1 Introduction

Chiba Univ. Preprint CHIBA-EP-171

May 2008

Magnetic monopoles and center vortices

as gauge-invariant topological defects

simultaneously responsible for confinement

Kei-Ichi Kondo,

Department of Physics, Graduate School of Science,

Chiba University, Chiba 263-8522, Japan

We give a gauge-invariant definition of the vortex surface in Yang-Mills theory without using the gauge fixing procedure. In this construction, gauge-invariant magnetic monopoles with fractional magnetic charges emerge in the boundary of the non-oriented vortex surface such that the asymptotic string tension reproduces the correct -ality dependence. We show that gauge-invariant magnetic monopoles and vortices are simultaneously responsible for quark confinement in four dimensional spacetime based on the Wilson criterion. These results are extracted from a non-Abelian Stokes theorem derived in the previous paper.

Key words: magnetic monopole, vortex, Wilson loop, non-Abelian Stokes theorem, quark confinement, N-ality, Yang-Mills theory,

PACS: 12.38.Aw, 12.38.Lg   E-mail: kondok@faculty.chiba-u.jp

## 1 Introduction

In understanding the non-perturbative phenomena in the infrared sector of Yang-Mills theory [1] and QCD such as quark confinement [2], chiral symmetry breaking and problem, some of topological configurations are believed to play the key role as the dominant dynamical degrees of freedom. Examples are magnetic monopoles, center vortices, instantons, merons, etc. Among them, chiral symmetry breaking and the problem can be explained e.g., by the Yang-Mills instantons [3, 4], although magnetic monopoles are not excluded as their mechanisms, see e.g. [5].

Quark confinement is believed to be explained by the condensation of Abelian magnetic monopoles [6] and/or center vortices [7, 8], since they realize the dual superconductor picture of QCD vacuum as the most promising scenario for confinement. Magnetic monopoles are topological objects of codimension 3: points for and closed loops for in -dimensional spacetime. On the other hand, center vortices are topological objects of codimension 2: closed loops for and closed surfaces for [9, 10, 11]. For example, the vortex in is in the first step identified with a closed thin tube of magnetic flux which can be thought of as the magnetic field generated by a toroidal solenoid in the limit of vanishing cross section, although the vortex must have a finite transverse extension to correctly reproduce the adjoint string tension and yield the finite action of vortices [12]. Recent lattice simulations exhibit both Abelian magnetic monopole dominance [13] and center vortex dominance [9] for the string tension. Moreover, if either Abelian magnetic monopoles or center vortices in Yang-Mills theory are removed from the ensemble of configurations, confinement is found to be lost. Moreover, chiral symmetry breaking is also lost. It is known according to numerical simulations that magnetic monopoles and vortices are strongly correlated. Incidentally, merons [16] can be also a candidate for confiners [4] and have something to do with magnetic monopoles due to numerical simulations on a lattice [17]. For recent reviews, see e.g., [14] for Abelian magnetic monopole and [15] for center vortices.

Yang-Mills instantons and merons are respectively self-dual Euclidean and non-self-dual Euclidean/Minkowski solutions of the gauge covariant Yang-Mills field equations derived from the Yang-Mills action. However, Abelian magnetic monopoles [18] or center vortices [7] in Yang-Mills theory have been obtained as gauge fixing defects by a partial gauge fixing where the gauge degrees of freedom is used to transform the gauge field variable (link variable on a lattice) as close as possible to a subgroup which is left unbroken: the maximal torus (Cartan) subgroup for the maximal Abelian gauge [18, 19] or the center subgroup for the maximal center gauge. Therefore, the current method of constructing Abelian magnetic monopoles and center vortices could not escape the charge of the gauge artifact.

In a series of recent papers, we have succeeded to give a gauge-invariant description of the dual superconductivity in Yang-Mills theory in the continuum [23, 24] and on a lattice [25, 26, 27, 28, 29] by developing the approach founded in [20, 21, 22]. Especially, the Wilson loop operator is expressed exactly in terms of a gauge-invariant magnetic current [33] and the magnetic monopole can be defined in a gauge-invariant way according to a non-Abelian Stokes theorem for the Wilson loop operator [30, 31, 32, 33]. These results enable us to clarify the role of magnetic monopole in confinement.

In this paper, we give a gauge-invariant definition of a vortex which can play the same role as the center vortex to sweep away distrust of gauge artifact. Indeed, this is achieved without relying on the (partial) gauge fixing such as the maximal Abelian gauge and the maximal center gauge. Consequently, the gauge-invariant magnetic monopoles emerge in the boundary of the closed vortex surface in . We show that both magnetic monopoles and vortices are necessary from a viewpoint of quark confinement based on the Wilson criterion such that the asymptotic string tension reproduces the correct -ality dependence. The result of this paper shows that the vortex and the magnetic monopole can be the alternative view of the one and the same fundamental dynamics in Yang-Mills theory defined in a gauge-invariant manner.

## 2 Wilson loop operator and magnetic monopoles

For the Yang-Mills connection (one-form) for a gauge group , the Wilson loop operator along a closed loop is defined by

 WC[A]:=tr[Pexp{ig∮CA}]/tr(1), (2.1)

where denotes the path-ordering prescription. The non-Abelian Stokes theorem (NAST) enables us to rewrite the Wilson loop operator into the surface integral form over the surface bounding (). A version of the non-Abelian Stokes theorem without any path or surface ordering is known as the Diakonov-Petrov version [30] of a non-Abelian Stokes theorem. The Diakonov-Petrov version of NAST was originally derived in [30] for case and later developed and extended to case in [31, 32]. Moreover, it has been shown [33] that the Wilson loop operator is rewritten in terms of two gauge-invariant conserved currents, the “magnetic-monopole current” and the “electric current” , defined by applying the exterior derivative , the coderivative (adjoint derivative) and Hodge star operation to :

 k:=δ∗f=∗df,j:=δf, (2.2)

where is the gauge-invariant two-form defined from the gauge connection by

 fμν(x)= ∂μ2tr(n(x)Aν(x))−∂ν2tr(n(x)Aμ(x)) +2tr(2(N−1)Nig−1n(x)[∂μn(x),∂νn(x)]), (2.3)

using the Lie-algebra valued field called the (normalized) color field defined by

 n(x)=nA(x)TA:=√2NN−1ξ(x)Hξ†(x),ξ(x)∈G/~H, (2.4)

with the stability group specified later and given by

 H:=Λ⋅H=r∑j=1ΛjHj, (2.5)

where () are the generators from the Cartan subalgebra of ( is the rank of the gauge group ) and -dimensional vector () is the highest weight of the representation in which the Wilson loop is considered. Indeed, both currents are conserved in the sense that and .

For the Wilson loop operator in the fundamental representation of , [33] and the Wilson loop operator is rewritten as

 WC[A]= ∫[dμ(ξ)]Σexp[i√N−12Ng∫Σf] = ∫[dμ(ξ)]Σexp{i√N−12Ng(k,ΞΣ)+i√N−12Ng(j,NΣ)}, (2.6)

where is the product of the Haar measure on over :

 [dμ(ξ)]Σ:=∏x∈Σdμ(ξ(x)), (2.7)

and is the -form and is the one-form defined by

 ΞΣ:=∗dΘΣΔ−1=δ∗ΘΣΔ−1,NΣ:=δΘΣΔ−1, (2.8)

with the -dimensional Laplacian (or the d’Alembertian in the Minkowski spacetime) and the two-form called the vorticity tensor as an antisymmetric tensor of rank two:

 ΘμνΣ(x):=∫ΣdSμν(X(σ))δD(x−X(σ)). (2.9)

Note that and are -forms, while and are one-forms for any in dimensional spacetime:

 (k,ΞΣ):=1(D−3)!∫dDxkμ1⋯μD−3(x)Ξμ1⋯μD−3Σ(x),(j,NΣ):=∫dDxjμ(x)NμΣ(x). (2.10)

For , in particular, arbitrary representation is characterized by an integer or a half-integer . The Wilson loop operator in the representation of obey the non-Abelian Stokes theorem:

 WC[A]=∫[dμ(ξ)]Σexp{iJg(k,ΞΣ)+iJg(j,NΣ)}. (2.11)

This agrees with (2.6) for a fundamental representation of SU(2).

We focus on the magnetic contribution defined by

 WmC=exp{i√N−12Ng(k,ΞΣ)}. (2.12)

For , it has been shown [33] that the magnetic charge defined by for obeys the quantization condition:

 qm=4πg−1n,n∈Z={⋯,−2,−1,0,+1,+2,⋯}. (2.13)

This follows from the condition that the non-Abelian Stokes theorem should not depend on the surface chosen for bounding the loop , since the original Wilson loop is defined for the specified closed loop . For an ensemble of point-like magnetic charges located at ()

 k0(x)=n∑a=1qamδ(3)(x−za),qam=4πg−1na,na∈Z, (2.14)

we have a geometric representation:

 WmC=exp{i12g4πn∑a=1qamΩΣ(za)}=exp{i12n∑a=1naΩΣ(za)},na∈Z, (2.15)

where is the the solid angle under which the surface shows up to an observer at the point . Therefore, a magnetic monopole with a unit magnetic charge in the neighborhood of the surface gives a non-trivial factor to the Wilson loop operator , since when is just below and above the surface . This is a nice feature of magnetic monopole for explaining quark confinement based on the Wilson loop.

For , however, we fall in a trouble about the magnetic monopole just defined. For , an ensemble of magnetic currents on closed loops ():

 WmC=exp{i12gn∑a=1qamL(C′a,Σ)}=exp{2πin∑a=1naL(C′a,Σ)},na∈Z, (2.17)

where is the linking number between the curve and the surface :

 L(C′,Σ)=L(Σ,C′):=∮C′dyμ(τ)ΞμΣ(y(τ)), (2.18)

where the curve is identified with the trajectory of a magnetic monopole and the surface with the world sheet of a hadron (meson) string representing a quark-antiquark pair. For case, see Fig. 1. However, such magnetic loops carrying the magnetic charge obeying the quantization condition (2.13) do not give non-trivial contributions to the Wilson loop, since and are integers. If the quantization condition (2.13) is true, the magnetic monopole can not be the topological defects responsible for quark confinement. In the following, we discuss how this dilemma is resolved.

## 3 Magnetic monopole and vortex

In the following, we extensively use the techniques developed by Engelhardt and Reinhardt [11] in constructing a continuum analogue of the maximal center gauge and center projection, but from a different angle in this paper.

We first consider the case. Suppose that the magnetic current has the support on the closed loop in dimensions:

 kμ(x)=kμ(x;C′):=Φ∮C′dyμδ4(x−y), (3.1)

where is a real number representing the magnetic flux carried by the magnetic charge to be discussed later in detail. The magnetic charge is defined by

 qm=∫d3~σμkμ, (3.2)

where denotes a parameterization of the 3-dimensional volume and is the dual of the 3-dimensional volume element :

 d3~σμ:=13!ϵμγ1γ2γ3d3σγ1γ2γ3,d3σγ1γ2γ3:=ϵβ1β2β3∂¯xγ1∂σβ1∂¯xγ1∂σβ1∂¯xγ1∂σβ1dσ1dσ2dσ3. (3.3)

First of all, we look for the field strength with the support , a two-dimensional surface bounding the closed loop , , so that (2.2) holds:

 ∂ν∗fμν(x;S)=kμ(x;C′). (3.4)

Such a solution is given by

 ∗fμν(x;S)=Φ∫S:∂S=C′d2σμνδ4(x−¯x(σ)). (3.5)

In fact, it satisfies the equation:

 ∂ν∗fμν(x;S)= Φ∫Sd2σμν∂xνδ4(x−¯x(σ)) = Φ∫Sd2σμν∂¯xνδ4(x−¯x(σ)) = Φ∮∂S=C′d¯xμδ4(x−¯x(σ)), (3.6)

where we have used the Stokes theorem in the last step.

Next, we proceed to obtain the gauge potential giving the field strength just obtained, i.e.,

 fμν(x;S)= ∂μbν(x;V)−∂νbμ(x;V), (3.7)

such that has the support only on the open set , the three-dimensional volume 111 The precise position of the open set is irrelevant for the value of the Wilson loop as shown below. In fact, the open set can be deformed arbitrarily by singular gauge transformations in such a way that its boundary representing the position of the magnetic flux of the vortex is fixed. whose boundary is : . Note that is cast into

 ∗fμν(x;S)= Φ∫V:∂V=Sd3σμνκ∂¯xκδ4(x−¯x(σ)) = Φϵμναβ13!ϵβγ1γ2γ3∫V:∂V=Sd3σγ1γ2γ3∂xαδ4(x−¯x(σ)) = ϵμναβ∂xα[Φ13!ϵβγ1γ2γ3∫V:∂V=Sd3σγ1γ2γ3δ4(x−¯x(σ))] = 12ϵμναβ{∂xα[Φ∫V:∂V=Sd3~σβδ4(x−¯x(σ))]−(α↔β)}, (3.8)

where we have used the Gauss (Stokes) theorem in the first equality. Therefore, the gauge potential is determined up to a gauge transformation:

 bμ(x;V)=Φ∫V:∂V=Sd3~σμδ4(x−¯x(σ)). (3.9)

This corresponds to an explicit (singular) gauge field representation of an ideal vortex configuration in 4 space-time dimensions [11].

In dimensions, the magnetic current is -form with the support on the -dimensional subspace in -dimensional spacetime:

 k(x)=k(x;C′D−3):=Φ∮C′D−3dD−3¯x(σ)δD(x−¯x(σ)). (3.10)

Then it is not difficult to show that an ideal vortex configuration should read

 bμ(x;V)= Φ∫VD−1:∂VD−1=SD−2dD−1~σμ δD(x−¯x(σ)). (3.11)

The gauge field for the vortex is not unique as mentioned above. Actually, the ideal vortex field can be gauge transformed to a thin vortex field which has the support only on the boundary of :

 bμ(x;V)=aμ(x;S)+iU(x;V)∂μU†(x;V), (3.12)

so that carries the same flux located on as that carried by . We adopt the gauge transformation:

 U(x;V)=exp[iΦΩV(x)], (3.13)

where is the solid angle taken up by the volume viewed from :

with being the area of unit sphere in dimensions. The solid angle defined in this way is normalized to unity for a point inside the volume . Note that the solid angle is defined with a sign depending on the orientation of as rays emanating from pierce . A deformation of keeping its boundary fixed leaves the solid angle invariant unless crosses . When crosses , the solid angle changes by an integer. 222 Whether the flux of a vortex is electric or magnetic depends on the position of the dimensional vortex surface in -dimensional spacetime. For example, in the vortex defined by the boundary of a purely spatial 3-dimensional volume carries only electric flux, which is directed normal to the vortex surface . On the other hand, a vortex defined by a volume evolving in time represents at a fixed time a closed loop and carries the magnetic flux, which is tangential to the vortex loop [11].

Now we show that the thin vortex field is obtained in the form:

 aμ(x;S)=Φ∫SD−2=∂VD−1dD−2~σμλ ∂λD(x−¯x(σ)), (3.15)

where is the Green function of the -dimensional Laplacian defined by

 −∂μ∂μD(x−¯x(σ))=δD(x−¯x(σ)), (3.16)

with the explicit form:

 D(x−y)=Γ(D/2−1)4πD/2[(x−y)2](D−2)/2. (3.17)

The ideal vortex field has support only on and hence it vanishes outside the vortex , i.e., for . Therefore, outside the vortex , i.e., , the thin vortex field is the pure gauge due to (3.12):

 aμ(x;S)=−iU(x;V)∂μU†(x;V),x∉V, (3.18)

which implies the vanishing field strength outside the vortex . This is reasonable, because the magnetic flux is contained in the vortex sheet . The magnetic field computed from the curl of can be localized on .

The derivation of (3.12) and (3.15) is as follows. By using the fact that the solid angle is rewritten as

 ΩV(x):=∫VD−1dD−1~σμ∂xμD(x−¯x(σ)), (3.19)

we obtain

 iU(x;V)∂μU†(x;V)=Φ∂μΩV(x)=Φ∫VD−1dD−1~σν∂μ∂νD(x−¯x(σ)). (3.20)

We have the decomposition:

 ∂μ∂ν=δμν∂2−(δμν∂2−∂μ∂ν)=δμν∂2−12ϵμραβϵνσαβ∂ρ∂σ, (3.21)

which is used to rewrite the pure gauge form into

 iU(x;V)∂μU†(x;V) = Φ∫VD−1dD−1~σμ∂2D(x−¯x(σ))−Φ∫VD−1dD−1~σν12ϵμραβϵνσαβ∂ρ∂σD(x−¯x(σ)) = −Φ∫VD−1dD−1~σμδD(x−¯x(σ))−Φ∫∂VD−1dD−2~σμλ∂λD(x−¯x(σ)), (3.22)

where we have used the definition of the Green function and the Stokes theorem in the last equality. In other words, the thin vortex field is the transverse part of the ideal vortex .

Finally, we find that the surface integral of over bounded by the Wilson loop is equivalent to the line integral of along the closed loop :

 ∫Σf(x;S)= ∫Σdb(x;V) = ∮∂Σ=Cb(x;V) = ∮C[a(x;S)+iU(x;V)dU†(x;V)] = ∮Ca(x;S), (3.23)

since the contribution from the last term vanishes for any closed loop . Then the line integral is cast into

 ∮Cdxμaμ(x;S)= ∫Σd2σνμ(x)∂νaμ(x;S) = Φ∫Σd2σνμ(x)∫∂VD−1dD−2~σμλ∂ν∂λD(x−¯x(σ)) = Φ−14∫ΣdD−2~σρσ(x)ϵμνρσ∫∂VD−1d2σαβ(¯x)ϵμλαβ∂ν∂λD(x−¯x) = Φ−12∫ΣdD−2~σαβ(x)∫∂VD−1d2σαβ(¯x)∂2D(x−¯x) +Φ∫ΣdD−2~σβσ(x)∫∂VD−1d2σαβ(¯x)∂α∂σD(x−¯x) = Φ12∫ΣdD−2~σαβ(x)∫∂VD−1d2σαβ(¯x)δD(x−¯x), (3.24)

where we have used the fact that the second term vanishes due to the Stokes theorem for obtaining the last result. Therefore, the line integral is rewritten as

 ∮Cdxμaμ(x;S)=ΦI(Σ,SD−2=∂VD−1), (3.25)

in terms of the intersection number between the world sheet of the hadron string and the vortex sheet . It is known that the intersection number is equal to the linking number between the Wilson loop and the vortex sheet , see Fig. 2:

 I(Σ,SD−2=∂VD−1)= L(C=∂Σ,SD−2=∂VD−1) = 12∫ΣdD−2~σαβ(x)∫Sd2σαβ(¯x)δD(x−¯x). (3.26)

Thus the thin vortex contributes to the Wilson loop in the fundamental representation:

 exp[i12g∫Σf(x;S)]=exp[i12gΦL(C,S=∂V)]=zL(C,S=∂V),z:=ei12gΦ. (3.27)

In general, is a complex number of modulus one, i.e., an element of U(1). If the magnetic charge obeys the Dirac quantization condition:

 Φ=2πg−1n (n∈Z), (3.28)

then reduces to the center element of SU(2):

 z=eiπn=±1,z1∈Z2. (3.29)

The quantization condition (3.28) will be called the fractional quantization condition which happens to agree with the Dirac one for SU(2), which is realized as a special case of the general quantization condition for discussed later. If the magnetic charge obeyed the quantization condition

 Φ=4πg−1n (n∈Z), (3.30)

then would be trivial, i.e., . Such a vortex can not give a non-trivial contribution to the Wilson loop. Therefore, the thin vortex carrying the fractional magnetic charge yields a center element under the fractional magnetic charge quantization (3.28). For , is a three-dimensional volume and its boundary is a closed two-dimensional surface . If is an oriented closed surface, then its boundary is empty, and hence the magnetic current does not exist in this case, since its support vanishes. Therefore, for the non-vanishing magnetic current to exist in the boundary of , the vortex surface must be non-oriented. Is there any relationship between the non-orientedness of the vortex surface and the fractional magnetic charge (3.28)? We will try to answer this question in the next section.

## 4 Non-orientedness of the vortex surface and fractional magnetic charge quantization

The above considerations suggest that in the continuum theory a smooth vortex surface consists of surface patches of different orientations , that is to say, the vortex surface is not globally oriented and that the magnetic monopole loops emerge at the boundaries of vortex surface patches where the magnetic flux direction on is defined by the orientation of the patch (and vice versa). Then we can understand only a fraction of the elementary magnetic charge dictated by the Dirac quantization condition is carried by the magnetic flux on an isolated patch of the vortex sheet . Hence the magnetic current of proper magnetic charges satisfying the quantization condition (3.30) is reproduced only when the open vortex surface patches are glued together to obtain a non-oriented closed vortex surface by combining their magnetic loops together. See Fig. 3. In fact, if two patches are glued together such that the surface orientation does not change across the boundary (the surface is globally oriented), the magnetic current at the boundary precisely cancel. For D=4, thus, the thin vortex defined on a non-oriented closed surface if linked to the Wilson loop , gives a non-trivial contribution to the Wilson loop average. See Fig. 2

Engelhardt and Reinhardt [11] have shown that the continuum center vortex configurations generate the Pontryagin index as self-intersections of the vortex network. It is well known in topology that the self-intersection number of closed, globally oriented two-dimensional surface in vanishes. This implies that the Pontryagin index vanishes for globally oriented vortex surfaces. Conversely, non-orientedness of the surfaces is crucial for generating a non-vanishing topological winding number. Therefore, the global non-orientedness of the vortex surfaces is necessary to generate a non-vanishing Pontryagin index originating from the vortex configuration. Moreover, it is well known in topology that the number of intersection points of two closed two-dimensional surfaces in is even. This implies that the number of self-intersection points of a closed surface is even, because the self-intersection number is defined by simply intersecting the surface with another surface infinitesimally displaced from it, i.e. a framing of the surface [11]. Thus, even if each self-intersection point of an center vortex surface configuration gives a contribution to the Pontryagin index, the number of such contributions is even and the Pontryagin index is integer-valued. Moreover, they have pointed out that a non-zero Pontryagin index requires the existence of magnetic current. In the lattice study [10], in fact, the orientability of the vortex surfaces was studied, with result that these surfaces are non-orientable and have non-trivial genus in the confinement phase. These considerations connect the gauge-invariant magnetic monopole and the meron by way of a vortex and they become topological objects simultaneously responsible for confinement.

## 5 N-ality dependence of asymptotic string tension

Now we show that the fractional charge quantization is crucial to understand the -ality dependence of asymptotic string tension. The asymptotic string tension depends only on the -ality of the quark charge (or the group representation of the Wilson loop). 333 The -ality of a given representation is obtained by the number of boxes in the corresponding Young diagram, mod . If is the matrix representation of a group element in a representation of -ality , and is an element in the center, then . This is because particles of zero -ality can never bind to a particle of non-zero -ality to form a color singlet and hence they can never break the string connecting two non-zero -ality sources.

Gluons belong to the adjoint representation of the gauge group and they have zero -ality, so that all center elements are mapped to the identity. Therefore, gluons can not break the string formed between a pair of quark and antiquark in the and representations of . Only other quarks, or other particles in the non-zero -ality representations, can do that. The Wilson criterion for quark confinement should be understood to imply the linear potential rising indefinitely in the limit in which the masses of any matter particles of non-zero -ality are taken to infinity.

For example, in gauge theory, the center group is and the representations can be divided into two classes: (half-integer) with -ality one, and (integer) with -ality zero. The asymptotic string tension of -ality=0 () must be zero, while the non-vanishing asymptotic string tension of all -ality=1 () must be the same:

 σJ={σ1/2(J=12,32,⋯)0(J=1,2,⋯). (5.1)

This expected result is easily derived from the considerations given in this paper: The vortex contribution to the Wilson loop operator in the representation reads

 exp[igJ∫Σf(x;S)]=exp[iJgΦL(C,S=∂V)]=zL(C,S=∂V), (5.2)

where

 z:=eiJgΦ=e2πJin={±1(J=12,32,⋯)1(J=1,2,⋯), (5.3)

provided that the magnetic flux or the magnetic charge obeys the same fractional quantization condition (3.28). For -ality=0, is trivial and the string tension is zero. For -ality=1, has the non-trivial value when the vortex sheet intersects the world sheet and the string tension is non-zero. Thus the above argument reproduces correctly the -ality dependence of the asymptotic string tension. The case will be discussed later.

## 6 Consistency with lattice magnetic charge

We consider how the above argument is consistent with the results of numerical simulations on a lattice, e.g., the magnetic monopole dominance in the string tension [26]. For this purpose, it is instructive to recall the definition of the magnetic charge and the magnetic current on a lattice for U(1) gauge theory [34]. The basic idea is: if a magnetic monopole is located inside an elementary spatial cube on the lattice, then the enclosed magnetic charge can be determined by measuring the total magnetic flux through the surface (a set of plaquettes) of this cube. For instance, the magnetic flux through the surface of a plaquette with an area lying in the -plane is related to the phase of the plaquette variable with being the component of the magnetic field in the direction perpendicular to the -plane.

The plaquette variable is given by the product of the four oriented link variables around the boundary of a plaquette : where . Hence, the phase (angle) associated with a given plaquette variable must satisfy

 −4π

However, it should be remarked that the plaquette variable is a periodic function of with a period . Therefore, the physical flux should be determined from , modulo , since the plaquette value remains unchanged by shifting by a multiple of . According to DeGrand and Toussaint [35], we decompose this angle into two parts:

 gΦP=2πnP+g¯ΦP,nP=0,±1,±2,−π

which covers the whole range of (6.1). If we add up the plaquette angles of the six plaquettes bounding an elementary cube , we will obtain a vanishing result, since each link is common to two plaquettes which give rise to the sum of two phases of the common link variable, equal in magnitude but of opposite sign, i.e., . Therefore, the magnetic charge as the magnetic flux through the closed surface bounding the elementary cube is given by

 Qm=∑P∈S=∂c¯ΦP=−∑P∈S=∂c2πgnP. (6.3)

Thus the magnetic charge is a multiple of . If a magnetic monopole is located in an elementary cube, then at least one of the plaquette angle must be larger in magnitude than , so that there is a Dirac string (line) crossing the corresponding surface. The Dirac string can be moved around by making a large gauge transformation such that a particular link variable is mapped out of the principal value . But the net number of such strings leaving the elementary volume will not be affected by the gauge transformation. Moreover, the number of magnetic monopoles contained in a volume is given by the sum of the magnetic monopole numbers of the elementary cubes making up the volume .

Next, we consider a lattice expression for the components of the magnetic current defined by . For example, the first component of the current is written in the form:

 k1=∂2F34+∂3F42+∂4F23=(∂2,∂3,∂4)⋅(F34,F42,F23). (6.4)

Therefore, integrating over the volume of an elementary cube with edges along the 2,3 and 4 directions is equivalent to computing the flux of through the surface of this cube due to the Gauss theorem: . In order to compute the three components of , we need to calculate the flux through the plaquettes of three cubes having one link directed along the 4-axis. This computation is carried out in a completely analogous way as described in the above for the magnetic flux. Hence, by construction, each component of the magnetic current on the lattice will be multiples of when measured in lattice units:

 kLμ(x)=2πg12ϵμνρσ∂νnρσ(x)=2πg∂ν∗nμν(x),nρσ(x)=0,±1,±2, (6.5)

where is the (forward) lattice derivative . The magnetic current satisfies the topological conservation law and constitutes the closed loop in the dual lattice where is the backward lattice derivative defined by .

Thus, it happens that the definition of the magnetic monopole due to DeGrand and Toussaint for the U(1) gauge theory is consistent with the observation made for the vortex in the continuum formulation for SU(2). The gauge-invariant magnetic charge defined on a lattice [26] reduces to the above one due DeGrand and Toussaint by taking a special gauge. Hence the result of [26] is consistent with the observation made in this paper.

## 7 Su(3) case

In the case of gauge group, it has been shown [33] that the magnetic charge measured by the Wilson loop is subject to the quantization condition:

 qm=2πg√2NN−1