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Magnetic Z(N) symmetry in hot QCD and the spatial Wilson loop

We discuss the relation between the deconfining phase transition in gauge theories and the realization of the magnetic Z(N) symmetry. At low temperature the Z(N) symmetry is spontaneously broken while above the phase transition it is restored. This is intimately related to the change of behaviour of the spatial 't Hooft loop discussed in hep-ph/9909516. We also point out that the realization of the magnetic symmetry has bearing on the behaviour of the spatial Wilson loop. We give a physical argument to the effect that at zero temperature the spatial Wilson loop must have perimeter law behaviour in the symmetric phase but area law behaviour in the spontaneously broken phase. At high temperature the argument does not hold and the restoration of magnetic Z(N) is consistent with area law for the Wilson loop.


Introduction
This paper is devoted to further study of theoretical aspects of the deconfining temperature phase transition in nonabelian gauge theories. It is an immediate continuation of our earlier work [1]. In [1] we showed that the deconfining phase transition in the pure Yang Mills theory is characterised by the change of behaviour of the 't Hooft loop operator V (C). In the "cold" phase the 't Hooft loop has a perimeter law behaviour < V (C) >∝ exp{−aP (C)}, while in the "hot" phase it has an area law behaviour < V (C) >∝ exp{−αS(C)}.
In the present paper we want to sharpen somewhat this observation and further discuss related questions. We wish to point out that V is in fact an order parameter which probes the breaking of a physical symmetry of the Yang Mills theory. The symmetry in question is the magnetic Z N symmetry discussed by 't Hooft [2]. The deconfining phase transition is therefore characterized by the change in the mode of realization of a global Z N symmetry: the symmetry is broken spontaneously in the "cold" phase while it is restored in the "hot" phase.
The previous two paragraphs may sound at first like a red herring. After all an order parameter for the deconfining phase transition as well as a related Z N symmetry have been discussed for many years. The order parameter in question is the free energy of an external static colour source in the fundamental representation: the Polyakov line P = TrP exp{ig β 0 dtA 0 }. The Z N symmetry is the transformation P → exp{i 2π N }P . We will refer to this transformation as the electric Z N . There is however a great difference between the physical nature of P and V and the associated Z N symmetries. The operator V is a canonical operator in the physical Hilbert space of the Yang Mills theory. The magnetic Z N symmetry similarly is a transformation that acts on quantum states in the physical Hilbert space. On the other hand P has a very different status. It is not an operator in the Hilbert space and as such not a canonical order parameter. It appears as an auxilliary object when projecting onto gauge invariant physical subspace of the Hilbert space. The "electric" Z N -the operation that transforms P by multiplying it by a phase -similarly is not a canonical symmetry. These issues were discussed in detail in [3]. There is no transformation of states in the physical Hilbert space that is related to this "symmetry", although it is indeed a symmetry of the Euclidean path integral representing the statistical sum. This is not to say of course that P and electric Z N are useless concepts. The standard effective action, defined by the constrained path integral exp −S ef f (P ) = DA 0 δ(P − P (A 0 )) exp −S(A) is gauge invariant. It is instrumental in computing the vortex expectation value. The way the electric Z(N ) symmetry is realized in S ef f is also related to the behaviour of the order parameter of the magnetic Z(N ). We will discuss this in detail in section 3 [1].
However if one wants to describe the deconfinement phase transition in terms of a canonical order parameter in the same way as the Ising transition is described in terms of magnetisation, one should zero in on V rather than on P and should study the magnetic Z N symmetry rather than electric Z N . This is what we intend to do in this paper.
The action of the magnetic Z N symmetry is very different in 2+1 and 3+1 dimensional cases. In 2+1 dimensions it acts very much like usual global symmetry in a scalar theory with the order parameter being a scalar vortex field. In 3+1 dimensions the symmetry acts not like a standard global symetry -its "charge" is an integral over a two dimensional spacelike surface rather than over the whole of the three dimensional space 1 . As a consequence its order parameter is not a local field but rather a magnetic vortex stretching over macroscopic distances.
It is therefore convenient to start the discussion with the three dimensional gauge theories and to present all the arguments in this case. The generalization of appropriate aspects of this discussion to 3+1 dimensions will be given in the last part of every section.
The plan of this paper is the following. In Section 2 we recap the definition of the 't Hooft loop and its 2+1 dimensional analog -the magnetic vortex operator. We formulate the arguments for the existence of the magnetic Z N symmetry in theories without fundamental matter fields. We also show by explicit construction that the generator of this symmetry in the pure gluodynamics is none other than the spatial Wilson loop.
In Section 3 we discuss the relation between the behaviour of the 't Hooft loop and the realization of the magnetic Z N in the ground state of the theory. We demonstrate that the mode of the realization of the symmetry changes at the deconfining phase transition, while spontaneously broken at low temperature the symmetry is restored above the phase transition.
In Section 4 we present in a toy model a simple physical picture explaining how the behaviour of spatial Wilson loop discriminates at zero temperature between the phases with broken and unbroken magnetic Z N . In the phase where the Z N symmetry is broken, W must have an area law while in the case of unbroken Z N it must have perimeter law. We explain why this argument does not generalize to the high temperature phase and thus why the area law behaviour of the Wilson loop in the hot phase is consistent with restoration of the magnetic Z N symmetry.
Finally in Section 5 we conclude with a short discussion. 2 The magnetic Z N symmetry and the 't Hooft loop operator.
In this section we discuss the notion of the magnetic Z N symmetry and its order parameter -'t Hooft loop, or magnetic vortex operator. Most of the material contained here is not new and, perhaps with the exception of explicit identification of the Z N generator with the spatial Wilson loop, is contained in [2,5,6]. At the risk of being repetitive we have decided nevertheless to include this extended introductory part, since we feel that the concept of magnetic Z N symmetry is not widely appreciated in the community. The Z N symmetry structure is the basis for our discussion of the deconfining phase transition in the following sections.
Let us start by recalling the argument due to 't Hooft that a nonabelian SU (N ) gauge theory with charged fields in adjoint representation posesses a global Z N symmetry [2].
We discuss the 2+1 dimensional case first. Consider a theory with several adjoint Higgs fields so that varying parameters in the Higgs sector the SU (N ) gauge symmetry can be broken completely. In this phase the perturbative spectrum will contain the usual massive "gluons" and Higgs particles. However in addition to that there will be heavy stable magnetic vortices. Those are the analogs of Abrikosov-Nielsen-Olesen vortices in the superconductours and they must be stable by virtue of the following topological argument. The vortex configuration away from the vortex core has all the fields in the pure gauge configuration Here the index α labels the scalar fields in the theory, h α are the constant vacuum expectation values of these fields, and U (x) is a unitary matrix. As one goes around the location of the vortex in space, the matrix U winds nontrivially in the gauge group. This is possible, since the gauge group in the theory without fundamental fields is SU (N )/Z N and it has a nonvanishing first homotopy group Π 1 (SU (N )/Z N ) = Z N . Practically it means that when going around the vortex location full circle, U does not return to the same SU (N ) group element U 0 , but rather ends up at exp{i 2π N }U 0 . Adjoint fields do not feel this type of discontinuity in U and therefore the energy of such a configuration is finite. Since such a configuration can not be smoothly deformed into a trivial one, a single vortex is stable. Processes involving annihilation of N such vortices into vacuum are allowed since N-vortex configurations are topologically trivial. One can of course find explicit vortex solutions once the Higgs potential is specified. As any other semiclassical solution in the weak coupling limit the energy of such a vortex is inversely proportional to the gauge coupling constant and therefore very large. One is therefore in a situation where the spectrum of the theory contains a stable particle even though its mass is much higher than masses of many other particles (gauge and Higgs bosons) and the phase space for its decay into these particles is enormous. The only possible reason for the existence of such a heavy stable particle is that it must carry a conserved quantum number. The theory therefore must possess a global symmetry which is unbroken in the completely higgsed phase. The symmetry group must be Z N since the number of vortices is only conserved modulo N . Now imagine changing smoothly the parameters in the Higgs sector so that the expectation values of the Higgs fields become smaller and smaller, and finally the theory undergoes a phase transition into the confining phase. One can further change the parameters so that the adjoint scalars become heavy and eventually decouple completely from the glue. This limiting process does not change the topology of the gauge group and therefore does not change the symmetry content of the theory. We conclude that the pure Yang-Mills theory also posesses a Z N symmetry. Of course since the confining phase is separated from the completely Higgsed phase by a phase transition one may expect that the Z N symmetry in the confining phase is represented differently. In fact the original paper of 't Hooft as well as subsequent work [5] convincingly argued that in the confining phase the Z N symmetry is spontaneously broken and this breaking is related to the confinement phenomenon.
The physical considerations given above can be put on firmer formal basis. In particular one can construct explicitly the generator of the Z N as well as the order parameter associated with it -the operator that creates the magnetic vortex [6]. We will now describe this construction.

The Abelian case
Consider first an Abelian gauge theory. In this case the homotopy group is Z and therefore we expect the U (1) rather than Z N magnetic symmetry. It is in fact absolutely straightforward to identify the relevant charge. It is none other than the magnetic flux through the equal time plane, with the associated conserved current being the dual of the electromagnetic field strength The current conservation is insured by the Bianchi identity. A group element of the U (1) magnetic symmetry group is exp{iαΦ} for any value of α. A local order parameter -a local field V (x) which carries the magnetic charge -is also readily constructed. It has a form of the singular gauge transformation operator with the singularity at the point x where Θ(x − y) is the polar angle function and J 0 is the electric charge density of whatever matter fields are present in the theory. The cut discontinuity in the function Θ is not physical and can be chosen parallel to the horizontal axis.
Using the Gauss' law constraint this can be cast in a different form, which we will find more convenient for our discussion where the integration goes along the cut of the function Θ which starts at the point x and goes to spatial infinity. The operator does not depend on where precisely one chooses the cut to lie. To see this, note that changing the position of the cut C to C ′ adds to the phase 2π g S d 2 x∂ i E i where S is the area bounded by C − C ′ . In the theory we consider only charged particles with charges multiples of g are present. Therefore the charge within any closed area is a multiple integer of the gauge coupling S d 2 x∂ i E i = gn and the extra phase factor is always unity.
The meaning of the operator V is very simple. From the commutation relation it is obvious that V creates a pointlike magnetic vortex of flux 2π/g. Despite its nonlocal appearance the operator V can be proven to be a local Lorentz scalar field [10]. The locality is the consequence of the fact that V (x) commutes with any local gauge invariant operator in the theory O(y) except when x = y. This is due to the coefficient 2π/g in the exponential which ensures that the Aharonov-Bohm phase of the vortex created by V and any dynamical charged particle present in the theory vanishes. Eqs. (3,5) formalize the physical arguments of 't Hooft in the abelian case.

2.2
The non-Abelian case at weak coupling.
Let us now move onto the analogous construction for nonabelian theories. Ultimately we are interested in the pure Yang -Mills theory. It is however illuminating to start with the theory with an adjoint Higgs field and take the decoupling limit explicitly later. For simplicity we discuss the SU (2) gauge theory. Consider the Georgi-Glashow model -SU (2) gauge theory with an adjoint Higgs field. where At large and positiveμ 2 the model is weakly coupled. The SU (2) gauge symmetry is broken down to U (1) and the Higgs mechanism takes place. Two gauge bosons, W ± , acquire a mass, while the third one, the "photon", remains massless to all orders in perturbation theory. The theory in this region of parameter space resembles very much electrodynamics with vector charged fields. The Abelian construction can therefore be repeated. The SU (2) gauge invariant analog of the conserved dual field strength is where n a ≡ H a |H| is the unit vector in the direction of the Higgs field. Classically this current satisfies the conservation equation The easiest way to see this is to choose a unitary gauge of the form H a (x) = H(x)δ a3 . In this gaugeF is equal to the abelian part of the dual field strength in the third direction in colour space. Its conservation then follows by the Bianchi identity. Thus classically the theory has a conserved U (1) magnetic charge Φ = d 2 xF 0 just like QED. However the unitary gauge can not be imposed at the points where H vanishes, which necessarilly happens in the core of an 't Hooft-Polyakov monopole. It is well known of course [9] that the monopoles are the most important nonperturbative configurations in this model. Their presence leads to a nonvanishing small mass for the photon as well as to confinement of the charged gauge bosons with a tiny nonperturbative string tension. As far as the monopole effects on the magnetic flux, their presence leads to a quantum anomaly in the conservation equation (10). As a result only the discrete Z 2 subgroup of the transformation group generated by Φ remains unbroken in the quantum theory. The detailed discussion of this anomaly, the residual Z 2 symmetry and their relation to monopoles is given in [6].
The order parameter for the magnetic Z 2 symmetry is constructed analogously to QED as a singular gauge transformation generated by the gauge invariant electric charge operator Explicitly One can think of it as a singular SU (2) gauge transformation with the field dependent gauge function This field dependence of the gauge function ensures the gauge invariance of the operator V . Just like in QED it can be shown [6], [10] that the operator V is a local scalar field. Again like in QED, the vortex operator V is a local eigenoperator of the abelian magnetic field B(x) =F 0 .
That is to say, when acting on a state it creates a pointlike magnetic vortex which carries a quantized unit of magnetic flux. The Z 2 magnetic symmetry transformation is generated by the operator and acts on the vortex field V as a phase rotation by π An operator closely related to U and which will be of interest to us in the following, is the generator of the magnetic Z 2 transformation only inside some closed contour C where the integration is over the area S bounded by C. The analog of the commutator eq.(16) for this operator is Taking the contour C to run at infinity U C becomes the generator of Z 2 .
We now have the explicit realization of the magnetic Z 2 symmetry in the Georgi-Glashow model.

The pure gauge theory.
Our next step is to move on to the pure Yang Mills theory. This is achieved by smoothly varying theμ 2 coefficient in the Lagrangian so that the coeficient of the mass term of the Higgs field becomes positive and eventually arbitrarily large. It is well known that in this model the weakly coupled Higgs regime and strongly coupled confining regime are not separated by a phase transition [7]. The pure Yang Mills limit in this model is therefore smooth.
In the pure Yang Mills limit the expressions eq. (9,12,17) have to be taken with care. When the mass of the Higgs field is very large, the configurations that dominate the path integral are those with very small value of the modulus of the Higgs field |H| ∝ 1/M . The modulus of the Higgs field in turn controls the fluctuations of the unit vector n a , since the kinetic term for n in the Lagrangian is |H| 2 (D µ n) 2 . Thus as the mass of the Higgs field increases the fluctuations of n grow in both, amplitude and frequency and the magnetic field operator B as defined in eq.(9) fluctuates wildly. This situation is of course not unusual. It happens whenever one wants to consider in the effective low energy theory an operator which explicitly depends on fast, high energy variables. The standard way to deal with it is to integrate over the fast variables. There could be two possible outcomes of this procedure. Either the operator in question becomes trivial (if it depends strongly on the fast variables), or its reduced version is well defined and regular on the low energy Hilbert space. The "magnetic field" operator B in eq.(9) is obviously of the first type. Since in the pure Yang Mills limit all the orientations of n a are equally probable, integrating over the Higgs field at fixed A µ will lead to vanishing of B. However what interests us is not so much the magnetic field but rather the generator of the magnetic Z 2 transformation U C of eq.(17). In the pure Yang-Mills limit we are thus lead to consider the operator The weight for the integration over n is the kinetic term for the isovector n a . As was noted before the action does not depend on n a in the YM limit. This term however regulates the integral and we keep it for this reason. This operator may look somewhat unfamiliar at first sight. However in a remarkable paper [11] Diakonov and Petrov showed that eq.(20) is equal to the trace of the fundamental Wilson loop along the contour C 2 .
We conclude, that in the pure Yang-Mills theory the generator of the magnetic Z 2 symmetry is the fundamental spatial Wilson loop along the boundary of the spatial plane.
There is a slight subtlety here that may be worth mentioning. The generator of a unitary transformation should be a unitary operator. The trace of the fundamental Wilson loop on the other hand is not unitary. One should therefore strictly speaking consider instead a unitarized Wilson loopW = W √ W W † . However the factor between the two operators √ W W † is an operator that is only sensitive to behaviour of the fields at infinity. It commutes with all physical local operators O(x) unless x → ∞. In this it is very different from the Wilson loop itself, which has a nontrivial commutator with vortex operators V (x) at all values of x. Since the correlators of all gauge invariant local fields in the pure Yang Mills theory are massive and therefore short range, the operator √ W W † must be a constant operator on all finite energy states. The difference between W andW is therefore a trivial constant factor and we will not bother with it in the following. Perhaps of more concern is the difference between W C andW C when the contour C is not at infinity. However here again the factor between the two operators W C W † C is only sensitive to physical degrees of freedom on the contour C and not inside it. Due to its presence the vacuum averages of W C andW C may differ at most by a factor which has a perimeter behaviour The question we will be interested in is whether < W C > has a perimeter or area behaviour. As far as the answer to this question is concerned W C andW C are completely equivalent, and we will not make distinction between them. In the rest of this paper we will therefore refer to W as the generator of Z 2 keeping this little caveat in mind.
Next we consider the vortex operator eq.(12). Again we have to integrate it over the orientations of the unit vector n a . This integration in fact is equivalent to averaging over the gauge group. Following [11] one can write n a in terms of the SU(2) gauge transformation matrix Ω.
The vortex operator in the pure gluodynamics limit then becomes This form makes it explicit thatṼ (x) is defined as the gauge singlet part of the following, apparently non gauge invariant operator The integration over Ω obviously projects out the gauge singlet part of V . In the present case however this projection is redundant. This is because even though V itself is not gauge invariant, when acting on a physical state it transforms it into another physical state 3 . By physical states we mean the states which satisfy the Gauss' constraint in the pure Yang-Mills theory. This property of V was noticed by 't Hooft [2]. To show this let us consider V (x) as defined in eq.(23) and its gauge transform V Ω = Ω † V Ω where Ω is an arbitrary nonsingular gauge transformation operator. The wave functional of any physical state depends only on gauge invariant characteristics of the vector potential, i.e. only on the values of Wilson loops over all possible contours.
Acting on this state by the operators V and V Ω respectively we obtain It is however easy to see that the action of V (x) and V Ω (x) on the Wilson loop is identical -they both multiply it by the centergroup phase (which stays unaffected by Ω) if x is inside C and do nothing otherwise. Therefore we see that for any physical state Ψ. Thus we have where the first equality follows from the fact that a physical state is invariant under action of any gauge transformation Ω and the second equality follows from eq.(26). But this equation is nothing but the statement that the state V |Ψ > is physical, i.e. invariant under any nonsingular gauge transformation.
We have therefore proved that when acting on a physical state the vortex operator creates another physical state. For an operator of this type the gauge invariant projection only affects its matrix elements between unphysical states.
Since we are only interested in calculating correlators of V between physical states, the gauge projection is redundant and we can freely use V rather thañ V to represent the vortex operator.
It is instructive to note that this propery is not shared by the Wilson loop. One can in fact represent the Wilson loop as a singlet gauge projection of a simple Abelian loop operator. The second exponential in eq.(20) can be written as Using eq.(21) we can rewrite the integral in eq.(20) -omitting the regulating kinetic piece-as: The Wilson loop is therefore the gauge singlet part of the Abelian loop The matrix elements of W C and U A C on physical subspace therefore are the same. However U A C as opposed to V does have nonvanishing nondiagonal matrix elements, that is matrix elements between the physical and the unphysical sectors. It therefore can not be used instead of W C in gauge theory calculations. For example non gauge invariant states will contribute as intermediate states in the calculation of quantities like the correlation function < U A C1 U A C2 >, while their contribution vanishes in similar correlators which involve the Wilson loop.
The generalization of the preceding discussion to SU (N ) gauge theories is straightforward. Once again one can start with the Georgi-Glashow like model, where the SU (N ) is higgsed to U (1) (N −1)4 . The construction of the vortex operator and the generator of Z N in this case is very similar and the details are given in [6]. Taking the mass of the Higgs field to infinity again projects the generator onto the trace of the fundamental Wilson loop. The vortex operator can be taken as where the hypercharge generator Y is defined as and the electric field is taken in the matrix notation E i = λ a E a i with λ a -the SU (N ) generator matrices in the fundamental representation.

Generalization to 3+1 dimensions
To conclude this section we discuss how the magnetic symmetry structure generalizes to four dimensions. The conserved Z N generator in the Georgi-Glashow model is defined through Although the definition of U contains explicitly the surface S through which the abelian magnetic flux is integrated, the operator in fact does not depend on S but is specified completely by its boundary. This is because changing S changes the phase of U by the magnetic flux through the closed surface. The only dynamical objects that carry magnetic flux in the theory are 't Hooft-Polyakov monopoles. Since their flux is quantized in units of 4π/g the change in the phase is always a multiple integer of 2π. In the pure Yang-Mills limit the operator U S again reduces to the trace of the fundamental Wilson loop along the boundary of S. Taking the contour to infinity defines the generator of magnetic Z N . As we have already noted, this charge is a little unusual in that it is defined as a surface integral rather than a volume integral. As a result the order parameter for this symmetry transformation is not a local but rather a stringy field. This is of course just a restatement of the fact that magnetic vortices in 3+1 dimensions are stringlike objects. The operator that creates a vortex can still be defined in a way similar to 2+1 dimensions. Skipping the intermediate steps which we went through in the previous discussion we give the final result for the pure Yang Mills SU (N ) gauge theory. The magnetic vortex along the curve C is created by the following operator of the "singular gauge transformation" 5 with ω C (x), the singular gauge function which is equal to the solid angle subtended by C as seen from the point x. The function ω is continuos everywhere, except on a surface S bounded by C, where it jumps by 4π. Other than the fact that S is bounded by C, its location is arbitrary. The vortex loop and the spatial Wilson loop satisfy the 't Hooft algebra where n(C, C ′ ) is the linking number of the curves C and C ′ . One can consider closed contours C or infinite contours that run through the whole system. For an infinite contour C and the Wilson loop along the spatial boundary of the system the linking number is always unity. The V (C) for an infinite loop is therefore an eigenoperator of the Z N magnetic symmetry and is the analog of the vortex operator V (x) in 2+1 dimensions. Any closed vortex loop of fixed size commutes with the Wilson loop if the contour C ′ is very large. Such a closed loop is thus an analog of the vortex-antivortex correlator V (x)V † (y), which also commutes with the global symmetry generator, but has a nontrivial commutator with U C if C encloses only one of the points x or y.
To summarize this section, we have shown that pure Yang Mills theory in 2+1 and 3+1 dimensions has a global Z N magnetic symmetry. The generator of the symmetry group in both cases is the trace of the fundamental Wilson loop along the spatial boundary of the system. The order parameter for this symmetry in 2+1 dimensions is a local scalar field V (x), while in 3+1 dimensions a stringlike field V (C). In both cases the field V is gauge invariant on physical states and is a bona fide canonical order parameter which distinguish in gauge invariant way the phases of the theory. In the next section we discuss the realization of the magnetic symmetry in the confined and the deconfined phases.
3 Hot and cold realizations of the magnetic Z N .
As with any global symmetry, it is important to understand what is the mode of realization of magnetic Z N in the ground state of the theory. This mode of realization depends of course on the parameters of the theory as well as on the temperature. The situation at zero temperature is well understood.

2+1 dimensions.
Again we start with three dimensions. There is a very general argument due to 't Hooft [2]  In fact in the weakly coupled Higgs phase this can be verified by the direct calculation of the expectation value of V [6]. This calculation maps very simply into the classic monopole plasma calculation of Polyakov and was discussed in detail in [6]. One can also explicitly construct the low energy effective Lagrangian in terms of the field V which realizes the spontaneously broken Z N symmetry and describes the low energy spectrum of the Georgi -Glashow vacuum.
A similar effective Lagrangian with some quantitative differences was argued to be valid also for the pure Yang-Mills theory in [12].
The application of the 't Hooft argument at finite temperature is somewhat less straightforward. Since at finite temperature the Lorentz invariance is broken, the temporal and spatial Wilson loops do not necessarily have the same behaviour and one has to be more careful. The original argument relates the behaviour of the vortex operator and the temporal Wilson loop. At finite temperature in the Euclidean formalism the extent of the system in the temporal direction is finite. As a result it is not possible to distinguish between the area and perimeter law for "asymptotically" large temporal loops. Instead the role of the temporal Wilson loop is taken over by the Polyakov line -the loop that winds around the total volume of the system in the temporal direction. Thus one expects that in the deconfining phase where the Polyakov line has a nonvanishing vacuum average, the vortex operator should have vanishing expectation value. Indeed this can be easily confirmed by the explicit calculation of the VEV of the vortex operator using the method of [1]. In [1] the calculation was performed in 3+1 dimensions, but adapting it to 2+1 dimensional case is trivial. We give below a brief outline.
Consider the equal time vortex-antivortex correlation function. At finite temperature is it given by the following expression The line integral can be taken along the straight line L connecting the points x and y. For definiteness we take x and y to be separated in the direction of the first axis. Introducing the imaginary time axis and the Lagrange multiplier field A 0 in the standard way this expression can be transformed to (38) where the "external field " a i is given by The delta function in time in front of the external field a i in eq.(38) appears for the following reason. As we saw in section 2. To evaluate the path integral eq.(38) we follow the standard procedure and integrate out all modes except for the Polyakov loop in a saddle point approximation [14], [13]. This leads to the effective action S ef f (q, a i ), where 1/2T rP = cos q To one loop order the effective action is given by The matrices τ a in A µ are the generators of SU (2) in the fundamental representation and are normalized according to trτ a τ b = 1 2 δ ab . The one loop effective potential U is related to a Bernoulli polynomial and can be read off the expressions in [13] [18]. The only property of U which is important to us is that it has two degenerate minima at q = 0, π.
To calculate the correlator we have to find the configuration of q which minimizes the action eq.(41). Qualitatively the form of the solution is clear. The considerations identical to those in [1] tell us that it must be the "broken" electric Z 2 domain wall: half a wall ( q → x2→∞ 0, q(x 2 = 0) = π 2 ) above the line L and half a wall ( q(x 2 = 0) = − π 2 , q → x2→−∞ 0,) below the line L separated by a discontinuity δq = π. The action of such a configuration is S ef f =α|x − y| whereα is the "Z 2 domain wall tension". The vortex correlator is thus given by As |x−y| become large the correlation function decreases exponentially, and thus the expectation value of the vortex operator vanishes. For the SU (N ) group this calculation trivially generalizes and gives the same result. The exponential decay is also obtained for the correlator of V m with any power m < N .
Recall that the vortex operator is the order parameter for the magnetic Z N symmetry. Moreover the powers of V exhaust all possible local order parameters 8 . Their vanishing is therefore an unambiguous indication that the magnetic Z N is restored in the high temperature deconfined phase.
In hindsight this is not very surprising. Indeed, we are dealing with physical discrete symmetry which is spontaneously broken at zero temperature. When the system is heated it is unavoidable that entropy effects take over and at some sufficiently high temperature the symmetry must be restored. A good qualitative guide here is the effective Lagrangian eq.(36). It describes a simple Z 2 invariant scalar theory. There is very little doubt that a system described by this Lagrangian indeed undergoes a symmetry restoring phase transition at some 8 The latter statement is correct modulo multiplication of V m by local gauge invariant and Z N invariant operators. These possible factors do not change the fact of the exponential decay of the corelators and are therefore unimportant for our discussion.
T c . Moreover the effective Lagrangian approach also suggests that this phase transition has deconfining character. As shown in [6,10,12] the charged states in the effective theory eq.(36) are represented by solitonic configurations of the vortex field V with unit winding number. The energy of any such state is linearly divergent in the infrared. The reason is that due to finite degeneracy of vacuum states, the minimum energy configuration looks like a quasi onedimensional strip across which the phase of V winds. The energy density inside this "electric flux tube" is proportional to the vacuum expectation value of V . When the VEV vanishes, so does the string tension. Stated in other words, when V vanishes, the phase fluctuations are large and the winding number is not a sharp observable. An external charge is thus screened easily by regions of space around it with vanishing V . The phase with < V >= 0 is therefore not confining. In the theory with several Higgs fields this phase exists even at zero temperature and corresponds to a completely Higgsed phase -where the gauge group is broken completely. In such a Higgs phase indeed the colour is screened rather than confined. In the pure Yang-Mills theory this phase is absent at zero temperature, but is realized as the deconfined phase at T > T c . We thus see that the behaviour of the vortex operator at high temperature does indeed match the simple intuition coming from a Z N invariant effective Lagrangian very well.

Extension to 3+1 dimensions
The 't Hooft argument now states that the vanishing vacuum average of the Polyakov line is incompatible with the area law behaviour of the spatial 't Hooft loop and vice versa. This means that in the confining phase the 't Hooft loop has perimeter law. In the high temperature deconfined phase the behaviour of the spatial 't Hooft loop must become area since the average of the Polyakov line is finite. Again this is confirmed by explicit calculation in [1].
A more subtle question is how the behaviour of the 't Hooft loop relates to the realization of the magnetic Z N symmetry. The Z N symmetry does not have an order parameter which is a local field defined at a point. The only order parameters in the strict sense (an eigenoperator with a nonvanishing eigenvalue) is a 't Hooft line V (C) which runs through the whole system [15].
In a system which is finite in the direction of the loop, but is infinite in the perpendicular directions everything is clear cut. In this case there are two possibilities: a) < V > = 0 and the magnetic Z N broken, or b). < V >= 0 and the nagnetic Z N restored.
In the system infinite in all directions C is necessarily an infinite line, and the expectation value < V (C) > clearly vanishes irrespective of whether Z N is broken or not. The 't Hooft loop along a closed contour on the other hand is never zero, since it is globally invariant under the Z N transformation. It is therfore impossible to find an operator whose VEV distinguishes between the two possible realizations of the magnetic symmetry by vanishing in one phase and not vanishing in the other. Nevertheless the behaviour of the closed loop does indeed reflect the realization mode of the symmetry, since it is qualitatively different in the two possible phases. Namely vacuum expectation value of a large closed 't Hooft loop (by large, as usual we mean that the linear dimensions of the loop are much larger than the correlation length in the theory) has an area law decay if the magnetic symmetry is spontaneously broken, and perimeter law decay if the vacuum state is invariant.
To understand the physics of this behaviour it is useful to think of the 't Hooft line as built of "local" operators -little "magnetic dipoles". Consider eq.(34) with the contour C running along the x-axis and the surface S chosen as the (x, y) plane. Let us mentally divide the line into (short) segments of length 2∆ centered at x i . Each one of these segments is a little magnetic dipole and the 't Hooft loop is a product of the operators that create these dipoles. The definition of these little dipole operators is somewhat ambiguous but since we only intend to use them here for the purpose of an intuitive argument any reasonable definition will do. It is convenient to define a single dipole operator in the following way where a ± i (x − y) is the c-number vector potential of the abelian magnetic monopole (antimonopole) of strength 4π/gN . The monopole field correpsonding to a i contains both, the smooth x i /x 3 part as well as the Dirac string contribution. The Dirac string of the monopole -antimonopole pair in eq.(43) is chosen so that is connects the points x − ∆ and x + ∆ along the straight line. The dipole operators obviously have the property This is because in the product the smooth field contribution of the monopole in D ∆ (x) cancels the antimonopole contribution in D ∆ (x + 2∆), while the Dirac string now stretches between the points (x − ∆) and (x + 3∆). When multiplied over the closed contour, the smooth fields cancel out completely, while the surviving Dirac string is precisely the magnetic vortex created by a closed 't Hooft loop operator. The 't Hooft loop can therefore be written as The dipole operator D(x i ) is an eigenoperator of the magnetic flux defined on a surface that crosses the segment [x i − ∆, x i + ∆]. Suppose the magnetic symmetry is broken. Then we expect the dipole operator to have a nonvanishing expectation value 9 < D >= d(∆). If there are no massless excitations in the theory, the operators D(x i ) and D(x j ) should be decorrelated if the distance x i − x j is greater than the correlation length l. Due to eq.(45), the expectation value of the 't Hooft loop should therefore roughly behave as where L is the perimeter of the loop. In the system of finite length L x , the vacuum expectation value of the vortex line which winds around the system in x-direction is therefore finite as in eq.(46) with L → L x .
On the other hand in the unbroken phase the VEV of the dipole operator depends on the size of the system in the perpendicular plane L y . For large L y it must vanish exponentially as d = exp{−aL y }. So the expectation value of V behaves at finite L y in the unbroken phase as: and vanishes as L y → ∞. Thus in a system which is finite in x direction, but infinite in y direction, the 't Hooft line in the x direction has a finite VEV in the broken phase and vanishing VEV in the unbroken phase.
In the limit of the infinite system size L x → ∞ the VEV obviously vanishes in both phases. This is of course due to the fact that V is a product of infinite number of dipole operators, and this product vanishes even if individual dipole operators have finite VEV 10 . However one can avoid any reference to finite size system and infinite vortex lines by considering closed 't Hooft loops.
For a closed loop with long sides along x axis at y = 0 and y = R the above argument leads to the conclusion that in the broken phase V must have a perimeter law, eq.(46). In the unbroken phase the correlation between the dipoles at y = 0 and dipoles at y = R should decay exponentially < D(0)D(R) >∝ exp{−α R l } and thus Thus the perimeter behaviour of < V (C) > indicates a vacuum state which breaks spontaneously the magnetic Z N while the area behaviour means that the magnetic Z N is unbroken.
The results of [1] then mean that in 3+1 dimensions as well as in 2+1 dimension the magnetic symmetry is restored above the deconfining phase transition, in the sense of eq.( 47).
In the next section we discuss what is the implication of this conclusion on the behaviour of the spatial Wilson loop.
carry the long range magnetic field of a dipole. However, the dipole field falls off with distance very fast. Therefore even though this fall off is not exponential the slight nonlocality of D should not affect the following qualitative discussion. 10 The VEV of the dipole D must be smaller than one since D is defined as a unitary operator.
4 Spatial Wilson loop at low and high temperature.
As we have shown in Section 2 the spatial Wilson loop is the generator of the magnetic Z N symmetry. We expect therefore that the mode of realization of the magnetic Z N is strongly linked to the behaviour of W . The argument is simplest to state for a toy model which exemplifies the basic physics in a very simple setting.
Rather than talk about nonabelian gauge theory, consider a scalar theory of a complex field φ with global Z N theory in 2+1 dimensions.
The generator of the Z N symmetry is given by where π = ∂ 0 φ * is the momentum conjugate to the field φ. Obviously with the canonical commutation relations between π and φ one has We will be interested in the behaviour of the operator which generates the Z N transformation only inside some region S of the two dimensional plane.
We will refer to this operator as the U-loop. Throughout this discussion we assume that there are no massless excitations in the spectrum of the theory and that the linear dimensions of the area S are much larger than the correlation length.
The statement we are aiming at is that at zero temperature in the phase with broken Z N the U-loop has an area law behaviour while in the phase with unbroken Z N this changes into the perimeter law behaviour.

U-loop in the broken phase
Consider the broken phase first. We are interested in the vacuum expectation value of U (S). This is nothing but the overlap of the vacuum state < 0| and the state which is obtained by acting with U (S) on the vacuum state |S >= U |0 >.
If the symmetry is broken, the field φ in the vacuum state is pointing in some fixed direction in the internal space. In the state |S > on the other hand its direction in the internal space is different -rotated by 2π/N -at points inside the area S. In the local theory with finite correlation length the overlap between the two states approximatelly factorises into the product of the overlaps taken over the region of space of linear dimension of order of the correlation length l where the label x is the coordinate of the point in the center of a given small region of space. For x outside the area S the two states |0 x > and |S x > are identical and therefore the overlap is unity. However for x inside S the states are different and the overlap is therefore some number e −γ smaller than unity.
The number of such regions inside the area is obviously of order S/l 2 and we thus In a weakly coupled theory this argument is confirmed by explicit calculation. The expectation value of the U-loop in the theory eq.(49) is given by the following path integral This expression directly follows from eq.(52) and integration over the canonical momentum in the phase space path integral. At weak coupling this path integral is dominated by a simple classical configuration. First, it is clear that the solution must be such that the phase of the field φ has a discontinuity of 2π/N when crossing the surface S since otherwise the action is UV divergent due to singular χ. Asymtotically at large distance from the surface the field should approach its vacuum expectation value. Since the source term χ vanishes outside S, eveywhere where φ is continuous it has to solve classical equations of motion. Also, for values of x 1 and x 2 which are well inside S the profile φ should not depend on these coordinates, but should only depend on x 0 . It is easy to see that a solution with these properties exists: it is given by the "broken" domain wall solution. Recall that the vacuum is degenerate and so there certainly exists a classical solution of the equations of motion which interpolates between two adjacent vacuum states φ → x0→∞ φ 0 and φ → x0→−∞ e i 2π N φ 0 . Breaking this classical solution along the plane x 0 = 0 and rotating the piece x 0 < 0 by 2π/N produces precisely the configuration with the correct boundary conditions and the discontinuity structure. The path integral in eq.(55) is therefore dominated by this classical configuration. Its action (up to corrections associated with the boundary effects of S) is αS where α is the classical wall tension of the domain wall which separates two adjacent Z N vacua. Thus we find that the expectation value of the U-loop is related to the domain wall tension of the Z N domain wall by < U (S) >= exp{−αS} (58)

U-loop in the unbroken phase
Now consider the unbroken phase. Again the U-loop average has the form of the overlap of two states which factorizes as in eq.(53). Now however all observables noninvariant under Z N vanish in the vacuum. The action of the symmetry generator does not affect the state |0 >. The state |S > is therefore locally exactly the same as the state |0 > except along the boundary of the area S. Therefore the only regions of space which contribute to the overlap are those which lay within one correlation length from the boundary. Thus where P (S) is the perimeter of the boundary of S. The absence of the area law is again easily verified by a perturbative calculation. In the unbroken phase the fluctuations of the field φ as well as the current density j 0 = i(πφ − π * φ * ) are small. To leading order in the coupling constant The possible area law contribution in the exponent is where G(p) is the Fourier transform of the charge density correlation function. The correlator of the charge densities however vanishes at zero momentum. This is because in the leading perturbative order the symmetry of the theory is actually U (1) and not just Z N as seen in eq. 55. Since the vacuum state is invariant it follows that the total charge Q = d 2 xj 0 (x) = j 0 (p = 0) on this state vanishes, and so does any correlation function that involves zero momentum component of the charge density. So the area contribution in eq.(61) is zero. Strictly speaking in the leading order in perturbation theory eq.(60) is not the complete result. The exact expression contains in the exponential also higher point correlators of the current density. Again however the possible area law contribution contains correlators of the total charge Q with powers of j 0 and therefore vanishes.

U-loop at high temperature
Let us see now how the argument changes at high temperature. The important difference is that the vacuum is not a pure state but rather a statistical ensemble. The average of the U-loop is therefore not given by a single matrix element but rather by Let us consider the theory in which the Z N symmetry is broken at zero temperature. For concreteness we will think about Z 2 symmetric theory, although qualitatively the discussion does not change for any N . The two degenerate vacuum states are characterized by the value of the condensate < φ >= ±µ.
In order to understand the behaviour of the U -loop we have to figure out what types of states contribute to the thermal ensemble. At zero temperature the only states that are of interst are those with finite energy. There are two towers of such states |n > µ and |n > −µ -constructed above each one of the degenerate vacua. These two towers of states do not talk to each other, not only because their overlap is zero, but also because they can not be connected to each other by action of any local (or semilocal) operator µ < n|O|n ′ > −µ = 0. An immediate corollary of this is that a superposition of the type |α, β >= α|n µ > +β|n > −µ violates clustering property of the correlators of local operators. For this reason at zero temperature in a spontaneously broken theory we are never interested in states which carry sharp quantum numbers of the broken symmetry.
At finite temperature however we are also asking after states with finite energy density, and therefore infinite energy. This part of the spectrum looks rather different if the energy density involved is high enough. The two vacuum configurations of the potential in eq.(49) are separated by a finite barrier. Let us call the hight of this barrier H. The states with energy density lower than H still separate into two towers. We will denote these states by |l >. However higher energy density states, with ǫ > H have different nature. Their wave function is not localized in the field space to the vicinity of one of the vacua, but rather is spread over distances larger than the distance between the two vacuum states 2µ. These states therefore naturally carry sharp quantum numbers with respect to the broken Z 2 symmetry. These states we will denote by |h >. In fact one expects that the higher the energy density the more these states look like the multiparticle states of a symmetric phase. That is to say as long as ǫ >> H it does not matter whether the potential has the double well structure or a single vacuum. These highly excited states should look like states with finite density of particles which carry the Z 2 charge.
At low temperatures, when the entropy effects are not important the contribution to the thermal ensemble comes only from the |l > -sector since the Bolzman factor for any of the |h > states vanishes exponentially in the infinite volume limit. As we have argued earlier, the average of U in each one of these states has an area law behaviour and so does the whole temperature average of U .
When the temperature reaches T C the phase transition occurs. The reason for the onset of the phase transition is that when the equilibrium energy density reaches critical threshold value, the |h > sector states start contributing to the thermal ensemble. The sudden change in the entropy due to these new channels drives the phase transition. Above the phase transition therefore there are two kinds of states that contribute to thermal averages. One can then write < U > T >Tc = n e −E l n /T < n, l|U |n, l > + s e −E h n /T < n, h|U |n, h > (63) where n stands for all other quantum numbers. In fact once the entropy effects become important enough to excite the |h >-sector, the contribution of |l > states to any physical observable becomes negligible. As discussed earlier each state in the second term gives a perimeter contribution exp{γ s P } to the average of the U -loop, so one could be tempted to conclude that the loop must have a perimeter law just like in zero temperature vacuum of a symmetric phase. This however is not necessarily the case. The reason is that < n, h|U |n, h > is not positive definite. In fact the number of states in which it is positive is roughly equal to the number of states on which it is negative. It is therefore very likely that the leading perimeter behaviour will cancel and the net result will again be an area law for < U >.
Indeed if the ensemble can be thought of as an ensemble of Z 2 charged free particles, the area law for U -loop follows immediately 11 . The U -loop in such an ensemble is where µ is the fugacity of a single particle, and the summation goes over all coordinates x i of the particles inside the area S and over all possible numbers of particles n. Assuming that particles have a finite size ∆, so that there are S/∆ possibilities to place one particle inside the area S in the dilute gas approximation the sum gives We stress that the thermal ensemble of particles is Z 2 invariant. The density matrix for such an ensemble can be written in the particle basis as The operator of the Z 2 transformation acts on the n-particle states as and so The explicit simple formula eq.(65) is derived in the dilute gas approximation. We expect that the physics will be similar as long as the interaction between the particles is short range. Whenever the interaction is long range the behaviour of < U (S) > can be different. For instance one does not expect area law behaviour if the particles are bound into pairs since in this case only Z 2 invariant states contribute to the thermal ensemble.
Our conclusion is that at finite temperature the behaviour of the U -loop is not strongly related to the mode of the realization of the Z N symmetry. It is rather more likely to have an area behaviour.
To reiterate, the physics involved is very simple. At zero temperature when acting on a state, the U -loop performs the Z N transformation inside the loop. At finite temperature however the thermal ensemble may even in the symmetric phase contain significant contributions from states with nonvanishing Z N charges. The U -loop therefore perturbs the thermal ensemble very significantly everywhere inside the area, and the natural outcome is an area law.
The argument is quite general and does not depend on the exact form of the Z N invariant potential and more generally on the field content of the theorywe could have added any number of extra fields to the theory eq.(49) without changing the conclusions. The same relation must exist between the mode of realization of the magnetic Z N symmetry and the behaviour of the Wilson loop in the pure Yang-Mills theory. The direct analogs of the scalar field φ(x) in eq.(49) and the U-loop of the scalar theory are correspondingly the vortex field V (x), and the spatial Wilson loop W (C).
As we have shown in the previous section, the magnetic Z N is restored at high temperature. The Wilson loop is nevertheless likely to have an area law as is indeed indicated by all existing lattice data. In this context we note that analytic strong coupling results also give area behaviour [8].

Wilson loop in 3+1 dimensions
The previous considerations generalize to the 3+1 dimensons. At zero temperature in the broken phase when acting with the Wilson loop W (C) on the vacuum one changes the state of those magnetic vortices which loop through C.
The number of such vortices which are present in a generic configuration in the broken phase is proportional to the minimal area subtending C. The number of the degrees of freedom that is changed by the action of W is thus proportional to the area S. Each of these degrees of freedom contributes a factor smaller than unity to the overlap with the vacuum state and so the VEV of W scales with the exponential of the area. In the unbroken phase the vacuum does not contain vortices of arbitrarily large size. The size of the vortices present in the vacuum is cutoff by the relevant correlation length. This is the case if the gauge group is completely broken by the Higgs mechanism. Therefore for contours C of linear dimension much larger than this length, the action of W (C) only disturbs degrees of freedom close to the contour C itself and the VEV must have the perimeter behaviour.
At high temperature even the symmetric thermal ensemble is populated by vortices. These vortices are not free since apart from them the ensemble also contains "free" charges. However unless this background of charges induces long range interactions between the vortices, the most probable result for the Wilson loop is the area law. A more detailed knowledge of vortex dynamics is necessary to draw a firm conclusion.
To close this section we note that the present considerations do not apply to Abelian theories. The magnetic symmetry does exist in this case too, but here it is the continuos U (1) group and the spectrum is massless. In this case there is no reason to expect the local factorization of the overlap and generically therefore the arguments of this section do not hold. In particular in the presence of long range correlations it is perfectly possible that the Wilson loop has a perimeter law even though the state is perturbed everywhere inside the area boundede by the loop 12 .
In this paper our aim was to point out two facts. First that the calculation of the VEV of the 't Hooft loop [1] implies the restoration of the magnetic Z N symmetry above the deconfinement transition. Second, that the mode of realization of magnetic symmetry is closely related to the behaviour of the spatial Wilson loop.
At zero temperature this relation is very rigid: spontaneous breaking of Z N implies the area law behaviour for W while unbroken Z N leads to perimeter law behaviour. At high temperature however even though the Z N symmetry is restored, the Wilson loop may have an area law. This is the consequence of the fact that even a Z N -invariant the thermal ensemble can contain a significant contribution of Z N nonsinglet states. The area law is particularly simply understood if the thermal ensemble at high temperature is well appproximated by an ensemble of weakly interacting magnetic vortices. The vortex gas argument has been previously brought up in favour of the area behaviour of the spatial Wilson loop in [19]. This behaviour is also confirmed by several lattice gauge theory calculations [16].
An alternative possibility is that due to the as yet unknown vortex dynamics, there is vortex -antivortex binding. To explore such a possibility it would be very interesting to measure on the lattice the free energy of a magnetic vortex.
In ref. ([15]) the behaviour of the free energy of magnetic and electric fluxes has been discussed in the low temperature phase. To be able to do it in the lattice framework one has to define the theory in a finite volume. As discussed by 't Hooft this can be achieved by imposing on the potentials periodic boundary conditions modulo a gauge transformation. As discussed in ref. [15] this admits the presence of vortices in 2+1 and of the vortex lines in 3+1 dimensions. 't Hooft's discussion was based on a Euclidean rotation identity for the twisted 4d path integrals valid for any temperature, and a factorization property of magnetic and electric fluxes. In the notation of ref. [15]: F ( e, m) = F e ( e) + F m ( m) Its validity at low T is very reasonable, but is inconsistent with the Euclidean rotation identity at high T. Based on this 't Hooft could prove (N ≤ 3) that in the confining phase, where the free energy of an electric flux is linear with the length (with the string tension ρ), the free energy of magnetic flux vanishes exponentially in the infinite volume limit. For a magnetic flux in, say the zdirection it is exp −ρL x L y . Thus the free energy of a magnetic flux is related to the behaviour of the Wilson loop. The free energy of an electric flux in the z direction in the hot phase vanishes exponentially like exp −αL x L y where α is the surface tension found in ref. [1]. So the next obvious question is how the magnetic flux free energy behaves in the hot phase. In a vortex gas picture this free energy should vanish exponentially in the infinite volume limit. Such a calculation in 2+1 dimensions has been performed and (modulo some uncertainty related to imperfect measurement of global vorticity) results are consistent with this expectation [20]. It would also be instructive to see how in the hot phase the additivity of electric and magnetic fluxes is broken.
We note that a recent lattice calculation [21] measures the monopole-antimonopole correlation. The results of [21] point to the screened behaviour of this correlation function for all temperatures. So in the hot phase it behaves like its electric partner, the correlator of Polyakov loops 13 . This via 't Hoofts argument, is consistent with the measured area behaviour of Wilson loops [16] and would imply that the magnetic flux free energy would fall off with an area law for all temperatures.
It is interesting to note that the vortex gas picture is equally applicable in high temperature confining and nonconfining gauge theory. In particular one can consider an SU (N ) gauge theory with sufficient number of adjoint Higgs fields, so that the gauge theory at zero temperature is broken completely. In this situation the magnetic Z N symmetry is unbroken in the vacuum and the Wilson loop has a perimeter law. Magnetic vortices are finite energy excitations with the mass of order M = M 2 v /g 2 , where M v is the vector boson mass. When the system is heated one expects that the thermal ensemble will contain a dilute gas of these vortices at any temperature. Therefore at any finite temperature the spatial string tension should be nonzero, although at low temperatures it will be exponentially suppressed if the theory is weakly coupled: