Wilson Loops, Worldlines, Monopole Plasma, and Confinement

The Wilson loop is the nonlocal observable that most sharply diagnoses how a gauge theory responds to external electric charges. It records the holonomy acquired by an infinitely heavy charged particle moving around a closed curve, and its behavior at large size distinguishes Coulomb forces, screening, and confinement.

The logic of this page has three layers. First, a charged particle in a gauge field naturally carries a worldline phase $e^{iq\int A}$. Second, a rectangular Wilson loop computes the static potential between an external charge and an external anticharge. Third, in compact gauge theory the perturbative photon is not the whole story: monopole events form a plasma, generate a mass for the dual photon, and turn the Wilson loop into an area law.

This is one of the course’s key recurring morals. Local perturbation theory sees the Lie algebra. Confinement is often controlled by the global topology of the gauge group and by nonperturbative configurations invisible in an expansion around $A_\mu=0$.

Required background. Compact gauge fields, Higgsing, solitons, and topological defects supplies compact $U(1)$ gauge fields, monopoles, and flux quantization.

Helpful background. Worldlines, worldsheets, and reparametrization gauge develops the first-quantized path integral used below.

From charged particles to Wilson lines

Euclidean Wilson loops and compact-QED normalization

Most formulas on this page are Euclidean. We write $A=A_\mu dx^\mu$ and $F=dA$. For a charge $q$,

$$W_q(C)=\left\langle \exp\left(iq\oint_C A\right)\right\rangle.$$

For a rectangular loop with spatial size $R$ and Euclidean time size $T$,

$$W(R,T)\sim e^{-T V(R)} \qquad (T\to\infty),$$

defines the static potential $V(R)$ of external probes. In Lorentzian signature the corresponding phase is $e^{-iT V(R)}$; the Euclidean expression follows after Wick rotation.

In the compact-QED discussion, numerical constants in the monopole action, fugacity, dual-photon mass, and string tension depend on the ultraviolet completion and on the normalization of $A$, $q$, and $e$. We write the monopole fugacity as $\zeta\propto e^{-S_0}$ and keep the core scale explicit when dimensional analysis requires it. The robust conclusions are that monopoles generate a dual-photon mass and an area law in three Euclidean dimensions.

A charged particle moving along a path $x^\mu(s)$ couples to a background gauge field through

$$S_{E,\rm int}=-iq\int A_\mu(x(s))\,{dx^\mu\over ds}\,ds =-iq\int_C A.$$

Thus its Euclidean path-integral weight $e^{-S_E}$ contains the phase

$$\exp\left(iq\int_C A\right).$$

If $C$ is an open path from $y$ to $x$, this object is not gauge invariant by itself. Under

$$A\mapsto A+d\alpha,$$

it transforms as

$$\exp\left(iq\int_y^x A\right) \mapsto e^{iq\alpha(x)} \exp\left(iq\int_y^x A\right) e^{-iq\alpha(y)}.$$

That is exactly what is needed to parallel-transport a charged field from $y$ to $x$. The gauge-invariant bilocal operator is not $\psi^\dagger(x)\psi(y)$, but rather

$$\psi^\dagger(x) \exp\left(iq\int_y^x A\right) \psi(y),$$

with the path specified. The path dependence is not a flaw. A gauge field measures parallel transport, so different paths can differ by the flux through the surface between them.

For a closed curve $C$, the endpoint phases cancel and the Wilson loop

$$W_q(C)=\left\langle e^{iq\oint_C A}\right\rangle$$

is gauge invariant.

A charged worldline carries the parallel-transport phase $e^{iq\int A}$. The open Wilson line transforms at its endpoints; a closed Wilson loop is gauge invariant.

For a non-Abelian gauge field, the same definition becomes

$$W_R(C)=\left\langle {1\over \dim R}\operatorname{tr}_R\,\mathcal P \exp\left(i\oint_C A_\mu^a T_R^a dx^\mu\right)\right\rangle,$$

where $R$ is a representation and $\mathcal P$ denotes path ordering. This page focuses mostly on Abelian compact $U(1)$, where the monopole mechanism can be derived explicitly. The conceptual role of the Wilson loop, however, is the same: it asks what energy is required to separate external charges.

Worldline representation of a charged propagator

The Wilson-line phase appears naturally in the first-quantized representation of a charged particle. For a scalar particle of mass $m$ in a fixed Abelian background $A_\mu$, the Euclidean propagator can be written schematically as

$$G_A(x,y)=\int_0^\infty d\tau\,e^{-m^2\tau} \int_{X(0)=y}^{X(\tau)=x}DX\, \exp\left[-\int_0^\tau ds\,{\dot X^2\over4} +iq\int_0^\tau ds\,\dot X^\mu A_\mu(X(s))\right].$$

The first term is the Brownian worldline action. The second term is the Wilson-line coupling. In a reparametrization-invariant form, closer to the heavy-particle limit, one writes

$$S[X]=m\int ds\sqrt{\dot X^2}-iq\int_C A.$$

The saddle is a classical path weighted by its proper length and by the gauge holonomy.

This representation also gives a quick way to see mass renormalization. If the gauge field is dynamical and Gaussian, integrating it out produces an effective interaction along the particle worldline:

$$S_{\rm eff}[X] =mL+{q^2\over2}\int ds\,ds'\, \dot X^\mu(s)\dot X^\nu(s')D_{\mu\nu}(X(s)-X(s'))+\text{local counterterms}.$$

At nearby points $s\simeq s'$, the photon propagator is singular. That short-distance part renormalizes the particle mass. At separated points on two different static worldlines, the same formula gives the Coulomb interaction.

For Gaussian electrodynamics, integrating out $A_\mu$ turns Wilson-line phases into pairwise interactions between worldline segments. The short-distance self-interaction renormalizes the mass; the long-distance interaction gives the static potential.

This is why Wilson loops are not merely formal gauge-invariant operators. They are the worldline observables of external charged probes.

The static limit and the rectangular loop

To isolate a potential, take a heavy charge and anticharge separated by distance $R$ and let them propagate for Euclidean time $T$. The two long vertical worldlines, joined at the beginning and end, form a rectangle $C_{R,T}$. Inserting the Wilson loop creates the sector with the two external charges.

The transfer-matrix interpretation gives

$$W(R,T)=\sum_n c_n(R)e^{-T E_n(R)}.$$

As $T\to\infty$, the lowest-energy state dominates:

$$W(R,T)\sim c_0(R)e^{-T V(R)}.$$

Therefore

$$\boxed{ V(R)=-\lim_{T\to\infty}{1\over T}\log W(R,T). }$$

Here and below, the $R$-independent ultraviolet self-energy of each infinitely heavy probe is understood to be subtracted from $V(R)$.

A rectangular Wilson loop of spatial width $R$ and Euclidean time height $T$ projects onto the lowest-energy state containing a static external charge and anticharge. An area law gives a linearly rising potential.

This formula is the operational definition of confinement for external probes. If

$$W(R,T)\sim e^{-\sigma RT},$$

then

$$V(R)=\sigma R.$$

The coefficient $\sigma$ is the string tension. It is the energy per unit length of the electric flux tube.

If instead

$$W(C)\sim e^{-\mu P(C)},$$

where $P(C)$ is the perimeter of the loop, then the logarithm is dominated by self-energies of the sources rather than by an energy proportional to their separation. This is a perimeter law. It describes screening, Higgs behavior, or short-range forces, depending on the theory.

There is also an intermediate Coulomb behavior. In four Lorentzian spacetime dimensions, a massless photon gives

$$V(R)\sim {q^2 e^2\over4\pi R},$$

up to sign depending on whether the probes have equal or opposite charges. In three Lorentzian spacetime dimensions, the Coulomb potential is logarithmic. In two Lorentzian spacetime dimensions, Maxwell theory has no transverse photon, and Gauss’s law gives a linear potential even without monopoles. That last case is kinematic, not the same mechanism as compact-QED confinement in $2+1$ dimensions.

Gaussian Wilson loops and Coulomb potentials

For noncompact Maxwell theory with action

$$S[A]={1\over4e^2}\int d^D x\,F_{\mu\nu}F_{\mu\nu},$$

the Wilson-loop expectation value is a Gaussian integral. Write the loop current as

$$J^\mu(x)=q\oint_C dx^\mu(s)\,\delta^{(D)}(x-x(s)).$$

Since the current is conserved,

$$\partial_\mu J^\mu=0,$$

the gauge-dependent longitudinal part of the photon propagator drops out. The result has the form

$$\left\langle e^{i\int J\cdot A}\right\rangle = \exp\left[-{e^2\over2} \int d^D x\,d^D y\,J_\mu(x)D_{\mu\nu}(x-y)J_\nu(y)\right].$$

For two long static worldlines, this reduces to the usual Coulomb kernel. In $d_s$ spatial dimensions,

$$V(R)\propto q^2 e^2\int {d^{d_s}k\over(2\pi)^{d_s}} {e^{i\mathbf k\cdot\mathbf R}\over \mathbf k^2},$$

with a self-energy subtraction understood. Equivalently,

$$\int {d^{d_s}k\over(2\pi)^{d_s}}{1-e^{i\mathbf k\cdot\mathbf R}\over \mathbf k^2} \sim \begin{cases} R, & d_s=1,\\ \log R, & d_s=2,\\ R^{2-d_s}, & d_s>2. \end{cases}$$

Thus ordinary noncompact Maxwell theory in $3+1$ dimensions gives a perimeter-plus-Coulomb law, not an area law. A perturbative photon does not confine in four dimensions.

The conclusion is subtle but important: a Wilson loop can show a linear potential either because Gauss’s law in one spatial dimension leaves no transverse spreading, or because a genuine nonperturbative mechanism squeezes flux into a tube. Compact QED in $2+1$ dimensions is the cleanest example of the second mechanism.

Area laws from strong-coupling surfaces

On a lattice, compactness is explicit. The Abelian Wilson action is

$$S=-\beta\sum_p \cos F_p,$$

where

$$F_p=\sum_{\ell\in\partial p}\epsilon_{p\ell}A_\ell, \qquad A_\ell\sim A_\ell+2\pi.$$

The Wilson loop is

$$W(C)=\left\langle e^{i\sum_{\ell\in C}A_\ell}\right\rangle.$$

At strong coupling, $\beta\ll1$, use the character expansion

$$e^{\beta\cos F_p}=\sum_{n_p\in\mathbb Z}I_{n_p}(\beta)e^{in_pF_p},$$

where $I_n$ is the modified Bessel function. Integrating over each link angle imposes an integer flux-conservation law: the plaquette integers $n_p$ form closed surfaces, except that in the presence of the Wilson loop their boundary must be $C$.

Let $N_{\min}(C)$ be the number of plaquettes in a minimal surface $\Sigma$ with $\partial\Sigma=C$. The leading configuration has $n_p=1$ on those plaquettes. Dividing by the vacuum partition function gives

$$W(C)\simeq \left({I_1(\beta)\over I_0(\beta)}\right)^{N_{\min}(C)}.$$

For small $\beta$,

$$I_0(\beta)=1+O(\beta^2), \qquad I_1(\beta)={\beta\over2}+O(\beta^3),$$

so

$$W(C)\sim\left({\beta\over2}\right)^{N_{\min}(C)} =e^{-\sigma A_{\min}^{\rm phys}(C)},$$

where $A_{\min}^{\rm phys}(C)=a^2N_{\min}(C)$ and the physical string tension is

$$\sigma\simeq {1\over a^2}\log {2\over\beta}.$$

In the strong-coupling expansion of compact lattice gauge theory, plaquette flux variables form surfaces. A Wilson loop forces an open surface ending on $C$, and the minimal surface gives the leading area law.

This strong-coupling result is elementary and powerful. It does not prove that every compact gauge theory confines at every coupling. It proves that at sufficiently strong coupling, the Wilson loop is dominated by fluctuating electric flux sheets and obeys an area law.

Compact QED in three Euclidean dimensions

Now consider continuum compact QED in three Euclidean dimensions. The smooth Maxwell action is

$$S={1\over4e^2}\int d^3x\,F_{\mu\nu}F_{\mu\nu}.$$

It is convenient to dualize the field strength to a vector

$$B_\mu={1\over2}\epsilon_{\mu\nu\rho}F_{\nu\rho}.$$

For a noncompact smooth gauge field, the Bianchi identity says

$$\partial_\mu B_\mu=0.$$

For a compact gauge field, this equation can fail at isolated points:

$$\partial_\mu B_\mu=2\pi\sum_a q_a\delta^{(3)}(x-x_a), \qquad q_a\in\mathbb Z.$$

These points are monopole instantons. They are not added by hand as external objects; compactness of the gauge group allows them as finite-action configurations once their cores are ultraviolet regulated. Because $e^2$ has dimensions of mass in three dimensions, the monopole core requires a scale $\Lambda_{\rm core}$, such as the inverse lattice spacing or the mass of a heavy field in a smooth completion. Its action has the form

$$S_0\sim c\,{\Lambda_{\rm core}\over e^2},$$

so their fugacity is exponentially small at weak coupling:

$$\zeta\sim \Lambda_{\rm core}^3 e^{-S_0}.$$

The power of $\Lambda_{\rm core}$ and the prefactor are ultraviolet dependent; the nonanalytic factor $e^{-S_0}$ is the semiclassical information that survives into the infrared.

Small is not the same as irrelevant. A dilute gas of rare monopoles can dominate the deepest infrared.

The monopole gas has the schematic grand-canonical partition function

$$Z_{\rm mon} =\sum_{N_+,N_-=0}^\infty {\zeta^{N_++N_-}\over N_+!N_-!} \int \prod_a d^3x_a\, \exp\left[-\sum_{a<b} q_aq_b\,V_m(x_a-x_b)\right],$$

where $q_a=\pm1$ for unit monopoles and $V_m(r)$ is the three-dimensional magnetic Coulomb potential,

$$V_m(r)\propto {1\over r}.$$

The neutrality condition is automatic in infinite volume or follows from the zero mode of the dual field.

From monopole plasma to dual photon mass

A Coulomb gas can be rewritten as a sine-Gordon theory. Introduce a dual scalar $\varphi$, often called the dual photon. With one common normalization,

$$B_\mu={e^2\over2\pi}\partial_\mu\varphi$$

in sectors without monopoles. The Maxwell action becomes

$${1\over2e^2}\int B_\mu B_\mu ={e^2\over8\pi^2}\int d^3x\,(\partial\varphi)^2.$$

A monopole insertion of charge $q$ is represented by $e^{iq\varphi(x)}$. Summing over unit monopoles and antimonopoles produces

$$S_{\rm dual}[\varphi] =\int d^3x\left[ {e^2\over8\pi^2}(\partial\varphi)^2 -2\zeta\cos\varphi \right].$$

Equivalently, after shifting the zero of the action,

$$S_{\rm dual}[\varphi] =\int d^3x\left[ {K\over2}(\partial\varphi)^2 +2\zeta(1-\cos\varphi) \right], \qquad K={e^2\over4\pi^2}.$$

Expanding around a minimum $\varphi=0$ gives

$$2\zeta(1-\cos\varphi)=\zeta\varphi^2+O(\varphi^4).$$

Thus the dual photon has a mass

$$m_D^2={2\zeta\over K} ={8\pi^2\zeta\over e^2} \propto e^{-S_0}.$$

The dilute monopole instanton gas is equivalent to a sine-Gordon theory for the dual photon $\varphi$. The monopole fugacity $\zeta$ generates a mass gap $m_D$.

This is the Debye-screening phenomenon in magnetic language. In an ordinary electric plasma, mobile charges screen electric fields and turn the Coulomb potential into a Yukawa potential. Here mobile monopole instantons screen magnetic fields in Euclidean spacetime. The dual photon mass is the inverse screening length.

A warning is worth making explicit. The mass gap by itself does not mean external electric charges are screened. The dual field is magnetic. Electric Wilson loops are more subtle: they insert a discontinuity in $\varphi$, and the sine-Gordon potential turns that discontinuity into a domain wall. That domain wall is the confining string.

Wilson loops as dual-photon domain walls

By Stokes’s theorem,

$$\oint_C A=\int_\Sigma F, \qquad \partial\Sigma=C.$$

In the dual formulation, inserting $e^{iq\oint_C A}$ can be represented as a singular background forcing the dual photon to jump across a surface $\Sigma$ bounded by $C$:

$$\Delta\varphi=2\pi q \qquad \text{across }\Sigma, \qquad q\in\mathbb Z.$$

In pure Maxwell theory, such a discontinuity can spread out. In the monopole plasma, $\varphi$ is pinned to one of the minima of $1-\cos\varphi$. A jump by $2\pi$ is a domain wall interpolating between adjacent equivalent vacua. The wall wants to minimize its area, so the Wilson loop becomes

$$W(C)\sim e^{-\sigma A_{\min}(C)}.$$

In the monopole plasma, a Wilson loop forces the dual photon to jump across a spanning surface. The sine-Gordon potential makes the jump a finite-tension domain wall, giving an area law.

Parametrically, the string tension is set by the stiffness of the dual photon times the inverse wall thickness. Since the wall thickness is $m_D^{-1}$, one finds

$$\sigma\sim K m_D \sim e^2 m_D \propto e^{-S_0/2}$$

at fixed $e^2$ and core scale, up to normalization-dependent powers and numerical factors. The essential point is the nonanalytic dependence $e^{-S_0/2}$: no finite order of perturbation theory around the photon vacuum can see this effect.

The monopole plasma therefore gives a complete mechanism for confinement in compact QED in $2+1$ Lorentzian dimensions, or equivalently three Euclidean dimensions:

$$\text{compactness}\quad\Longrightarrow\quad \text{monopole instantons}\quad\Longrightarrow\quad \text{dual-photon mass}\quad\Longrightarrow\quad \text{Wilson-loop area law}.$$

Two-dimensional Maxwell theory as a useful contrast

In one spatial dimension, electric flux has nowhere to spread. Gauss’s law for a charge $+q$ at $x=0$ and charge $-q$ at $x=R$ gives a constant electric field between them and zero field outside:

$$\partial_x E=e^2q\bigl[\delta(x)-\delta(x-R)\bigr].$$

The energy is

$$V(R)={1\over2e^2}\int dx\,E^2 ={e^2q^2\over2}R.$$

The Wilson loop obeys an area law,

$$W(R,T)\sim e^{-\sigma RT},$$

but this is not caused by a monopole plasma. It is a consequence of the absence of transverse directions. This example is a good antidote to a common slogan: “area law” means “linear potential for external charges,” but the physical mechanism behind that linear potential depends on dimension and field content.

Four-dimensional compact gauge theory

The role of monopoles changes with dimension. In three Euclidean dimensions, monopoles are pointlike instantons. In four Euclidean dimensions, monopoles are worldlines. A dilute gas of massive monopole particles need not destroy the Coulomb phase. Instead, compact $U(1)$ lattice gauge theory in four dimensions can have distinct phases: a Coulomb phase at weak coupling and a confining phase at strong coupling.

The confining phase is often described as a dual superconductor. Magnetic objects condense, the electric field is squeezed into flux tubes, and electric Wilson loops have an area law. The magnetic disorder operator, or ’t Hooft loop, gives the complementary diagnostic.

In three Euclidean dimensions monopoles are instantons. In four Euclidean dimensions they are worldlines. Their proliferation disorders electric Wilson loops and is naturally described as a dual-superconductor mechanism.

For non-Abelian Yang–Mills theory, confinement is harder. There is no weakly coupled dilute-monopole derivation in ordinary four-dimensional pure Yang–Mills analogous to compact QED in three dimensions. Still, Wilson loops remain the central gauge-invariant diagnostic, and the area-law language survives:

$$W_R(C)\sim e^{-\sigma_{k(R)} A(C)}$$

for large loops in a confining regime. Here $k(R)$ is the representation’s center charge, or $N$-ality for $SU(N)$. Even in pure Yang–Mills theory, gluons can screen a representation with trivial center charge, so the asymptotic string tension is controlled by $k(R)$ rather than by the full representation. Dynamical matter can screen additional center charges and cause string breaking.

Screening, string breaking, and what Wilson loops really diagnose

A Wilson loop with external charges in representation $R$ measures the energy of the flux configuration sourced by those charges. If the theory contains dynamical matter with the same charge or representation, the flux tube can break. Pair creation screens the external probes, and at sufficiently large $R$ the potential saturates rather than rising forever.

Therefore the cleanest confinement criterion uses pure gauge theory or probes carrying charges that cannot be screened by dynamical matter. In non-Abelian gauge theory, the center of the gauge group often controls this distinction. Representations with nontrivial center charge cannot be screened by adjoint matter, while representations with trivial center charge can be.

In compact Abelian theories, the same caution appears in simpler form. A Wilson loop area law for external unit charges is a statement about the response of the gauge field vacuum. Adding light unit-charged matter may change the asymptotic large-loop behavior to a perimeter law because the external sources are screened.

Common pitfalls

A few confusions are especially easy to make here.

First, compactness is not visible in perturbation theory. Expanding

$$1-\cos F={1\over2}F^2+O(F^4)$$

suggests Maxwell theory, but the compact theory has additional flux sectors and monopoles. The quadratic action sees only the tangent space of the gauge group.

Second, a mass gap does not automatically imply confinement. An ordinary Higgs phase has a massive gauge boson and screened forces, not a confining string between fundamental charges. In compact QED in three dimensions, the dual photon mass matters because Wilson loops force a dual-domain wall.

Third, the surface $\Sigma$ used in

$$\oint_C A=\int_\Sigma F$$

is not physical by itself. Different choices of $\Sigma$ differ by a closed surface. The compactness and charge quantization ensure that properly defined Wilson loops are independent of this arbitrary choice.

Fourth, “linear potential” is not always the same mechanism. In one spatial dimension, Gauss’s law gives a linear potential because there are no transverse directions. In compact QED in $2+1$ dimensions, the linear potential comes from monopole-induced disordering of electric flux.

Summary

Wilson loops turn the qualitative word “confinement” into a gauge-invariant test. For a rectangular loop,

$$V(R)=-\lim_{T\to\infty}{1\over T}\log W(R,T).$$

A perimeter law means the dominant cost is localized near the probe worldlines. A Coulomb law means flux spreads through space. An area law means the cost grows with the minimal surface spanning the loop, or equivalently with the length of the electric flux tube between static probes.

The strong-coupling lattice expansion gives an elementary area law: Wilson loops are boundaries of fluctuating plaquette surfaces. Compact QED in three Euclidean dimensions gives a more continuum, semiclassical mechanism: monopole instantons form a plasma, the plasma is equivalent to a sine-Gordon theory for the dual photon, and Wilson loops become domain walls of the dual photon.

The next page turns from gauge-theory confinement and monopoles toward strings, branes, sigma models, and the geometric language suggested by sums over paths and surfaces.

Exercises

Exercise 1: Rectangular Wilson loops and static potentials

Suppose a rectangular Wilson loop has the large-$T$ behavior

$$W(R,T)=A_0(R)e^{-T V_0(R)}+A_1(R)e^{-T V_1(R)}+\cdots,$$

where $V_0(R)<V_1(R)<\cdots$ and $A_0(R)\neq0$. Show that

$$V_0(R)=-\lim_{T\to\infty}{1\over T}\log W(R,T).$$

Solution

Factor out the lowest exponential:

$$W(R,T)=A_0(R)e^{-T V_0(R)} \left[1+{A_1(R)\over A_0(R)}e^{-T(V_1(R)-V_0(R))}+\cdots\right].$$

Taking the logarithm gives

$$\log W(R,T)=\log A_0(R)-T V_0(R) +\log\left[1+O(e^{-T(V_1-V_0)})\right].$$

Dividing by $-T$ and taking $T\to\infty$ removes the finite prefactor and the exponentially small corrections:

$$-\lim_{T\to\infty}{1\over T}\log W(R,T)=V_0(R).$$

Exercise 2: Linear potential in one spatial dimension

In one spatial dimension, place charges $+q$ at $x=0$ and $-q$ at $x=R$. Let the electrostatic energy be

$$E_{\rm field}={1\over2e^2}\int dx\,E^2(x),$$

and impose Gauss’s law

$$\partial_xE=e^2q\bigl[\delta(x)-\delta(x-R)\bigr].$$

Assuming $E=0$ outside the interval $[0,R]$, find $V(R)$.

Solution

Integrating Gauss’s law across $x=0$ gives a jump $E(0^+)-E(0^-)=e^2q$. Since $E=0$ for $x<0$, we get $E=e^2q$ for $0<x<R$. Across $x=R$, the charge $-q$ brings the field back to zero.

Therefore

$$V(R)=E_{\rm field}={1\over2e^2}\int_0^R dx\,e^4q^2 =\frac{e^2q^2}{2}R.$$

The force is constant. This is the one-dimensional Coulomb law.

Exercise 3: Strong-coupling area law for compact U(1)

Use

$$e^{\beta\cos F_p}=\sum_{n_p\in\mathbb Z}I_{n_p}(\beta)e^{in_pF_p}$$

and the link integral

$$\int_0^{2\pi}{dA\over2\pi}e^{imA}=\delta_{m,0}$$

to explain why the leading strong-coupling contribution to a Wilson loop is a plaquette surface bounded by the loop.

Solution

After expanding every plaquette factor, the path integral contains products of phases

$$\exp\left(i\sum_p n_pF_p+i\sum_{\ell\in C}A_\ell\right).$$

Since each plaquette angle $F_p$ is a signed sum of link angles, the exponent can be rewritten as

$$i\sum_\ell A_\ell\left(J_\ell+\sum_p \epsilon_{p\ell}n_p\right),$$

where $J_\ell$ is the integer current supported on the Wilson loop. Integrating over each link imposes

$$J_\ell+\sum_p\epsilon_{p\ell}n_p=0.$$

Thus the plaquette integers $n_p$ must form an integer-valued surface whose boundary is $C$. The cheapest nonzero choice has $n_p=1$ on a minimal surface and $n_p=0$ elsewhere. Its weight is

$$\left({I_1(\beta)\over I_0(\beta)}\right)^{N_{\min}(C)}.$$

For $\beta\ll1$, this is approximately $(\beta/2)^{N_{\min}}$. Since $A_{\min}^{\rm phys}=a^2N_{\min}$, this is the area law $e^{-\sigma A_{\min}^{\rm phys}}$ with $\sigma\simeq a^{-2}\log(2/\beta)$.

Exercise 4: Dual-photon mass from the monopole fugacity

Consider the dual sine-Gordon action

$$S=\int d^3x\left[{K\over2}(\partial\varphi)^2+2\zeta(1-\cos\varphi)\right].$$

Expand around $\varphi=0$ and find the dual-photon mass.

Solution

For small $\varphi$,

$$1-\cos\varphi={\varphi^2\over2}+O(\varphi^4).$$

Thus

$$2\zeta(1-\cos\varphi)=\zeta\varphi^2+O(\varphi^4) ={1\over2}(2\zeta)\varphi^2+O(\varphi^4).$$

The quadratic action is

$$S_2=\int d^3x\left[{K\over2}(\partial\varphi)^2+{1\over2}(2\zeta)\varphi^2\right].$$

Therefore

$$m_D^2={2\zeta\over K}.$$

With $K=e^2/(4\pi^2)$, this gives

$$m_D^2={8\pi^2\zeta\over e^2}.$$

Exercise 5: Why a dual-domain wall gives an area law

Assume that a Wilson loop forces the dual photon to jump by $2\pi$ across a surface $\Sigma$ bounded by $C$. Suppose the sine-Gordon kink tension per unit area is $\sigma$. Show that the leading semiclassical behavior is

$$W(C)\sim e^{-\sigma A_{\min}(C)}.$$

Solution

The Wilson loop imposes a boundary condition, not a local perturbative source. The dual field must interpolate between adjacent minima of the sine-Gordon potential across some surface $\Sigma$ with boundary $C$.

For a large smooth loop, the wall thickness is microscopic compared with the loop size. The action of a wall configuration is approximately tension times area:

$$S_{\rm wall}[\Sigma]\simeq \sigma A(\Sigma).$$

The path integral is dominated by the least-action wall, hence by the minimal-area surface:

$$S_{\rm saddle}\simeq \sigma A_{\min}(C).$$

Therefore

$$W(C)\sim e^{-S_{\rm saddle}} =e^{-\sigma A_{\min}(C)}.$$

For a rectangle, $A_{\min}=RT$, so $W(R,T)\sim e^{-\sigma RT}$ and $V(R)=\sigma R$.

Further reading

• J. B. Kogut, “An introduction to lattice gauge theory and spin systems,” Reviews of Modern Physics 51 (1979), 659–713.
• J. B. Kogut, “The lattice gauge theory approach to quantum chromodynamics,” Reviews of Modern Physics 55 (1983), 775–836.
• A. M. Polyakov, Gauge Fields and Strings, Contemporary Concepts in Physics, vol. 3 (Harwood Academic Publishers, 1987), Chapters 3–5 and 7.
• M. D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, 2014), chapters on gauge invariance, Wilson lines, lattice gauge theory, and effective actions.
• M. Srednicki, Quantum Field Theory (Cambridge University Press, 2007), chapters on gauge theory, solitons, monopoles, and instantons.
• A. Zee, Quantum Field Theory in a Nutshell, 2nd ed. (Princeton University Press, 2010), chapters on vortices, monopoles, instantons, duality, Yang–Mills theory, and lattice gauge theory.