tiny value of $g^{2}N$ corresponds to small $\Lambda LN$ and implies that the corrections $k^{ab}$ are negligible.

We shall begin our study of confining strings with values of $g^{2}N$ such that the effect of $k^{ab}$ is negligible, i.e. we shall first take $K^{ab}=\delta^{ab}$. Later on, we shall also use the one-loop correction (7) as a probe of the $L$-dependence of the confining strings, while keeping the theory still well in (or at least on the “safe” side of) the calculable regime.

Following the usual terminology, we often call the scale $m$ the “dual photon mass.” We should, however, keep in mind that $m$ is really the mass of the heaviest of the $N-1$ dual photons, whose mass spectrum is given by $m_{k}\sim m\sin^{2}{\pi k\over N}$, $k=1,...,N-1$.¹⁵¹⁵15We cannot help but mention that the EFT (3), in addition to providing a semiclassically calculable framework to study confinement, exhibits many other, perhaps more unusual, phenomena. For example, in the abelian large-$N$ limit Cherman:2016jtu , the 3D theory (3) becomes effectively four dimensional by generating an emergent latticized dimension, with many features similar to $T$-duality of string theory. Further, as shown in Aitken:2017ayq ; Anber:2017ezt , the theory generates doubly exponentially small nonperturbative scales, $\sim e^{-e^{1/g^{2}}}$, which are manifested in the existence of exponentially weakly bound states of the dual photons of (3)—the 3D remnant of 4D glueball supermultiplets. The widths of DWs are generally determined by the lightest dual photon mass.

Of special interest to us here are the discrete chiral and center global symmetries of SYM:

1. 1.

   A 0-form chiral symmetry, ${\mathbb{Z}}_{2N}^{(0)}$. This is the usual discrete $R$-symmetry of SYM that acts on the fermionic superpartners of (1) (and on the superpotential), by a phase rotation. However, of most relevance to us is that it also shifts the dual photons:¹⁶¹⁶16Notice that ${\mathbb{Z}}_{2N}$ acts as ${\mathbb{Z}}_{N}$ on the dual photon and the superpotential.

   	$\displaystyle{\mathbb{Z}}_{2N}^{(0)}:~{}\bm{\sigma}$	$\displaystyle\rightarrow$	$\displaystyle\bm{\sigma}+{2\pi\bm{\rho}\over N}~{},$		(11)
   	$\displaystyle e^{\bm{\alpha}_{a}\cdot\bm{x}}$	$\displaystyle\rightarrow$	$\displaystyle e^{i{2\pi\over N}}e^{\bm{\alpha}_{a}\cdot\bm{x}}~{},~{}a=1,\ldots,N,$

   where we also show the action of ${\mathbb{Z}}_{2N}^{(0)}$ on the terms appearing in the superpotential (4). Here, $\bm{\rho}$ denotes the Weyl vector, $\bm{\rho}=\bm{w}_{1}+\ldots+\bm{w}_{N-1}$. As we shall see, this symmetry is spontaneously broken to ${\mathbb{Z}}_{2}$ (fermion number), leading to the existence of $N$ ground states and DWs between them.

2. 2.

   A “0-form” center symmetry ${\mathbb{Z}}_{N}^{(1),{\mathbb{S}}^{1}_{L}}$. The notation is chosen to emphasize that this is the dimensional reduction of the ${\mathbb{S}}^{1}_{L}$-component of the 4D 1-form center symmetry. As explained in detail in Anber:2015wha ; Cherman:2016jtu ; Aitken:2017ayq , the 0-form center symmetry acts on the dual photons and their superpartners:

   	$\displaystyle{\mathbb{Z}}_{N}^{(1),{\mathbb{S}}^{1}_{L}}:~{}\bm{x}$	$\displaystyle\rightarrow$	$\displaystyle{\cal P}\bm{x},$		(12)
   	$\displaystyle e^{\bm{\alpha}_{a}\cdot\bm{x}}$	$\displaystyle\rightarrow$	$\displaystyle e^{\bm{\alpha}_{a+1({\rm mod}N)}\cdot\bm{x}}~{},$		(13)

   and is an exact unbroken global symmetry of the theory.

   In a basis independent way, the operation denoted by $\cal{P}$ in (12) is the product of Weyl reflections with respect to all simple roots Anber:2015wha , while in the standard $N$-dimensional basis for the weight vectors it is a ${\mathbb{Z}}_{N}$ cyclic permutation of the vector’s components Cherman:2016jtu ; Aitken:2017ayq . This clockwise action is evident from the way ${\mathbb{Z}}_{N}^{(1),{\mathbb{S}}^{1}_{L}}$ acts on the $e^{\bm{\alpha}_{a}\cdot\bm{x}}$ terms in the superpotential, eqn. (13).

3. 3.

   A 1-form center symmetry ${\mathbb{Z}}_{N}^{(1),{\mathbb{R}}^{3}}$, acting on Wilson line operators in ${\mathbb{R}}^{3}$ by multiplication by appropriate ${\mathbb{Z}}_{N}$ phases.

The 0-form and 1-form center symmetries discussed above are parts of the ${\mathbb{Z}}_{N}^{(1),{\mathbb{R}}^{4}}$ 1-form center of the ${\mathbb{R}}^{4}$ theory. As recently realized Gaiotto:2014kfa , SYM on ${\mathbb{R}}^{4}$ has a mixed ’t Hooft anomaly between the 0-form ${\mathbb{Z}}_{2N}^{(0)}$ chiral symmetry and the ${\mathbb{Z}}_{N}^{(1),{\mathbb{R}}^{4}}$ 1-form center symmetry. This ’t Hooft anomaly persists upon an ${\mathbb{R}}^{3}\times{\mathbb{S}}^{1}$ compactification and anomaly matching is saturated by the spontaneous breaking of the discrete chiral symmetry, ${\mathbb{Z}}_{2N}^{(0)}\rightarrow{\mathbb{Z}}_{2}^{(0)}$, as in (14) below. Anomaly inflow on the resulting DWs implies that the DW worldvolume is nontrivial, supporting the phenomena of quark deconfinement (and braiding of Wilson lines, for DWs with three-dimensional worldvolume). Aspects of this inflow has been studied in many works Gaiotto:2017yup ; Gaiotto:2017tne ; Argurio:2018uup ; Bashmakov:2018ghn ; Anber:2018jdf ; Anber:2018xek . In the semiclassical small-$L$ setup of this paper, anomaly inflow and the deconfinement of quarks on DWs were studied in Cox:2019aji . For a recent discussion of these symmetries see also Poppitz:2020tto .

As already stated, our SYM EFT (3,4) has $N$ vacua, determined by solving for the stationary points of the superpotential $W(\bm{x})$. These are labelled by $k=0,\ldots,N-1$. In the $k$-th vacuum, the dual photon field $\bm{\sigma}$ has a nonzero expectation value $\langle\bm{\sigma}\rangle_{k}$, while $\bm{\phi}$ is fixed at the center symmetric point:

	$\displaystyle\langle\bm{\sigma}\rangle_{k}$	$\displaystyle=$	$\displaystyle{2\pi k\over N}\bm{\rho},\;\langle\bm{\phi}\rangle_{k}=0,\;{\rm such\;that}\;$		(14)
	$\displaystyle\langle e^{\bm{\alpha}_{a}\cdot\bm{x}}\rangle_{k}$	$\displaystyle=$	$\displaystyle e^{i{2\pi k\over N}},a=1,\ldots N,\;{\rm and}~{}\langle W\rangle_{k}\equiv W_{k}=Ne^{i{2\pi k\over N}}~{}.$

We also showed the expectation value $W_{k}$ of the superpotential in the $k$-th ground state. The $N$ vacua (14) are interchanged by the action of the spontaneously broken ${\mathbb{Z}}_{2N}^{(0)}\rightarrow{\mathbb{Z}}_{2}^{(0)}$ symmetry (11), while the ${\mathbb{Z}}_{N}^{(1),{\mathbb{S}}^{1}_{L}}$ symmetry is unbroken.¹⁷¹⁷17This follows from ${\cal{P}}{2\pi\bm{\rho}\over N}={2\pi\bm{\rho}\over N}-2\pi\bm{w}_{1}$. In words, the action of the 0-form center ${\mathbb{Z}}_{N}^{(1),{\mathbb{S}}^{1}_{L}}$ on the vacua (14) is a weight-lattice shift of $\langle\bm{\sigma}\rangle_{k}$, which is an identification, as per (1). The 1-form ${\mathbb{Z}}_{N}^{(1),{\mathbb{R}}^{3}}$ symmetry is also unbroken in the bulk of SYM, corresponding to the confinement of quarks.

There are DWs between the various $N$ vacua. A $k$-wall is one that connects vacua $k$ units apart, i.e. $\bm{\sigma}$ changes from $\langle\bm{\sigma}\rangle_{0}$ to $\langle\bm{\sigma}\rangle_{k}$. The BPS $k$-walls are the ones of lowest tension among all DW connecting the same two vacua. The tension of a BPS $k$-wall depends only on the asymptotics of the superpotential on the two sides of the DW, $W_{q}$ and $W_{q+k({\rm mod}N)}$, and is independent on the details of the Kähler metric, see ref. Hori:2000ck . The BPS DW tensions in SYM are

$\displaystyle T_{k-{\rm wall}}=Mm\;2N\sin{\pi k\over N}~{},~{}k=1,\ldots,N-1~{}.$

(15)

These BPS $k$-walls and their tensions play an important role in the constructions of candidate minimum-tension double-string confining configurations.

To study confinement, we add a static source term to the action for quarks of weights²¹²¹21We stress that since the long-distance dynamics is abelian, Wilson loops can be classified by the weights $\vec{\lambda}$. These are $SU(N)$ weight lattice elements, representing the charges of the quarks under the unbroken $U(1)^{N-1}$. These abelian sources can be given a fully gauge invariant description using Wilson loops representing static sources in $SU(N)$ representations $R$, extending in ${\mathbb{R}}^{1,2}$, but “decorated” by the insertion of appropriate powers of the ${\mathbb{S}}^{1}$ holonomies. In the abelianized phase where the holonomy has a ${\mathbb{Z}}_{N}^{(1),{\mathbb{S}}^{1}_{L}}$-preserving expectation value, individual weights can be selected by a discrete Fourier transform, see Poppitz:2017ivi for details. $\pm\vec{\lambda}$ at fixed positions $\vec{r}_{1,2}\in{\mathbb{R}}^{2}$:

$\displaystyle S_{{\rm source}}=\int dt\;\vec{\lambda}\cdot(\vec{A}_{0}(\vec{r}_{1})-\vec{A}_{0}(\vec{r}_{2}))~{}.$

(19)

We denote, recalling footnote 9, the ${\mathbb{R}}^{2}$ coordinates as $(z,y)$, and take $\vec{r}_{2}$$=$$(z$$=$$0$, $y$$=$$0)$ and $\vec{r}_{1}$$=$$(z$$=$$0$, $y$ $=$ $R)$. Then, using the duality relation (6), we find

	$\displaystyle S_{\rm source}$	$\displaystyle=$	$\displaystyle-\int dt\int\limits_{0}^{R}dy\;\vec{\lambda}\cdot\vec{F}_{0y}(0,y)={g^{2}\over 4\pi L}\;(\vec{\lambda})^{a}K_{ab}\;\int dt\int\limits_{0}^{R}dy\;\partial_{z}\sigma^{b}(z,y)|_{z=0}$		(20)
		$\displaystyle=$	$\displaystyle-{g^{2}\over 4\pi L}(\vec{\lambda})^{a}K_{ab}\int dtdzdy\;\partial_{z}\delta(z)\int\limits_{0}^{R}dy^{\prime}\delta(y-y^{\prime})\sigma^{b}(z,y)~{}.$

We then add the action of the EFT (3), $S_{\rm EFT}=\int dtdzdyL_{\rm EFT}$, to the source action $S_{\rm source}$ and introduce dimensionless coordinates $\tilde{t},\tilde{z},\tilde{y}$, via $t={\tilde{t}\over m},z={\tilde{z}\over m},y={\tilde{y}\over m}$, to obtain, after dropping the tilde from the dimensionless coordinates²²²²22Here, $\vec{\nabla}$ denotes the gradient operator in ${\mathbb{R}}^{2}$ and not a vector in the group lattice.

	$\displaystyle S_{\rm EFT+source}$	$\displaystyle=$	$\displaystyle{M\over m}\int dtdzdy\left[K_{ab}(\partial_{t}x^{a}\partial_{t}\bar{x}^{b}-\vec{\nabla}x^{a}\vec{\nabla}\bar{x}^{b})-{1\over 4}{\partial W\over\partial x^{a}}K^{ab}{\partial\bar{W}\over\partial\bar{x}^{b}}\right.$		(21)
			$\displaystyle\left.\qquad\qquad\qquad-4\pi(\vec{\lambda})^{a}K_{ab}\;\partial_{z}\delta(z)\int\limits_{0}^{R}dy^{\prime}\delta(y-y^{\prime}){\rm Im}[x^{b}(z,y)]\right]~{}.$

We numerically solve the static equation of motion²³²³23This equation implies that $\bm{\sigma}$ exhibits a $2\pi\bm{\lambda}$ jump upon crossing the horizontal ($y$) axis in the positive $z$ direction, for $y$ between $0$ and $R$. See also the original derivation in Polyakov:1976fu ; Polyakov:1987ez , also given in Poppitz:2017ivi . following from (21)

$$\nabla^{2}x^{a}=\frac{1}{4}K^{ab}K^{cd}\;\frac{\partial W}{\partial x^{c}}\frac{\partial^{2}\bar{W}}{\partial\bar{x}^{b}\partial\bar{x}^{d}}+2\pi i(\vec{\lambda})^{a}\partial_{z}\delta(z)\int_{0}^{R}dy^{\prime}\delta(y-y^{\prime})~{}.$$

(22)

Distance scales in the equations above are all dimensionless numbers, which are converted to physical units upon multiplication by $1/m$, i.e. the (heaviest, at $K_{ab}=\delta_{ab}$) dual photon Compton wavelength, with $m$ defined in (10). Likewise, the dimensionless static energy,

$\displaystyle\tilde{E}=\int dzdy\left[K_{ab}\vec{\nabla}x^{a}\vec{\nabla}\bar{x}^{b}+{1\over 4}{\partial W\over\partial x^{a}}K^{ab}{\partial\bar{W}\over\partial\bar{x}^{b}}\right]~{}$

(23)

is converted to the physical energy $E=M\tilde{E}$. A dimensionless string tension (or a DW tension (15)) $\tilde{T}$ computed from the simulation, is converted to physical units as $T=Mm\;\tilde{T}$. When we study the effect of changing $L$ at fixed $\Lambda$ (i.e. changing $g^{2}N$ in (5) according to (8)), being mindful of these scalings and the definitions of $M$ and $m$ in terms of $L$ and $\Lambda$ will be important.

The double-string confinement mechanism is illustrated on Fig. 2 and is explained in more detail in the caption. As already alluded to, the confining string consists of two DWs between the $N$ vacua of SYM. Its essence is that due to confinement, the chromoelectric flux of color sources can only propagate along one dimensional lines, the DWs of SYM theory. As explained at length below, the fluxes carried by the DWs are such that one needs two DWs in order to absorb the quark fluxes.

We shall consider quark sources in different representations of $SU(N)$ and discuss what kind of double-string configurations are expected. But first, we remind the reader of the results of Cox:2019aji , which are essential for our arguments. In that paper, the electric fluxes carried by BPS $k$-walls in SYM theory were determined. It has long been known, see Hori:2000ck , that there are $N\choose k$ BPS walls between vacua $k$ units apart. Ref. Cox:2019aji showed that the BPS $k$-walls fall into two classes, labeled by $I$ and $II$ below, which have monodromies of $\vec{\sigma}$, or electric fluxes $\bm{\Phi}^{k,I}_{i_{1},...,i_{k}}$ and $\bm{\Phi}^{k,II}_{i_{1},...,i_{k-1}}$, given by:

	$\displaystyle\bm{\Phi}^{k,I}_{i_{1},...,i_{k}}$	$\displaystyle=$	$\displaystyle 2\pi{k\over N}\bm{\rho}-2\pi(\bm{w}_{i_{1}}+\ldots+\bm{w}_{i_{k}}),\;{\rm all}\;k\;\bm{w}_{i}\;{\rm different};\;{\left(\begin{array}[]{c}N-1\cr k\end{array}\right)}\;{\rm walls},$		(26)
	$\displaystyle\bm{\Phi}^{k,II}_{i_{1},...,i_{k-1}}$	$\displaystyle=$	$\displaystyle 2\pi{k\over N}\bm{\rho}-2\pi(\bm{w}_{i_{1}}+\ldots+\bm{w}_{i_{k-1}}),\;{\rm all}\;k-1\;\bm{w}_{i}\;{\rm different};\;{\left(\begin{array}[]{c}N-1\cr k-1\end{array}\right)}\;{\rm walls}.$		(29)

The total number of BPS $k$-walls is, as already stated, ${N\choose k}={N-1\choose k}+{N-1\choose k-1}$. We stress that this is because the weights $(\vec{w}_{i_{1}},...,\vec{w}_{i_{k}})$ are to be taken all different, with indices ranging from $1$ to $N-1$; likewise all weights in the second set, $(\vec{w}_{i_{1}},...,\vec{w}_{i_{k-1}})$, are to be taken different.²⁴²⁴24For $k=1$, the second set in (26), $\bm{\Phi}^{1,II}$, consists of a single wall carrying flux $\frac{2\pi\bm{\rho}}{N}$, hence the total number of $k=1$ walls is $N$. The above spectrum of BPS $k$-wall fluxes is invariant under $k\rightarrow N-k$ up to reversal of the overall sign of the electric flux (a parity transformation, as in Bashmakov:2018ghn ).