Entropic order parameters for the phases of QFT

We propose entropic order parameters that capture the physics of generalized symmetries and phases in QFT's. We do it through an analysis of simple properties (additivity and Haag duality) of the net of operator algebras attached to space-time regions. We observe that different types of symmetries are associated with the breaking of these properties in regions of different non-trivial topologies. When such topologies are connected, we show that the non locally generated operators generate an Abelian symmetry group, and their commutation relations are fixed. The existence of order parameters with area law, like the Wilson loop for the confinement phase, or the 't Hooft loop for the dual Higgs phase, is shown to imply the existence of more than one possible choice of algebras for the same underlying theory. A natural entropic order parameter arises by this non-uniqueness. We display aspects of the phases of theories with generalized symmetries in terms of these entropic order parameters. In particular, the connection between constant and area laws for dual order and disorder parameters is transparent in this approach, new constraints arising from conformal symmetry are revealed, and the algebraic origin of the Dirac quantization condition (and generalizations thereof) is described. A novel tool in this approach is the entropic certainty relation satisfied by dual relative entropies associated with complementary regions, which quantitatively relates the statistics of order and disorder parameters.


Introduction
Transcending the weak coupling regime has been a recurring theme in the context of QFT in the past decades. Many pressing reasons motivate this interest. We have the everlasting confinement problem in gauge theories [1,2], examples of non-Fermi liquid behaviour at low temperatures in condensed matter theory [3], electromagnetic dualities in QFT [4], and the holographic duality [5].
In the quest of understanding strong coupling phenomena, it is natural to seek for sufficiently robust features that remain valid at any value of the coupling. This includes looking for alternative descriptions, or new structures, which may be studied in a controllable manner. The present article is framed within the Haag-Kastler algebraic approach to QFT [6,7]. This approach has been fruitful for progress at the conceptual level. As described below, it can be considered a minimalistic approach, that only assumes very general and basic properties about the way operator algebras are assigned to space-time regions. Moreover, it is the natural approach for the description of entanglement entropy and other statistical measures of states.
Structures that transcend the perturbative regime are generally connected to symmetries, whether space-time or internal ones. Examples are conformal symmetry, supersymmetry, global and local symmetries, and the recently introduced generalized global symmetries [8]. However, most of the time the way these symmetries are considered is linked to the Lagrangian QFT definition, relying on a weak coupling regime.
There are two notable exceptions. For the case of global symmetries, a first principle algebraic approach was carried out by Haag, Doplicher, and Roberts [9][10][11][12]. They sought to find the imprint of the symmetry already in the neutral (observable) sector of the theory. They found that the superselection sectors arising by including charged operators in the model were seen to be in correspondence with certain endomorphisms of the observable algebra. Having identified the imprint, one can try to reverse the logic. Given a structure of endomorphisms with certain defining properties, called in the literature DHR superselection sectors, one seeks to derive the symmetry group itself. This problem was completed leading to the reconstruction theorems [13]. For the case of conformal symmetries, a first principle approach started with the works of Polyakov, Ferrara, Grillo, and Gatto [14,15], known as the conformal bootstrap, and which is being used with great success at present [16].
One would like to extend the algebraic approach to other kinds of symmetries, such as local ones. This extension turns out to be more complicated. The reason is that for local symmetries, the associated charged operators cannot be localized in a ball. One can measure their charge at arbitrarily long distances employing local operators only. An example is an electric charge which can be measured by the electric flux at infinity. Some modifications of the DHR formalism were proposed in this regard. One considers sectors which, instead of being localizable in balls, are localizable in cones that extend out to infinity [17,18]. This approach departs from the local QFT philosophy it started with, due to the infinite cones. It would be better to understand all kinds of symmetries already in a compact region of Minkowski space-time and keep aligned with the local QFT attitude. We would also like to include symmetries associated with higher dimensional cones. Presumably, these would be related to the generalized symmetries introduced more recently in [8]. But from the algebraic perspective, the higher dimensional cones would represent superselection charges with infinite energy in an infinite space, and they have been discarded in that regard.
In this article, we propose a unified approach to symmetries in QFT which is fundamentally local. We do not want to resort to a Lagrangian or any local current, and we want to be able to frame the description talking about the vacuum state on subregions of flat topologically trivial Minkowski spacetime. To connect with more conventional approaches, we seek to define order parameters that signal the presence and breaking of the symmetries, allowing a broad characterization of phases in QFT's. As it will be clarified through the text, order parameters in this context are naturally defined using information theory, and we call them entropic order parameters. These can be related to operator order parameters, though not the standard singular line operators that are usually considered. In particular, these operators cannot be renormalized arbitrarily.
Quite surprisingly, such a path to symmetries in QFT has a simple and geometrical starting point, based on causality. In QFT, causality is enforced by the requirement of commutativity of operators at spatial distances. This is summarized by where A(R) is the algebra of operators localized in a certain region R, R is the set of points causally disconnected from R, and A is the commutant of the algebra A. Naively, one could be inclined to believe that this relation might be saturated in a QFT, i.e, we would have the equality in (1.1). Such saturation is called Haag duality, or duality for short. 1 It turns out that something more interesting can happen. The previous inclusion does not need to be saturated. Indeed, as we will describe, it is precisely in the difference between both algebras where generalized symmetries may appear. This difference consists of operators that cannot be locally generated in R but are still commuting with operators in R . Therefore, if we include them in the algebra of R to restore duality, we introduce a violation of additivity, the property stating that operators in a region are generated as products of local operators in the region. The tension between duality and additivity in these theories cannot be resolved.
The observation that global symmetries entail violations of Haag duality was known a long time ago, see for example [7]. The reason is that one can form observables out of the product of local charged operators. If one chooses a region R which is disconnected, so that it has non-trivial homotopy group π 0 , then Haag duality will not hold due to the existence of chargeanticharge operators localized at different disconnected patches. These non-local operators are called intertwiners. This type of breaking of Haag duality was studied in full detail for twodimensional conformal field theories in [19], where the structure of the algebra was unraveled and shown to be controlled by the structure of superselection sectors. In higher dimensions the analysis was complemented in [20], by describing the breaking of duality in the region R complementary to R. This region has a non-trivial π d−2 homotopy group, and the violation of duality is due to the existence of twist operators, that implement the symmetry locally. This will be described in more detail below.
While the relation between duality violation for topological non-trivial regions and global symmetries was appreciated, the starting point for the algebraic derivation of global symmetries was the DHR endomorphisms. In this paper, we take the breaking of duality as the fundamental physical feature, from which the symmetries could be derived. This seemingly mild change of perspective eases the way to generalizations. We will be able to discuss all kinds of symmetries by focusing on the "kinematical" properties of algebras and regions in the vacuum. For this purpose, we can avoid studying superselection sectors, which may have a dynamical input, or introduce infinite cones for their description. We observe that different types of symmetries are related to the breaking of duality for regions of different topology. While global symmetries entail the breaking of duality for regions with non-trivial π 0 or π d−2 , we observe that local symmetries fit nicely as QFT's in which duality is broken for regions with non-trivial π 1 or π d−3 . Going up in the ladder, in QFT's with generalized symmetries, duality is broken for regions with non-trivial π i or π d−2−i . We argue that for any i ≥ 1, the symmetries are bound to form an Abelian symmetry group. Finally, we show that the breaking of duality of the complementary regions π i and π d−2−i is due to the existence of non-local operators with specific commutation relations between themselves. Physically, these dual non-local operators correspond to order and disorder parameters, and their behavior characterizes the phases of the theory.
As a by-product of this analysis, it follows that the Dirac quantization condition nicely fits into the algebraic framework. It turns out to be simply originated when enforcing causality of the net of algebras. Although this might sound trivial, causality of the net becomes threatened in situations where the inclusion (1.1) is not saturated. Enforcing duality and causality directly provides the generalized quantization condition.
Having identified the connection between the failure of duality and the symmetries in QFT's, in the second part of the article we proceed to construct order parameters that sense their presence and their breaking. We start by showing that the non-local order-disorder operators that violate additivity are the only ones that can display area laws, typical of confinement of electric or magnetic charges in gauge theories. Conversely, the breaking of duality in the appropriate region is seen as a necessity for the existence of order parameters with area law behavior, like the Wilson loop in pure gauge theories.
The choice of operator order parameters is not unique. Indeed there is an infinite number of possibilities. This is somewhat in contrast with the previous inclusion of algebras (1.1), which is robust and completely unambiguous. Natural order parameters should arise from such inclusions. To accomplish this, we find most natural to resort to information theory. In fact, given an inclusion of the previous type, entropic order parameters can be defined as the relative entropy between the vacuum and a state in which we have sent to zero all expectation values of nonlocal operators. This relative entropy is a well-defined notion of uncertainty for the algebra of non-local operators, and it will play a central role in the article. The entropic approach to global symmetries recently developed in [20], which in turn was inspired by the work [21] concerning free fermions in two dimensions, is here generalized to regions of different topology.
On one hand, the choice of relative entropy is convenient because it is robust and standard. But more importantly, it allows us to quantitatively relate the physics of order and disorder parameters. This is due to a general property of relative entropies called certainty principle [20,22]. In the present light, this relates the entropic order parameter with the entropic disorder parameter, for complementary geometries. In other words, quoting a specific example, the statistics of Wilson loops and t' Hooft loops in complementary regions, are precisely related to each other by the certainty relation.
We will compute the entropic order and disorder parameters for symmetries and phases in QFT's in several cases of interest. In some instances, we can check compatibility with the certainty principle, or use this relation to understand their behavior. We will start with QFT's with global symmetries, and consider scenarios with conformal symmetry and with spontaneous symmetry breaking. Both phases will be seen to be distinguished already at a qualitative level by the order parameters, as they should. Similarities with the phase structure of gauge theories that arise from the present approach will be highlighted. Interestingly, for scenarios with spontaneous symmetry breaking, the computations are related to the solitons/instantons of the theory, as could have been anticipated. We then move to the case of gauge theories. We will first analyze the case of the Maxwell field, which can be done in great detail, and where the match between the order and disorder approaches will be confirmed with surprisingly good accuracy. We then analyze several interesting constraints that appear in gauge theories with conformal symmetry in four dimensions. In this scenario, a specific relative entropy becomes enough constrained to be determined analytically. We finally move to the Higgs phase, which as explained by t' Hooft in [23], is dual to the confinement scenario, and where semiclassical physics can be used to study the entropic order parameters.
A final remark is in order. One of the initial motivations for this work was to understand issues about entanglement entropy in gauge theories. Several specific regularizations of entropy were proposed in the literature, which pointed to some UV ambiguities of entropy in gauge theories [24][25][26][27][28]. As explained in [20,29,30], such ambiguities do not survive the continuum limit. In this paper, we find that for specific QFT's (the ones with generalized symmetries), there is more than one possible algebra for a region of specific topology. These multiple choices are macroscopic and physical and pertain to the continuum model itself. They have no relation with regularization ambiguities, nor with the description in terms of gauge fields. Corresponding to the multiplicity of algebras there are multiple entropies for the same region. These entropies measure different quantities and therefore should not be understood as ambiguities. The relative entropy order parameters introduced in this paper are precisely well-defined notions of the differences between these entropies.
In the algebraic approach, a QFT is described by a net of von Neumann algebras. This is an assignation of an operator algebra to any open region of space-time. The particular QFT model is determined by how the algebras in the net relate to each other and with the vacuum state.
We will restrict to consider only causal regions, which will be typically denoted by R below. Causal regions are the domain of dependence of subsets of a Cauchy surface. In this paper, we will be interested in the properties of algebras assigned to causal regions based on the same (arbitrary) Cauchy surface C. These regions will have in general non-trivial topologies whose properties are the same as the ones of subregions of C (typically the surface t = 0) in which they are based. Hence, we will often make no distinction between a d − 1 dimensional subset of C and its causal d-dimensional completion.
The algebras A(R) attached to regions R satisfy the basic relations of isotony and causality, A(R) ⊆ (A(R )) , (2.2) where R is the causal complement of R, i.e. the space-time set of points spatially separated from R, and A is the algebra of all operators that commute with those of A. We always have A = A.
Extensions of these relations are expected to hold for sufficiently complete models but are not granted on general grounds. For example, (2.2) could be extended to the relation of duality (also called Haag's duality) A(R) = (A(R )) , and we could also expect a form of additivity where R 1 ∨ R 2 = (R 1 ∪ R 2 ) , A 1 ∨ A 2 = (A 1 ∪ A 2 ) are the smallest causal regions and von Neumann algebras containing R 1 , R 2 and A 1 , A 2 respectively. We will call a net complete if it satisfies (2.3) and (2.4) for all R based on the same Cauchy surface. The main focus of the paper concerns nets that are not complete in this sense and how this is related to generalized symmetries in the QFT.
The de Morgan laws Conversely, additivity follows from unrestricted validity of duality and the intersection property. Therefore the intersection property is another aspect of duality and additivity.
It is expected that algebras for topologically trivial regions R, such as a ball, satisfy duality, and that additivity holds for topologically trivial regions whose union is also topologically trivial. This last statement means the algebra of R is generated by the algebras of any collection of balls (of any size) included in R and whose union is all R. This accounts for the idea that the operator content of the theory is formed by local degrees of freedom. We will assume this additivity property that can be summarized in that any localized operator of the theory is locally generated (in a topologically trivial space). 2 However, a different question is whether any operator of a certain algebra A(R) is locally generated inside R itself when the region is topologically non-trivial. We will see several examples in the next sections that will show the existence of non locally generated operators in such A(R) is not an uncommon phenomenon.
Let us be more precise. Given a net, we can always construct an additive algebra for a region R as A add (R) = B is a ball , B⊆R A(B) . (2.8) This gives us a minimal algebra, in the sense that it contains all operators which must form part of the algebra because they are locally formed in R. The assignation of A add (R) to any R gives the minimal possible net and if A add (R) A(R) it follows that there are more than one different net.
In this freedom of choosing the operator content of different regions, the greatest possible algebra of operators that can be assigned to R and still satisfies causality must correspond to a minimal one assigned to R , A max (R) = (A add (R )) .
Evidently if A add (R) A max (R) it follows that the additive net does not satisfy duality. In this situation one can enlarge the additive net by adding non locally generated operators, to generate a net satisfying duality (2.3). In general, this may be done in multiple ways. We will call such nets Haag-Dirac (HD) nets for reasons that will become apparent later on. By construction, Haag-Dirac nets satisfy duality A HD (R) = (A HD (R )) , (2.10) but in general will not satisfy additivity. Therefore, there is a tension between duality and additivity which cannot be resolved in these incomplete theories. Notice that for a global pure state the entropy of an algebra A is equal to the one of its algebraic complement A . The present discussion shows this does not translate to an equality of entropies for complementary regions, except for an HD net.
To be more concrete, let us call a ∈ A max (R) to a collection of non locally generated operators in R such that In the same way we have operators b ∈ A max (R ) non locally generated in R such that Evidently, the dual sets of operators {a} and {b} cannot commute to each other. Otherwise it would be A max (R) ⊆ (A max (R )) = A add (R) and these operators would be locally generated. Given the existence of non locally generated operators a in R, the necessity of the existence of dual complementary sets of non locally generated operators b in R is due to the fact that for two different algebra choices A 1,2 for R there are two different choices A 1,2 associated to R . The later cannot coincide because of the von Newman relation A = A.
Since the dual non locally generated operators {a} and {b} do not commute, when constructing Haag-Dirac nets A HD (R) satisfying duality, we have to sacrifice some operators of A max (R) or A max (R ), to keep the net causal. The assignation A max (R) for all R does not form a net. A possible choice is A max (R) for R and A add (R ) for R or vice-versa, and usually there are some intermediate choices. In particular, if the topologies of R and R are the same, both of these choices are not very natural, and may break some spatial symmetries. 3 An important remark is the following. Even if some non locally generated operator is excluded from the algebra of R it does not mean it does not exist in the theory. All non locally generated operators that could be assigned to R are always formed locally in a ball containing R and thus its existence cannot be avoided. They will always belong to the algebra of this ball. In particular, the full operator content of the different nets is the same. . We assume both A add(R) and A add(R ) have no center (are factors), see [32]. This is an expected property in QFT. A center would produce an irreducible sector unrelated to non locality.
. We will describe several specific examples below.
In the applications of this paper, these dual fusion rules are associated with group representations and their conjugacy classes. This brings in the idea of symmetries. In the specific models we analyze, duality is seen to fail when the algebras are constructed as the invariant operators under certain symmetries. Examples are orbifolds of a global symmetry and gauge-invariant operators for some gauge theories. We will see that the particular topology of R where duality or additivity fails depends on the type of symmetry involved. Orbifolds show algebra-region "problems" when one of the homotopy groups π 0 (R) or π d−2 (R) are non-trivial. The case of ordinary gauge symmetries might give problems for regions with non-trivial π 1 (R) or π d−3 (R). Higher homotopy groups correspond to the case of gauge symmetries for higher forms gauge fields. Notice that in these examples the gauge symmetry plays an auxiliary role in the construction of the models, but does not play a direct role in the final theory. However, the algebra of the non locally generated operators does play a fundamental role. It can be interpreted as a generalized symmetry in the sense of [8].
We consider the subalgebra O of a theory F, consisting of operators invariant under a global symmetry group G acting on F. The theory O = F/G is called an orbifold. These models were treated in more detail in [20]. In this case, we take regions R with non-trivial π 0 (R), that is, disconnected regions. The complement R will have non trivial π d−2 (R ). The simplest example is two disjoint balls R 1 , R 2 , and its complement S = (R 1 ∪ R 2 ) , which is topologically a "shell" with the topology of S d−2 × R.
Let ψ i,r 1 , ψ i,r 2 be charge creating operators in R 1 , R 2 in the theory F, corresponding to the irreducible representation r, and where i is an index of the representation. The corresponding to this representation is invariant under global group transformations and belongs to the neutral theory O. See figure 1. It commutes with operators in O add (S) but cannot be generated additively by operators in O(R 1 ) and O(R 2 ) since the charged operators ψ i,r belong to the field algebra F but not to O.
In a dual way, there are twist operators τ g implementing the group operations in R 1 and acting trivially in R 2 . These commute with O(R 1 ) and O(R 2 ), that is, uncharged operators in R 1 or R 2 , but do not commute with the intertwiners, which have charged operators in R 1 . The twists can be chosen to satisfy 6 This shows explicitly that, retaining additivity, duality fails for the two component region R 1 R 2 and for its complement S. The reason is the existence in the model of operators (twists and intertwiners) in these regions, which cannot be additively generated inside the same regions by operators localized in small balls. However, the intertwiners and twists can be generated additively in O but in bigger regions with trivial topology.
For finite groups, the number of independent twists coincides with the number of intertwiners. This is because the number n C of conjugacy classes of the group is equal to the number of irreducible representations. For Lie groups, there is an infinite number of irreducible representations, and the same occurs for conjugacy classes. In this case, as described in more detail below when discussing gauge theories, it is the duality between "electric" and "magnetic" weights the one ensuring that both sets of operators run over dual lattices.
As shown in appendix A), we can choose the intertwiners to satisfy a closed algebra. More concretely we get the fusion algebra wherer is the representation conjugate to r, and n r 3 r 1 r 2 are the fusion matrices of the group representations,
giving the number of irreducible representations of type r 3 in the decomposition of the tensor product of r 1 and r 2 . Because n r 3 r 1 r 2 = n r 3 r 2 r 1 the algebra (2.18) is Abelian. The same can be said of the algebra of the twists. From (2.14) we get with m c 3 c 1 c 2 the fusion coefficients of the conjugacy classes.
The two Abelian algebras of twists and intertwiners do not commute with each other. For finite groups they can be embedded in the non-Abelian matrix algebra of |G| × |G| matrices, see appendix A. A similar embedding works for Lie groups but the embedding algebra needs to be infinite-dimensional. For Abelian symmetry groups, the commutation relations take a very simple form where χ r (g) is the group character.
The DHR theory of ball localized superselection sectors gives examples of the failure of additivity-duality for regions with non-trivial π 0 , π d−2 for any dimension. The theory shows that under quite general conditions for these types of sectors, and for d ≥ 3, the fusion algebras arise from a group, as described above [9,10,12,13]. More general fusion rules may appear in d = 2 [7,18,32]. As shown by the reconstruction theorem in such papers, starting with the model O with this type of duality failure a new theory F exists where charged operators cure these duality and additivity problems. The symmetry group is globally represented in F acting on the charged fields. This reconstruction does not modify the theory (the correlation functions) in the subalgebra O. This does not seem to have a transparent analog in gauge theories.
In this section, we move our focus towards theories that violate duality for regions having nontrivial π 1 (R). From the dual perspective, these theories will also show problems for regions with non-trivial π d−3 (R ). The failure of duality or additivity for these types of regions gives place to a failure of the intersection property for topologically trivial regions A, B, with an intersection R or R . See figure 2. 7 The main working example in this situation will be that of gauge theories. However, before describing the specific non-local operators associated to gauge theories, we want to show how the structure arising from a failure of duality-additivity is rather fixed on general grounds, without referring to gauge fields. In particular, it is possible to show that the dual non-local operators form dual Abelian groups, and the commutation relations are fixed.
For gauge theories, these features appear when there is a subgroup of the center of the gauge group which leaves invariant all matter fields. For pure gauge theories, as we will show below, the non-local operators correspond to t' Hooft and Wilson loops associated respectively to the center Z of the gauge group and its dual Z * , the group of its characters (which is isomorphic to Z). All other Wilson and t' Hooft loops are locally generated. Any finite Abelian group can be formed in this way with a gauge theory because the cyclic group Z n is the center of SU (n) a b R R A B Figure 2: Left: Duality and additivity cannot be valid simultaneously for the region R (here a ring-like region or a solid torus). The operator a is not additive in R. The interlocked operator b is again not additive in the complement R . a and b do not commute with each other. For dual algebras attached to R and R either a or b have to belong to the respective algebra (not both of them at the same time), and additivity is lost. Right: Violation of the intersection property. The figure shows a section of two spherical cap regions A and B intersecting in the ring R (here d = 4). A non additive operator in R is additive in both the topologically trivial regions A and B. It then necessarily belongs to the intersection of the algebras of A and B. This implies that additivity for R cannot be maintained at the same time than the intersection property.
and any finite Abelian group is a product of cyclic groups. In d = 4 R and R have the same topology of S 1 and both the Wilson and t' Hooft loops are now non-local operators of the same ring R. For pure gauge fields, the group of non-local operators is then Z × Z * . Adding matter fields several subgroups of Z × Z * can be realized. We describe the non-local operators for a Maxwell field and non-Abelian ones. In the appendix B we show explicitly these properties for arbitrary gauge fields in a lattice.

The non-local operators form Abelian groups
Now we show that the dual algebras of non-local operators correspond to dual Abelian groups, and the structure of the commutation relations is fixed. We keep the discussion as simple as possible. A mathematically precise proof would follow the ideas of the DHR analysis for global symmetries, see [7]. Some natural assumptions have to be made. Borrowing the terminology of this analysis, an underlying assumption is that the non-local operators are transportable. This just states that the non-local sectors are preserved by deformations. More precisely, for any two homotopic regions R 1 and R 2 with the same topology there is a one to one correspondence between the non local sectors [a] 1 and [a] 2 between R 1 and R 2 . This correspondence has two steps. First, any non local operator a for a region R is a non local operator for an homotopic region R ⊆R. Then, the tube of homotopy R 12 connecting R 1 and R 2 has the same topology, and includes R 1 and R 2 . Therefore non-local operators in either R 1 or R 2 give non-local operators in R 12 , and the classes can be matched.
A simple property is that given two arbitrary regions R 1 and R 2 , if R 1 is included in a topologically trivial region disjoint from R 2 , any non locally generated operators based in R 1 and R 2 must commute to each other. This follows from the assumption that non locally generated operators in a region R 1 become locally generated in a topologically trivial region containing R 1 . In this case, we say that R 1 and R 2 are not linked.
First, we refer to the upper panel in figure 3. A loop operator of class a (dashed curve) can be converted into the product of two loops of class a using operators that are additive (local) in the shaded region (see appendix B for the explicit construction in the lattice). This operation cannot change the class of each of the loops at the extremes. To prove that this should be possible consider the algebra of non-local operators in the two rings R 1 and R 2 . This is the tensor product of the algebras of non-local operators in R 1 with the ones in R 2 . It is not difficult to see that the original one-component loop of type a has the same action on this algebra as the product of the two independent loops. Then they belong to the same class and must be locally related. This is an important step in showing that the algebra is Abelian. The lower panel shows why this fails in the case of twist operators for non-Abelian global symmetries. 8 What in the previous case were two spatially separated rings is converted into four spatially separated balls It is no longer the case that the algebra of non-local operators in the four balls is the tensor product of the non-local operators (intertwiners) in B 1 B 2 with the ones in B 3 , B 4 . We can cross intertwiners between B 2 and B 3 for example. In the non-Abelian case, the twist on the left does not have the same action on this algebra as the product of two twists on the right.
We conclude we can glue and split loops associated with the same representation in this form. Now let us take a simple ring R as in figure 4. Inside the ring, we can place an elongated loop of class a, which is a folded version of the loop in the left upper panel of the figure (3). This is locally generated inside R since its topology can be shrunk inside R. If we glue the extremes as in the right-hand side of the figure we obtain two loops of conjugate classes inside R. These two loops must, therefore, be equivalent to the trivial class since they are locally generated. This gives [n] a aā = 0 , a = 1 .

(2.22)
These fusion rules can be seen to arise from an Abelian group. Let us see how this comes about. The product of classes is associative and commutative. We already have the unit and the inverse,  [n] a 3 a 1 a 2 can only be non zero for just one class [a 3 ]. This defines an Abelian group G a for the product of classes. The elements of the group are just the classes, which contain an inverse and an identity, and the product in the group is the product of classes. All this argument runs in the same way for the classes [b] of the non-local operators in R which for a group G b . Below we will show how to choose actual operators of the theory representing the abstract fusion of classes. In other words, we will find loop operators representing the Abelian symmetry group.
This argument does not hold in this generality for regions with non-trivial π 1 (R) in d = 3, as shown by the examples of global symmetries having non-Abelian groups discussed in the preceding section. The reason is, as explained above, that in d = 3 (two spatial dimensions) the operation of figure 3 does not hold in general. Still, for pure gauge theories in d = 3 the proof holds (see appendix B), and we have an Abelian group for the non-local sectors.
The same proof of Abelianity should work for sectors corresponding to regions with the topology of spheres S k for 0 < k < d − 2. The conclusion is that living aside the case of dimensions 0 and d−2, which include the case of global symmetries, in all other cases the product of a class [a] and its inverse [ā] is an operator that is locally generated on the appropriate region.
A slightly different chain of arguments is as follows. We can imagine we started with a different and bigger set S of sectors s. These abstract sectors could run for example over all the irreducible representations of a certain non-Abelian group, whether of discrete or Lie type, as it is the case of Wilson loops for non-Abelian gauge theories. To run the argument we only assume these sectors satisfy some generic notion of fusion rules s * s = s N s ss . (2.24) Here the fusion coefficients might be associated with a non-abelian symmetry group, or to a more general structure. We only ask the fusion algebra to be Abelian N s ss = N s s s , which follows from the locality principle in QFT.
But crucially, not all the sectors s ∈ S are non locally generated in the region R. All the sectors being produced in the fusion of arbitrary products of ss are locally generated, for the same reason as above. Let us call the set of sectors appearing in arbitrary products of ss by S 1 . By construction S 1 defines a subcategory of the category S. The true classes associated with the violation of Haag duality arise as the quotient of the whole set S by the sectors in S 1 . In the literature of tensor categories, see [36], this is called the universal grading of S, and the associated group the universal grading group. Grading of a category S by a group G is a partition of S of the form S = g∈G S g , (2.25) such that for any s g ∈ S g and any s h ∈ S h the product s g * s h belongs to S gh . The universal grading, as its name suggests, can always be found, and it is associated with S 1 being formed by arbitrary products of ss. For symmetric fusion rings, like the ones we are considering, the resulting universal braiding group G, shown to be associated with the breaking of Haag duality, is necessarily Abelian.
An analogous result holds for theories with k-form symmetries [8]. The proof of Abelianity in such work relies on the Euclidean continuation of the QFT, in particular the Euclidean continuation of the generators of the generalized global symmetry. Here we did not invoke a particular Hamiltonian and no relativistic symmetry was necessary for the argument. Abelian nature just follows from the physical requirement that the true non-trivial classes should be not locally generated. This directly forces us to consider the universal grading of the original fusion rules alluded above, which is necessarily an Abelian group.

Algebra of non local operators
We have shown that the classes of non locally generated operators in R form an Abelian group. We want to show we can take operator representatives of these classes providing the actual group operations. An Abelian group G is a product of cyclic subgroups Z n 1 ⊗ Z n 2 · · · . If we have operators with the cyclic subgroup law under product, then it is enough to take representatives for each of the factor cyclic subgroups in different spatially separated rings inside the region to get representatives for the full group.
Then, let C a = {[a k ]}, k = 0, · · · , n−1 be a cyclic factor of order n of the group G a associated to the classes [a] in R. Letã be a representative of [a], an actual operator in the theory. The unitary operatorâ =ã/ √ãã † =ã/|ã| belongs to the same class [a]. We haveâ n = U , with U unitary, commuting withâ, and U ∈ [1]. All the spectral projections of U belong to the algebra of locally generated operators and commute withâ. Using the spectral decomposition we can construct V = U −1/n by taking the n th root of the eigenvalues, with the same spectral projections. With these observations there are now many choices for V . Any of them will do the work. Define a = Vâ. We have a k belongs to the class [a k ], and a 0 = a n = 1. This gives the operator representation a k−1 of the group element associated to the cyclic subgroup C a . The same can be done for the other cyclic subgroups in R and also for the operators b ∈ [b] with the group operation laws of G b for the dual classes [b] in R .
Having constructed the operator representations of the symmetry, consider the unitary transformation x → bxb −1 . It maps A add (R ) into itself, since for x in this additive algebra bxb −1 is in the identity class. Then it also maps its commutant A max (R) = A(R) ∨ a into itself. We also observe that this automorphism of A max (R) does not depend on the precise choice of representatives b. This is because any other choice arises from b as products of locally generated operators in R , and these operators commute with all x ∈ A max (R).
It will be more useful to define the following maps of A max (R), associated to each irreducible representation of G b , The third equation just follows by direct evaluation.
A not so transparent property of the previous map is that E r (a) ∈ [a]. The reason is that we can imagine to choose representatives a of G a in R, such that they are actually supported in a smaller ringR ⊂ R. Then we can move the b operators inside R but outsideR. The map is then composed by locally generated operators in R, which cannot change the class [a]. Finally, from the last equation in (2.26) it is clear that for E r (a) = 0 we have E r (a) = 0 for r = r .
The previous observations imply there is a one to one correspondence of representations of G b with non local classes [a]. It has to be one to one since otherwise there would be linear combinations of elements of different classes which vanish or the b operators would not be linearly independent. Therefore we can label the representations of G b by the label a such that E a (a) ∈ [a] = 0. Further, we can show that E a (a 1 ) = a 1 , for any a 1 of [a]. First defineã = E a (a) for which E(ã) =ã andã ∈ [a]. Now, any element a 1 of [a] can be written by takingã and multiplying by arbitrary products of locally generated operators. Therefore Basically, the intuition is that the previous map is a projection of A max (R) into its different classes [a]. In the context of Von Neumann algebras, projections are often associated to conditional expectations, which we will describe below in detail. In this case, E a is not a conditional expectation since the target space is actually not an algebra because the non trivial classes [a] = [1] do not contain the identity by construction. The map is better seen as a projection in a vector space.
In any case, using both (2.26) and (2.28) it follows that Finally, in order to construct a maximal causal net satisfying duality, we have to take subsets of dual operators {a}, {b}, such that they satisfy causality and close under fusion. This is equivalent to take maximal sets of pairs of non local operators M = {(a i , b j )} such that (2.31) These maximal causal nets were called Haag-Dirac nets in the introduction above exactly for this reason. The generalized Dirac quantization condition χ a (b) = 1 arises from the algebraic perspective by requiring Haag duality and causality.
To summarize, we conclude that the number of elements in {a} and {b} is the same. Besides, {a} is the group of characters of {b}, or, equivalently, the other way around. The dual Abelian groups arising from the breaking of Haag duality are Pontryagin duals of each other. The commutation relations are fixed to be (2.29), and the phases χ a (b) in this relation form the table of characters of the symmetry group. The Dirac quantization condition arises by enforcing causality of the net. Remarkably, these features are simply inescapable consequences of the violation of Haag duality for regions with non-trivial π 1 and π d−3 in local QFT. In particular, we have not defined the dual operators, say the b's, by their commutation relations with the a's, as it is usually done since 't Hooft's original work [23]. Also, we have not assumed any symmetry group structure and charged operators to start with.

Standard non local operators
Interestingly, given a region R with non-local operators, there is a standard way to obtain representatives of the non-local operators. The construction generalizes the Doplicher-Longo construction of standard twists [33,34]. 9 These standard operators are uniquely defined by the condition where J R is the vacuum Tomita-Takesaki reflection corresponding to A add (R).
The existence of these operators is a simple consequence of a theorem that states that any automorphism of a von Neumann algebra with a cyclic and separating vector is implementable by a unitary operator, and one can choose the unitary to be invariant under the conjugation J (see [7] theorem 2.2.4). In the present case, the algebra is (A add (R)) , the automorphism is the one induced by the non-local operators of type a (which is independent of the representative), and the vector state is the vacuum. Then we get a unitary a invariant under the modular conjugation of (A add (R)) which is the same as the modular conjugation of A add (R), and hence (2.32). By construction, the algebra of the standard operators a and b is the expected one. By the same reason a belongs to (A add (R )) but not to A add (R), and it is a non local operator in R.
Further interesting properties follow from the fact that the standard operator leaves the natural cone P of vectors invariant. This cone is defined as generated by all vectors of the form OJO|0 for O in the algebra [7]. The important point here is that vectors in the natural cone include the vacuum and have a positive scalar product. If follows that a i |0 ∈ P and 0|a i |0 > 0. This last equation also entails 0|a i a j |0 > 0.
This interesting construction gives, for example, standard non-local Wilson and t' Hoof loops defined exclusively by the vacuum and the geometry of the chosen region. In particular, they enjoy all the symmetries that these regions and the vacuum may have.

Maxwell field
A simple example of these scenarios is the Maxwell field in d = 4. This is the Gaussian theory of the electric and magnetic fields, with equal time commutation relations Equivalently, the theory can be described by the normal oriented electric and magnetic fluxes Φ E , Φ B on two-dimensional surfaces with boundaries Γ E and Γ B . For such fluxes, we have a commutator proportional to the linking number of Γ E and Γ B , (2.34) We will always assume these fluxes to be smeared over positions of Γ E and Γ B such that the flux operators are well defined (and not operator-valued distributions). If the smearing region for Γ E and Γ B lies inside a region with the topology of a ring R and its complement R respectively, and the integral of the smearing function adds up to one (which we will also assume in the following), eq. (2.34) still holds for the smeared fluxes. In d = 4 the topology of R is the same as the topology of R. It is S 1 × R 2 and it has non-trivial π 1 (R).
Because ∇E = ∇B = 0 the fluxes are conserved, and then the surface over which they are computed can be deformed keeping the boundary fixed. By deforming the surface of the flux we can take it away from some local operator lying in the original surface, and this implies that the fluxes will commute with the locally generated operators associated with the complementary ring.
We can write a bounded electric flux operator (t' Hooft loop) T g = e igΦ E , and a magnetic operator (Wilson loop) W e = e iqΦ B , for any g, q. The commutation relations for these simply linked loops follows from (2.34) This non-commutativity implies these operators cannot be locally generated in the ring in which they are based. For example, if T g were locally generated in R (where its boundary lies) this would imply, by the arguments given above, it necessarily commutes with W q based on the complementary ring. But this is not possible according to (2.35). Notice this is an explicit example of relation (2.29).
Therefore, the algebra of a ring R and its complement R (also a ring) cannot be taken additive without violating duality. The reason is that the commutant of the additive algebra of the ring contains both the electric and magnetic loops of any charge based on R , and this is not additive. We have and analogously by interchanging R ↔ R . Here we have written W q R , T g R for Wilson and t' Hooft loops based on R .
One can repair duality at the expense of additivity by defining the algebras for rings to contain, on top of locally generated operators, some particular non locally generated ones that commute with other selected non locally generated operators in the complement. A natural condition is to select operators with electric and magnetic charges (q, g), which are the same for any ring, such that our choice does not ruin translation and rotation invariance. Given two dyons (q, g), (q , g ) in the same ring, the one formed by their product, (q + q , g + g ), and the conjugate (−q, −g), should also be present to close an algebra. Therefore the set of all dyons should be an additive subgroup of the plane, giving a lattice (q, g) = n(q 1 , g 1 ) + m(q 2 , g 2 ) , (2.37) where n, m ∈ Z, and (q 1 , g 1 ) and (q 2 , g 2 ) are the generating vectors of the lattice. Locality between a would be "dyon" (q, g) in R and another one (q,g) in R (i.e. the vanishing of the phase in (2.35)) gives the Dirac quantization condition 10 for an integer k. This is compatible with (2.37) provided q 1 g 2 − g 1 q 2 ∈ 2πZ. If we want to construct a Haag-Dirac net we need to take a maximal set of charges that satisfy (2.38). This forces us to choose q 1 g 2 − g 1 q 2 = 2π . (2.39) This is the most general condition for a U (1) symmetry. But for the case of the relativistic Maxwell field, in solving for the space of solutions of the previous equation we need to take into account that there is a duality symmetry (see for example [4]) Then, there is a hidden free parameter in the solution of (2.39) that moves us between isomorphic Haag-Dirac nets. This freedom can be eliminated by writing the different solutions as where q 0 and θ ∈ [0, 2π) are parameters, g 0 = 2π/q 0 , and n e , n m are integer numbers. Writing the two real parameters as a single complex one τ = θ/(2π) + 2πi/q 2 0 , the Haag-Dirac nets verifying duality and causality are determined by this parameter, A HD (τ ). In this parametrization there is a residual duality symmetry, since nets with τ = τ + 1, τ = −1/τ are isomorphic.
Nets with θ = 0, π are not time reflection symmetric. Notice that in a specific model describing electric charges and monopoles, adding a topological θ term to the Lagrangian, or equivalently considering the θ vacua, the lattice of charges is changed according to the Witten effect [37]. We see such effect here as arising from the previous freedom we encountered in describing the lattice of charges instead.
The nets constructed in this way will satisfy duality, but of course, they are not additive. Additivity can be recovered if we couple the theory to charged fields. For example, if we have a field ψ of electric charge q we can now consider Wilson line operators of the form Taking products of consecutive Wilson lines, and allowing for the fusion of the fields with an opposite charge at the extremes of the lines we want to join, the Wilson loop W q in R (with the specific charge q), becomes an operator in the additive algebra of R. In the same way, if we have magnetic charges g, T g corresponding to this charge should be additive in R, and with a dyon (q, g) we can break the operators T g W q . For the theory to still satisfy locality the charges have to satisfy (2.38). This is now converted into the Dirac-Schwinger-Zwanziger (DSZ) quantization condition for the charges. As mentioned before, this condition is seen here as a consequence of causality in the net of algebras for the theory without charges. In this way, by adding a full set of charged fields with charges corresponding to a HD net, we can make the theory "complete", in the sense of both satisfying duality and additivity. If we do not add charged operators for a full lattice, there will still be some problems of algebras and regions, which can be studied by taking a quotient by the new locally generated loops.
Let us close this section with an important remark. In the presence of charged fields, the flux operators T g W q continue to exist even if (q, g) does not belong to the lattice. But they now depend on a surface rather than a closed curve. Since ∇E = ∇B = 0 is modified, the fluxes on a surface cannot be deformed to other surfaces with the same boundary. Then, in this scenario, the operator belongs to a topologically trivial region and cannot be associated with a ring.

Non Abelian Lie groups
In this section, we consider the case of non-Abelian Lie groups, whose features can be described directly in the continuum limit. 11 We start with pure gauge theories, without charged matter.
Later we will consider the effect of adding matter. The objective is again to understand the failure of duality and additivity for these theories.
For generic gauge theories, as for the Maxwell field, the set of gauge-invariant non-local operators, with the potential of being non additively generated, is given by the Wilson and t' Hooft loops [23,38,39]. Although Wilson loops are one dimensional in all dimensions, 't Hooft loops are only one dimensional in four dimensions, where they were originally defined. In other dimensions t' Hooft operators are defined in d − 3 dimensions, see Appendix (B) for the explicit construction. We conclude that Wilson loops are the right candidates to violate additivity in regions with non-trivial π 1 , while the dual 't Hooft operators are the right candidates to violate additivity in regions with non-trivial π d−3 . In d = 4 both operators contribute to the violation of duality in ring-shaped regions.
Let us start with the Wilson loops. These are defined for each representation r as where C is a loop in space-time and P the path ordering. As shown in Appendix (B), they can be chosen in order to satisfy the fusion rules of the representations of the gauge group Therefore, there is one independent Wilson loop per irreducible representation of the gauge group.
We now want to find whether Wilson loops are unbreakable or not. A Wilson loop of representation r could be certainly broken in pieces if there are charged fields φ r transforming in representation r. With this charged field we can construct Wilson lines where we assume the field φ r transforms according to representation r. These lines decompose the Wilson loop into a product of operators localized in segments. Although we are considering pure gauge theories without charges, we cannot escape the fact that, for non-Abelian gauge fields, the gluons are charged themselves. They are charged under the adjoint representation. Indeed we can form the following Wilson line, terminated by curvatures, where all fields are in the adjoint representation of the Lie algebra. We conclude that a loop in the adjoint representation can be generated locally by multiplying several of these lines along a loop.
Since the adjoint Wilson loop is locally generated, the same can be said for all representations generated in the fusion of an arbitrary number of adjoint representations. Therefore, the truly non-local Wilson loops, those violating Haag duality, are labeled by the equivalence classes that arise when we quotient the set of irreducible representations by the set of representations generated from the adjoint. 12 To understand in precise terms what we mean by the last statement we need to invoke several notions from the theory of representations of Lie groups. Since introducing and describing them in detail would take some time and space, and it will certainly interrupt the flow of the presentation, we will assume here knowledge of such topic, and refer to the references [40][41][42][43][44][45][46][47] for more details. For the present context, the most important notions we need are the weight and root lattices. For a Lie algebra g, a Cartan subalgebra h is a maximal Abelian subalgebra. If h is generated by l elements, the Lie algebra is said to have rank l. Since h is Abelian, it can be diagonalized in every irreducible representation of the algebra. A weight associated with certain eigenvector in certain irrep is defined as the l-component vector formed by the eigenvalues of the Cartan subalgebra generators. It turns out that the weights form a lattice generated by arbitrary linear combinations with integer coefficients of a set of fundamental weights ω (i) . The number of fundamental weights is equal to the rank. Physically, this lattice contains the information of all the representations of the algebra. In this lattice, each irreducible representation is labeled by a dominant weight. In the weight lattice, such dominant weights are in one to one correspondence with orbits of the Weyl group, so that These equivalence classes label all the irreducible representations, and therefore all the Wilson loops. Now, for every Lie group, there is a universal representation called the adjoint representation. It is the representations in which the Lie algebra transforms itself. The weights of the adjoint representation are called roots. The roots also form a lattice, called the root lattice It is generated from a set of l fundamental roots α (i) . Physically, while the weight lattice contains all possible weights, and therefore all weights appearing in arbitrary products of fundamental representations, the root lattice contains all weights appearing in arbitrary products of the adjoint representation.
The dominant weights appearing in the root lattice can be isolated in the same way as before, employing the Weyl group The non locally generated classes of Wilson loops are then labeled by where Λ Z is equivalent to Z * , the group of representations of the center Z of G. These representations form the dual of the Abelian group Z, which is isomorphic with Z. 13 One can construct actual representatives of such non-additive classes using the generic construction described in the previous section. We conclude we can find non additively generated operators in a ring that satisfy the algebra of the characters of the center of the group. This quotient is an example of the universal grading alluded to in the previous section.
An exactly similar discussion goes for t' Hooft loops, when one starts with the dual description in terms of the dual GNO group [39,48]. Then, this results in a non additively generated t' Hooft loop (violating duality for regions with non-trivial π d−3 ) per element of the center of the gauge group, as originally defined in [23]. Construction of such non-additive 't Hooft loops that does not use the dual description is provided in Appendix (B). Such construction also allows constructing the non-additive Wilson loops using the dual group. One can also start by labeling the t' Hooft loops with the equivalence classes of the gauge group. These equivalence classes are in one to one correspondence with orbits of the Cartan subalgebra under the Weyl group [40], as it is the case for magnetic monopoles [48]. But again, labeled in this way, not all 't Hooft loops are non-locally generated.
We conclude that the physical symmetry group violating Haag duality in pure gauge theories is Z * ×Z, where Z * is generated by the non-breakable Wilson loops and Z by the non-breakable 't Hooft loops. We thus find the algebraic origin of the generalized global symmetries described in [8]. Haag-Dirac nets can now be constructed by enforcing causality of the net. These conditions were studied in [49] and the lattices found there are seen here as isomorphic HD nets. 14 Finally, let us mention how these features change with the inclusion of matter. For d > 4 matter fields will break non-local operators only if they are charged for the center Z of the group (electrically charged fields) or the dual Z * of the center (magnetically charges fields). Let us call M e ⊆ Z * and M m ⊆ Z to the electric and magnetic charges with respect to Z and Z * respectively. These can fuse and then M e , M m are subgroups. These charges have to satisfy the generalized Dirac quantization condition (2.31) by causality. All operators in Z which do not commute with M e cannot be considered any more operators in a ring, and are now just operators that exist in balls. Then the remaining t 'Hooft loops in the ring are given by (Z * /M e ) * ⊆ Z, which is a true subgroup of Z. M m is included in this subgroup by the Dirac quantization condition, and the loops in M m are now locally generated, broken by magnetic charges. Then the remaining non locally generated t' Hooft loops in the ring are given by (Z * /M e ) * /M m . Analogously the non local Wilson loops will be (Z/M m ) * /M e . In a complete theory, these isomorphic groups should be trivial. For d = 4 the group of non locally generated operators for a pure gauge theory is Z × Z * , which is now naturally isomorphic to Z * × Z. Let the subgroup of dyons be D ⊆ Z × Z * and its isomorphic image D * ⊆ Z * × Z. We have the group of non-local operators given by ((Z * × Z)/D * ) * /D.

Generalizations
The same arguments apply for QFT's in which the violation of Haag duality appears for regions R with non-trivial π n and regions R with non-trivial π d−n−2 . There are special instances that need to be taken with special care. But in general, a violation of duality due to certain charges which can be localized in regions with non-trivial π n , for n ≥ 1, d−n−2 ≥ 1, is such that it gives rise to an Abelian group. The reason is the same as before. A region with such properties is connected. Therefore, if an operator of certain representation r (with such topological properties) can be created inside a ball, then the representation rr * can be generated additively inside the region. This implies that the set of non-additively generated operators corresponds to the universal grading of the associated tensor category. Such grading gives rise to an Abelian group.
Examples of these types of symmetries should come from p-form gauge fields A µ 1 µ 2 ··· , but we will not consider explicit examples in this paper. This construction again connects with the generalized global symmetries described in [8]. We remark though, that the Abelianity of the sectors we have discussed is rooted in the analysis of Haag duality, and it can be proven without the necessity of going to Euclidean space.
Among the zoo of possible situations, there are some special ones in which R and R are "ring" like regions sharing the same topology. This occurs for n = (d − 2)/2, in which both regions have non trivial π (d−2)/2 . This possibility only appears for even dimensions. In particular, for d = 4 we have both Wilson and 't Hooft loops violating duality of the ring R and its complement R , which is also a ring. In this case, the groups G a and G b of complementary (simple laced) regions are not only dual to each other but there is also a natural isomorphism arising from transporting the non-local operators from R to R by deformations. These situations have the additional interest that for special geometries one can construct conformal transformations mapping the complementary regions, as we further discuss below.
It is an interesting program to understand what kind of non-local algebras could appear more generally in different topologies, under some simple assumptions such as that the algebras do not have centers and the sectors are homotopically transportable. For example, regions with knots would not be necessarily equivalent to other topologically equivalent ones without them. This general analysis may reveal interesting new cases depending on the assumptions.
A different simple example that is not covered by ordinary gauge theory is the case of higher helicity fields. The free (linearised) graviton is described by a field h µν with gauge invariance h µν → h µν +∂ µ ξ ν +∂ ν ξ µ . Gauge invariant operators are generated by the curvature tensor R αβγδ . This is conserved in all indices. This conservation should give rise to flux operators across twodimensional surfaces which are non locally generated operators on the one-dimensional boundary. However, in contrast to the gauge theories described above, the non-local operators are indexed with space-time indices. We might anticipate from this observation a breaking of Lorentz symmetry for a HD net.

Entropic order parameters
We now use the lessons we have learned about additivity and duality in QFT's to construct entropic order parameters that capture the physics of generalized symmetries. In other words, we seek to find natural entropic order parameters that can distinguish, from a unified perspective, the different phases of QFT's. For example, we seek entropic order parameters capturing the essence of the confinement, Higgs, and massless phases in quantum gauge theories.
Taking as a starting motivation the confinement phase, it is well known that the Wilson loop of a fundamental representation was initially devised as an order parameter for it [38]. The expectation value of such a fundamental Wilson loop can decay exponentially fast with the area of the loop. This behavior is indicative of confinement since it implies a linear quark-antiquark potential. On the other hand, a perimeter law scaling of the Wilson loop excludes the possibility of confinement, at least at large distances.
However, in theories such as QCD, whose matter content includes charged fields in the fundamental representation, the Wilson loop has a perimeter law even if quarks are confined. Moreover, even in the absence of charged matter fields, the same holds for the Wilson loops in the adjoint representation. It seems no coincidence that these two examples concern precisely line operators that are locally generated in the ring.
These observations trigger the following discussion. The right order parameters in QFT's, characterizing the phase of some generalized symmetry, should be the appropriate non-additive operators discussed in the previous section. These are the operators that violate Haag duality in the appropriate region. In turn, right entropic order parameters are those able to capture the physics of such non-additive operators. The objective of this section is to build on this idea, define the right entropic order parameters, and study them in different phases of different systems.
We start by setting the idea that non-additive operators are the right order parameters on firmer ground. To do so we argue that for any general QFT it is not possible to construct a loop order parameter displaying an area law by employing only operators that are locally generated in the ring. We can wave only a sub-perimeter law behavior (perimeter law, or even a constant law). This implies that the existence of a confinement order parameter requires a non locally generated operator and the failure of the additivity property for ring-like regions.
Associated with this failure of additivity, and as discussed in the previous section, there will be multiple choices of nets of algebras. We will use this multiplicity to define natural "blind" entropic order parameters, which do not rely on a particular operator, but just on the algebraic structure of the net of algebras. We will show that such entropic order parameters can be defined both for order parameters, such as intertwiners and Wilson loops and for disorder parameters, such as twists and 't Hooft loops. It turns out that both perspectives, order vs disorder, are related through the entropic certainty relation [22].
We will finally use all these tools to analyze different known phases in QFT's, such as spontaneous symmetry breaking scenarios, Higgs and confinement phases, and conformal ones as well.

An area law needs non locally generated operators
Let us first recall that the exponential decay of the expectation value of a (appropriately smeared) line operator with size is always bounded from below by an area law [51]. To explain this we refer to Figure 5, which shows four rectangular loop type operators. These are formed by products of two half-loops (labelled 1 and 2) reaching just to a plane of reflection, and their reflected CRT images (labelled1 and2 respectively). The application of reflection positivity in the Euclidean version, or CRT positivity in real time, 15 leads to , with x, y the two sides of the rectangle, it follows from this relation, and the analogous one in the y axis, that the potential V (x, y) must be concave Figure 5: The construction of Bachas [51] that shows the convexity of the quar-antiquark potential.
Then, the slopes ∂ x V (x, y), ∂ y V (x, y) never increase. They will converge to a fix value in the limit of large size. If these values are non zero we have an area law. If they are zero we have a sub-area law behaviour. No loop operator expectation value can go to zero faster than an area law ∼ e −c A as the size tends to infinity. This calculation holds for any loop, whether locally or non locally generated in the ring, provided they are locally generated in the plane. The derivation can be justified more rigorously in a lattice model [51]. Now we think only in loop operators formed additively in a ring. It is more convenient to use circular loops for our present purposes. As the loops are locally generated we can imagine forming a partial operator W (l 1 , l 2 ) in an arc (l 1 , l 2 ) of the ring of longitudinal size l = l 2 − l 1 . The idea is that we construct now a loop of a certain size not by increasing the size of a smaller loop as above, but by increasing the size of an operator in an arc until the arc closes into a ring.
Assume rotational invariance and define the potential We can use CRT positivity again in this case as shown in figure 6. The result is Therefore the slope of V (l) is non increasing and If the loops are formed as products of small pieces in a rotationally symmetric way we can form loops of larger radius starting with the same cross-section. For such a sequence of loops of different radius, we have the same V (0), independently of the radius. Eq. (3.5) gives a perimeter law, or more precisely a sub-perimeter law behavior. In particular, this excludes the possibility of an area law or any law increasing more than linearly in the perimeter.
The application of the same idea to the case of non locally generated loop operators in the ring fails. The reason is that we cannot define the partial, non-closed, line operators. Using a non-gauge-invariant Wilson line introduces several problems when some gauge fixing is chosen. If we do not gauge fix, the expectation value of this line is zero, and the potential infinity. This prevents the calculation to give any useful bound.
Analogous results are expected to hold for spherical shells of different dimensions k. General operators should have an area or sub-area law behavior V R k+1 . The argument goes along the same lines as for (3.2). Besides, additive operators will not be able to display "area law" (V ∼ R k+1 ). Their expectation values will be restricted to have sub-perimeter law behavior (V R k ), and they would not be appropriate order parameters. The argument leading to this statement should parallel the one deriving (3.5). In appendix C we make some remarks about how these arguments could be made mathematically precise.

Definition and general properties of entropic order parameters
Let's move now towards constructing sensible entropic order parameters signaling the presence or absence of generalized symmetries in QFT's. As discussed until now, these generalized symmetries manifest themselves through the violation of duality and additivity for different regions with different topologies. In all the cases considered, these violations have been related to certain non-additive "order" operators a (such as intertwiners, Wilson loops or high dimensional generalizations) and certain non-additive disorder operators b (such as twists and 't Hooft loops). These operators close specific algebras, typically given by fusion algebras associated to a certain group of generalized symmetries.
Since both, order and disorder operators, when properly chosen, generate self consistent von Neumann algebras {a}, {b}, the first obvious information theoretic notion coming to mind is the Von Neumann entropy where for a given algebra M, the symbol tr M means to take the canonical trace associated to it [54]. Although we will implicitly study these quantities as well, it turns out there are better suited entropic order parameters for the characterization of symmetries. In particular, notice that the quantities (3.6) have an unpleasant dependence on the choice of non local operators.
To motivate the new order parameters, we follow the logic described in [22]. Let us first remind that for a finite d-dimensional Hilbert space, the Von Neumann entropy can be written equivalently as where τ = 1/d is the maximally mixed density matrix and S M (ω | τ ) is a quantity known as relative entropy. The relative entropy is defined for two quantum states, and in these finite dimensional scenarios it is defined by Intuitively, the relative entropy measures the distinguishability between the two input states. Relation (3.7) expresses that the uncertainty measured by the Von Neumann entropy is measured as well by the distance between the state and the state with maximal uncertainty. Since there is a minus sign in the previous relation, the higher the relative entropy, the smaller the uncertainty on M.
Using relative entropy is better for several reasons. First, relative entropy displays monotonicity under general quantum channels and restrictions onto subalgebras [54]. This property will be used in the applications. Second, relative entropy is well-defined across different types of algebras, including the type III von Neumann algebras appearing in QFT. This will ease the application to QFT, as it avoids many potential issues just from the start.
Finally, using relative entropy suggests certain generalizations. Notice that in (3.7), the maximally mixed state τ can be equivalently written as the composition of ω with a map ε : M → 1, defined by ε(m) := 1 d Tr(m)1. Rewriting the relative entropy as suggests a couple of generalizations to this notion of uncertainty. First, the map ε : M → 1 is one example of a whole space of such maps, as we describe below. Second, instead of the identity as the target algebra, we could choose any subalgebra N ⊂ M. The maps ε : M → N are called conditional expectations [54,55]. They are positive, linear, and unital maps from an algebra M to a subalgebra N . They leave the target algebra invariant and they further satisfy the following bimodule property These maps are the mathematical definition of what restricting our observational abilities means, see [54] for an extensive review.
Examples are tracing out part of the system 11) or retaining the neutral part of a subalgebra under the action of a certain symmetry group From a general standpoint, if M = N ∨ Q is the algebra generated by N and certain algebra Q, we say the conditional expectation "kills" Q.
An important further property of these maps, that we will use continuously, is that they can be used to lift a state in N to a state in M, The generalization we are seeking for is thus If M = N ∨ Q, this quantity measures the uncertainty of Q in the state ω, given the knowledge of N . The fact that side correlations with an algebra N are taken into account in this quantity will be very important for the QFT applications.
We now define the order-disorder entropic parameters in the following manner. If the algebra of the non additive operators a lives in a certain region R, this provides us with a natural inclusion of algebras Associated to this inclusion we should have a space of conditional expectations leading to the following entropic order parameter This entropic order parameter was considered in [20] for the case of global symmetries, inspired by ideas in [21].
A parallel story works for the disorder parameters b. We remind they live in the complementary region R , and they provide us with the following inclusion of algebras with its associated space of conditional expectations and the following entropic disorder parameter An important difference between these order parameters versus the Von Neumann entropies (3.6) is that these relative entropies are purely geometric objects depending only on R. The other difference is that they include the side correlations between the order-disorder operators with the appropriate additive algebras. This turns out to be important, as we now describe.
The order-disorder algebras do not commute between themselves. These commutation relations are completely fixed, as shown in Eq. (2.21). These commutation relations imply fundamental uncertainty principle type bounds between the two algebras. Such implications arising from quantum complementarity can be accommodated in the entropic formulation. This problem was considered in detail in [22], inspired by a result in [20]. To analyze it, we notice there is a natural way to understand quantum complementarity in this context. Given a generic inclusion of algebras N ⊆ M, and the space of conditional expectations between them ε : M → N , there is a natural complementarity diagram In this diagram, going vertically takes the algebras M and N to its commutants M and N respectively. Going horizontally in the arrow direction means restricting to the target subalgebra. If M = N ∨ Q and N = M ∨Q then ε kills Q ⊂ M, and the dual conditional expectation ε killsQ ⊂ N .
Notice that while N commutes with M , the algebras M and N do not commute with each other. The only operators which do not commute with each other are the ones in Q andQ, the ones killed by the appropriate conditional expectations. These algebras Q andQ are called complementary observable algebras (COA), see [22]. They generalize the notion of complementary operators to operator algebras.
As a simple example, take M as the Abelian algebra X generated by the position operator. Then choose a conditional expectation that kills the full M = Q = X . In other words ε : X → 1. The complementarity diagram becomes in this case As expected, we conclude that the COA of X is P, the algebra generated by the momentum operator, and viceversa.
The case of interest to us, and that will be a recurring theme in the following sections, concerns the one associated to order-disorder parameters in QFT. This is Let's now continue with the general case (3.21). Associated to such diagram we have an entropic order parameter for the upper side, namely S M (ω|ω • ε), and one entropic order parameter for the lower side, namely S N (ω|ω • ε ). In [22] the following relation between those was derived λ is a certain fixed number called the algebraic index of the conditional expectation ε, which is equal to that of the dual conditional expectation ε .
In the examples of this paper λ will be log |G|, for a group G. This relation was called entropic certainty relation in [20], where it was first derived for the case of global symmetry groups. The original references defining the algebraic index are [56][57][58]. For the study of the index in a generic inclusion of finite-dimensional algebras see [59,60].
To prove such a relation, a fundamental step is to understand the space of conditional expectations ε in a generic inclusion of algebras N ⊂ M. The study of such space has been carried out in different scenarios. To our knowledge, the first references studying it were [61][62][63][64]. In the context of the inclusion of factors in type III algebras, it was considered in [58]. In [22] it was recently analyzed from a somewhat more physical perspective. Intuitively the result is the following. Let us denote the space of conditional expectations from M to N as C(M, N ). Then, if the target algebra N has a center spanned by projectors P N j , then any ε ∈ C(M, N ) is of the form where ε j ∈ C(M j , N j ). But now the inclusion N j ⊂ M j contains no center in the target algebra, and one can prove that for such inclusions, space C(M j , N j ) is isomorphic to the space of states in the relative commutant M j ∩ N j . Notice that if the original target algebra is a factor, so that it has a trivial center, and the relative commutant is also trivial, then space C(M j , N j ) contains only one element. This is typically the case in continuum QFT. 16 It can also be proven [61][62][63][64] that there always exists a conditional expectation preserving the trace. In this case tr(m) = tr( (m)) , For these trace preserving conditional expectations, in the finite dimensional case, it was proven in [20] that where in the right hand side we have von Neumann entropies. An important example of trace perserving conditional expectations are group averages. They will play a role below, although the framework is more general. The relative entropy S M (ω | ω • ε), as an entropic order parameter, is a well-defined version of the subtraction of two cutoff entropies. It also teaches us that this subtraction is monotonic with the region. Related to this same monotonicity, the continuum limit in a cutoff theory is independent of the details of the cutoff.
Finally, applying the generic certainty relation (3.24) to the case of entropic order-disorder parameters, characterized by the diagram (3.23), we obtain (3.28) From this expression and the positivity of relative entropy we obtain the individual bounds Also, from the entropic certainty relation (3.28), other bounds can be obtained using monotonicity of relative entropy under quantum channels or algebra restrictions.
The way the certainty relation is realized is easily guessed in certain limits. If the expectation values of the a operators tend to zero, for example when the region R is very thin, the state with and without the conditional expectation will not be easily distinguished, and the order parameter goes to zero while the dual disorder one saturates the bound Analogously, when the expectation values of b tend to zero we have Interesting physical information about the phase of the theory can be learned from the geometric setup in which these limits are achieved, and from the subleading terms in these expressions.
Summarizing, symmetries are associated with the appearance of two different algebras for the same region, the additive algebra A add and the maximal algebra A max . Entropic order parameters of these symmetries are then naturally suggested from the fact that two different states can be produced out of the vacuum for the same algebra. The relative entropy between these states further satisfies a surprising relation that ties the statistics of complementary dual nonlocal operators. Another natural geometrical order parameter would be produced by expectation values of the standard non-local operators described in section 2.2.3. Indeed, the definition of these operators also uses the full algebra in the region R. We will not study these operators further in the present paper.

Improving bounds by including additive operators
In the previous section, we have defined entropic order parameters associated with generic regions R and their complements R . The relative entropy is a measure of distinguishability and here we essentially compare the vacuum on the additive and non-additive algebras. If we restrict attention to just one set of non-local operators the relative entropy is smaller than the optimal, and the certainty relation is not saturated. Including the information on the additive algebra, or, equivalently, on the multiplicity of equivalent non-local operators in R, should improve the bounds obtained from (3.28). Before treating specific QFT examples, in this section, we want to get more intuition about these features. We will see how including larger and larger sets of additive operators improves the results of the computations characteristically.
To motivate the calculation below, consider the order parameter for the case of a global symmetry. We have a region formed by two single component regions as in figure 7, and consider the case in which these regions are very near each other. In the limit in which they touch each other, the twist is squeezed between the two regions and its expectation value goes to zero. Then, we expect the relative entropy over the intertwiners to tend to the maximal value log |G| [20]. We want to have a handle on how this limit is approached. To produce a lower bound to this relative entropy, we can compute it in the algebra of any of the intertwiners drawn along the surface in figure 7. These, however, will have some specific expectation values which may differ significantly from the maximal one I = 1. Large expectation values ∼ 1 are needed to achieve a maximal relative entropy log |G|. To find such improved "fat" intertwiner may be a complex task. The idea is then to use an algebra of many intertwiners along the surface, that can be taken to be uncorrelated to each other in good approximation, to improve the bound. We remark that we are not enlarging the number of independent intertwiners. That is indeed impossible. There is only one independent intertwiner per irreducible representation of the symmetry group. In other words, all the "new" intertwiners we are adding into the game can be written by multiplying the original one by additive operators in the two regions. It is the contribution of the additive operators which greatly improves the result of the computation.
More concretely, let us take dual sets of non-local operators belonging to dual Abelian groups G a and G b . They obey bab −1 = χ a (b)a. The corresponding conditional expectation killing the operators a is E(x) = |G| −1 b bxb −1 . We are interested in understanding the sub-leading terms in the approach of the relative entropy S A add ∨{a} (R)(ω|ω • E) to the saturation limit log |G|.
With the knowledge of the expectation values of the a operators, we can produce a lower bound just restricting the calculation of the relative entropy to this algebra. As explained in more detail in the next section, it is convenient to use the orthogonal projectors Now suppose we have in A add ∨ {a} a series of N commuting operators a i , i = 1, · · · , N , and assume they have identically distributed uncorrelated expectation values. That is, we have the Abelian algebra G ⊗N a generated by the projectors P i b with Since h −1 P b h = P hb 1 , the state ω • E in this algebra is given by the mixture of N states ω  We want to understand the relative entropy in the limit of large N . We can reason as in the operational interpretation of the classical Shannon relative entropy [65]. Let us call β = P 1 b 1 · · · P N b N , to a generic sequence of projectors labelled by the sequence {b 1 , · · · , b N }. The state ω has a probability distribution that is highly picked around the set of projectors where β contains each P b a number ∼ p b N of times. Let us call β p to this set of projectors where the fraction of each P b is determined by the probability distribution p(b). According to Shannon's theorem, this set of projectors form a fraction ∼ e N H(p) of the total number e N log |G| of projectors and have all the same probability ω(β p ) ∼ e −N H(p) . On the other hand, consider the set of projectors β q corresponding to sequences having a different fraction of elementary projectors determined by the probability distribution q(b). These are ∼ e N H(q) projectors, and have a probability where H(q, p) is the classical relative entropy between the "one particle" probability distributions [65]. Then the probability of the β q sequences is exponentially suppressed with respect to β p by the relative entropy between these distributions. As a consequence, of all states in the mixture (3.34) only the one with h = 1 will have significant overlap with ω and we expect that the relative entropy converges to maximum values log |G| exponentially fast.
To see this in more detail we have to compute (3.37) For h ∈ G b and p a probability distribution over the group elements, we call hp to the distribution hp(b) = p(hb). Separating the sum in the different probability distributions q we have This already tells us that assuming hp = p for all h we will have an exponential decay with N . If hp = p for some h, the a operator expectation values do not break all the b symmetries, and cannot provide the maximum value log |G| asymptotically. We exclude this case.
Evidently, in the large N limit, for each q, the sum inside the logarithm will be dominated by a particular h such that H(q, hp) = H(h −1 q, p) is minimal. Replacing this sum by the best h, the saddle point approximation gives This distribution satisfies H(q (p,h) , p) = H(q (p,h) , hp). Therefore we have If we have many decoupled algebras {a i } with different expectation values we expect a sum over the single classical relative entropies in the exponent, Let us check the previous calculation in the simplest scenario of G b = Z 2 . In this case we have only one a = 1 and the projectors are P ± = (1 ± a)/2. We take A collection of uncorrelated equally distributed a i gives us the state over the multiple projectors where the indices s α are ±. The conditional expectation E acts by averaging a projector P 1 + P 2 − · · · with the one where the indices have changed signs. Hence the state transformed with the conditional expectation is (3.44) The relative entropy is with The sum cannot be done analytically. Numerically, the formula (3.45) agrees with the saddle point calculation to leading order, and gives a sub-leading logarithm term, There is an interesting corollary to this calculation. For a gapped theory with topological contributions to the entropy, the topological term in the mutual information appears when the distance between the regions is smaller than the correlation length. However, we can also take regions separated to each other more than the correlation length and the topological term will appear even in this case if the regions are exponentially large. This is because we can achieve saturation if we can take a sufficiently large number of uncorrelated intertwiners even if the intertwiner's expectation value is exponentially small.

Global symmetry
In this section, we study theories with global symmetries. For the problems of interest in this paper, the algebra structure of such theories was described previously in section (2.1). Summarizing that discussion, in these theories there is a breaking of Haag duality in a pair of disconnected regions due to the existence of certain intertwiners (2.13). These are neutral operators formed by a charged operator on one side and a compensatory anti-charge operator on the other side. There is one intertwiner per irreducible representation. In the complementary region, which has the topology of a spherical shell, there is also a breaking of duality due to the existence of twists operators (2.14), representing the symmetry group locally, and which do not commute with the intertwiners (2.21).
We take two disconnected regions R 1 and R 2 and their complement, the "shell" S = (R 1 R 2 ) . As described above the are two choices for the algebra of R 1 R 2 , namely the additive algebra O R 1 R 2 and the additive algebra plus the intertwiners O R 1 R 1 ∨ I. Similarly we have two algebras for S, the additive one O S and the additive one plus the twists O S ∨ τ . The quantum complementarity diagram reads in this case The associated entropic certainty relation involving the two dual order parameters is (3.50) In this case the index is |G|, the total dimension of the group.
It is worth remarking that for global symmetries, the orbifold theory O comes together with the theory F containing charged operators. Using this theory F we can produce another order parameter for a single component region R where E F is the conditional expectation produced by the twists that kills the charged operators. This order parameter vanishes if ω is invariant under the group, and hence under the conditional expectation, as is the case of an unbroken symmetry vacuum (see [20]). In this last case the intertwiner order parameter can be written, using the theory F, as where on the right-hand side appears the difference between the mutual information on the two models.
These order parameters were studied at length in [20], with a focus on the topological contributions to the entropy. This requires that R 2 is the complement of R 1 , except for a thin regularization region between them. In this case S O R 1 R 2 ∨I (ω|ω • E I ) can be understood as the difference between regularized entropies between the model with charges F and the orbifold O.
Here our focus is on these relative entropies as order parameters for phases of the theory, and for that, we add an understanding of the sub-leading terms at saturation, which are important in distinguishing phases. The opposite geometry of far separated balls will also be useful.
To this end, we start by explaining how to put bounds on the entropic order parameters. We then analyze the order parameters in symmetric and broken symmetry phases.

Bounds from operator expectation values
The algebra of a fixed set of twists was described in section (2.1). We have where U (g) is the global symmetry operation. For non-Abelian groups, the twists are not observables, since they are not invariant under the symmetry group. We can produce invariant combinations by averaging over the conjugacy classes 17 This generates a closed algebra with fusion coefficients given by the fusion of conjugacy classes, see (2.20). This is an Abelian algebra. For computing the entropies it is convenient to diagonalize this algebra and define the projectors They are labeled by the set of irreducible representations. We have P r P r = δ r,r P r , We are interested in the conditional expectation E τ in this algebra, which is the dual one to the conditional expectation for intertwiners. This was shown in [20] to be E τ (τ g ) = δ g, 1 (3.57) Therefore, from (3.55) we get Any state ω in this abelian algebra is determined by the probabilities of the different sectors q r = P r . (3.59) The relative entropy becomes This is the formula for the twists relative entropy presented in [20]. We have S τ (ω|ω • E τ ) ∈ [0, log |G|], which follows by taking into account |G| = r d 2 r . Notice that this expression is not equal to the von Neumann entropy of the twist algebra if the group is non abelian. Now we want to compute the relative entropy in the algebra of the intertwiners. In [20] this computation was approached by enlarging the theory to include charged operators. Although the quantitative final result is bound to be the same, it proves useful for later use in gauge theories to have an approach based only on the neutral sector. We should then work only with the algebra of intertwiners. As shown in appendix (A), this is again an Abelian algebra represented by the fusion matrices N (r) . These matrices can be simultaneously diagonalized. It is convenient to name the basis vectors where the matrices are diagonalized with the conjugacy classes of the groupḡ. The number of conjugacy classes is the same as the number of irreducible representations. We have that the expression of the matrices in this basis is given by (3.61) These have the right algebra because of the formula expressing the decomposition in characters of the product of characters χ r (g)χ r (g) = r n r rr χ r (g) . (3.62) From this formula and the orthogonality of characters it is simple to derive where dḡ is the number of elements in the conjugacy classḡ. The matrix S that diagonalizes This matrix is in fact unitary. In term of these matrices 18 which is Verlinde's formula for group representations. The projectors over the diagonal are now the associated relative entropy becomes The interest of the formulas (3.60) and (3.70) is that they relate entropic quantities with operator expectation values.
The certainty relation plus monotonicity of relative entropy can be used to constrain the order parameters by using expectation values of operators. We have, (3.71) 18 There is a dual version of these formulas in the basis of representations, where twists are diagonal while the intertwiners are non diagonal. These are based on the fusion rules of the conjugacy classes of the group (2.20), see [66] pag. 404.

Symmetric phase
Below we will only consider finite groups. For Lie groups see [20]. Let us consider the case of a CFT and two nearly complementary regions, a ball R 1 of radius R, and the complement R 2 of a ball of radius R + . The two regions are separated by a thin shell of width . As argued in [20], the twist expectation values in a thin shell in a symmetric phase will be exponentially small in the area of the shell The twist does not change the vacuum state in the bulk of the region, and only a local contribution from the boundary arises. The constant c 0 depends on the precise twist. The previous law applies for any twist except for the identity.
With this information we can use the formulas in the previous section to put bounds to the entropic order parameters. We obtain where c r is some constant that will not play a role in what follows. We have definedq r ≡ d 2 r G , and from the normalization of probability we have that r δq r = 0. This implies the last term in (3.60), coming from the non-Abelian nature of the group does not contribute to the correction. Introducing such probabilities in (3.60) and expanding in δq r the first order correction vanishes. The correction appears at second order in δq r and we find (3.74) To understand the origin of this formula in terms of the intertwiners we can apply the ideas of section 3.3. To have such an exponential approach to saturation we would have to find intertwiners with expectation values exponentially near 1. Instead of that we just need to locate many independent intertwiners along the surface of the shell. The number of almost uncorrelated intertwiners will be proportional to the area. In the limit of small separation , they can be separated enough between themselves to have small cross-correlations. This implies (3.75) From (3.74), (3.75), and the certainty relation we get for the shell order parameter where c 1 < c < c 2 . The exact dimensionless coefficient c depends on the theory and in general, it is not easy to compute. To match the terminology of line operators in gauge theories, we will call this a perimeter law for the shell order parameter, awkward as it may seem in the present case. We are taking the convention that for a non-local operator for a region of topology S k × R d−k , with a fixed width, and large size r of the S k , we will always call perimeter law if the expectation value decays with the exponential of r k . As we have discussed above this is the maximal rate of decay for local operators. We will call area law if it decays exponentially with r k+1 . This is the maximal possible decay rate of non-local operators.
From this result, the certainty relation (3.50) gives for the intertwiner order parameter the form This is the dual version of the area law.
In the vein of adding more intertwiners to improve the lower bound to the intertwiner relative entropy, one could ask why not to locate intertwiners all over the region. Doing this we obtain a number of intertwiners proportional to the volume of the region N ∼ V / d−1 . However, for these intertwiners located all over the region, with one charged operator on each side of the shell, the expectation values decay as we get further from the shell, since the charge anti-charge pairs get more separated. These expectation values will decay with a certain power depending on the conformal dimension of the charged operators (3.78) The relative entropies over each individual intertwiner will go as ∼ 4∆ R 4∆ . We can make a rough estimate considering these intertwiners uncorrelated (which is hardly the case in a CFT) using (3.41). To overcome the area term we need ∆ < 1/4, which is beyond the unitarity bound for d ≥ 3. Therefore, the scaling already arises just by considering a set of intertwiners close to the entangling surface, consistent with the twist result.
In the opposite limit, in which the distance L between the regions R 1 and R 2 grows large in comparison with their size R, the roles of intertwiners and twists are qualitatively interchanged. In this scenario, the intertwiners decay as where ∆ is the scaling dimension of the lowest dimensional operator charged under the group. This gives us a lower bound to the intertwiner parameter. Noticing that corrections to saturation on the relative entropy only come at second order, this will scale as (R/L) 4∆ . However, from (3.52) and the results about the mutual information for well-separated regions [67] in a CFT, we get that this is in fact the correct scaling, This is a logarithmic law. We will also say this is a sub-area law, while the area law in this case would correspond to an exponent linear in L.
This enforces the following behavior for the twists and implies the potentially useful fact that, in principle, it is possible to obtain the leading charged conformal dimension of the theory from the behavior of the expectation value of the best wide twists.
For two balls in a CFT, these relative entropies are functions of the cross-ratio η, and we have quite different unrelated behavior in the two limits of η → 0 and η → 1. There is nothing that relates the behaviors in the two limits.
In the massive case, the changes are quite obvious. The area law for thin shells is the same except we take m 1, in which case the intertwiners cannot contribute and we get where the small sub-leading terms are expected to be quite independent of the size of the shell for fixed width. The twist operators can be chosen such that τ ∼ 1 is m 1, even if the size is very large with respect to . This is a constant law for the twist parameter.
For separated balls we get in the same vein This is an area law for the intertwiner parameter. This area law, simple as it is, can also be induced from a dual point of view, noticing that we can insert many uncorrelated twists in between the line that separates the two charged operators.
Summarizing, we see the characteristic pairs area vs constant laws, and perimeter vs sub-area laws for the dual order parameters of thin regions, which have a rational explanation in terms of the certainty relation.

Spontaneous symmetry breaking
From the previous discussion, one can anticipate that something qualitatively different is going to happen for scenarios with spontaneous symmetry breaking. In these cases, the correlation functions of intertwiners do not go to zero at large distances. The reason is that the one-point functions of charged operators in the vacuum do not vanish. Choosing charged operators bigger than the SSB scale we can make them approach I ∼ 1 as much as we want. We have for R 1 , R 2 larger than the SSB scale This holds for intertwiners in the large distance limit. It corresponds to a constant law for this order parameter.
We then expect for thin shells, using a volume worth of different approximately uncorrelated intertwiners In other words, we expect the approach to saturation to be exponentially fast in the volume of the region enclosed by the twists. This is an area law for the shell order parameter (in the terminology adapted to the loop operators). Again we have area vs constant laws for dual order parameters, because of the certainty relation.
To study these features in more detail consider the simple case of a Z 2 broken symmetry φ → −φ of a real scalar field with a double-well potential V (φ). Call the two vacua | ± v . We start with the twist. We should first find its expectation value. Taking a large ball R, the expectation value is the path integral in Euclidean space with a boundary condition (3.86) When analyzing this expectation value, there are going to be subtleties coming from the regularization at the borders. The operator has to be smeared there. But this smearing will contribute with a term proportional to the boundary in the effective action, which is going to be superseded by the volume contribution, as long as the region is sufficiently large. Accordingly, we will think in the large volume limit and neglect boundary terms.
To compute the expectation value we use a semi-classical limit. We thus need to find solutions to the classical equations of motion with the appropriate boundary conditions. In this limit, such path integral is computing a configuration where the field goes to +v at infinity in every direction. To achieve that, and the boundary condition, the field should be φ( x) = 0 at R and grow positively as we move away from t = 0 in the time direction. This is a configuration with a non-trivial field around t = 0 that has the form of an instanton interpolating between −v and v in the time direction, but where we change the sign of the negative part of the trajectory. This is still a solution to the equations of motion because of the action of the twist (see figure 8, left panel). The action is the same as the one of the instanton. 19 Thus, we are computing an instanton corresponding to the tunneling from one vacuum to the other inside the spherical region. This has finite action if we keep the volume large but finite. Analogously, we are computing an overlap, τ ≡ v|τ |v V = −v|v V of the two vacua in the region of volume V . In the large volume limit, this transition amplitude is just originated from a translation-invariant solution, for which which is an instanton in one dimension. One can alternatively think of it as a domain wall. Call the corresponding one-dimensional action of this one-dimensional instanton S I , which however has d − 1 dimensions in energy. This is the usual instanton action of a non-relativistic degree of freedom φ in a double-well potential V (φ); see [68] for specific examples and general features. We are ignoring subleading corrections from fluctuations around the saddle point. The total action has a factor of the volume and the amplitude τ ∼ e −S I V . (3.89) This allows us to compute the coefficient of the volume term of the expectation value of the twist, which does not depend on the shape of the region, as far as this region is large enough.
The entropic order parameter in the twist algebra is then given by We remind that the factors of 2 in the exponents appear because the correction to formula (3.60) comes at second order. We thus find an order parameter scaling with the volume. But as opposed to the conformal scenario, the leading coefficient of the exponent can be explicitly computed.
We should be able to find a volume scaling in the intertwiner relative entropy as well, connecting (3.84) with (3.85). To find such a contribution we need to understand how to choose our intertwiner. First, we define a homogeneously smeared operator φ A over a region A. Doing the spectral decomposition, we define projectors on the space of positive and negative eigenvalues, P A + , P A − respectively. Since the global symmetry acts as Associated with these projectors there is a charged operator V A = P A + − P A − . This transforms as φ itself. There are similar projectors P B ± , associated with homogeneous smearing of the scalar field for an outside region B. With those, we can find an analogous charged operator V B in that region. With these charged operators we define the following intertwiner I ≡ V A V B , I 2 = 1.
Considering the region B outside large enough, and φ = v positive, we can set P B − = 0, P B + = 1. Then the probability of the projectors P ± = (1 ± I)/2 are P + = 1 − P A − /2, To compute this expectation value we again turn to the path integral. Now we are in the situation of figure 8, right panel. The value of the field is negative at t = 0 inside the region, because of the insertion of the projector, and the classical solution will prefer to sit at φ = −v there. The solution is now formed by two consecutive instantons taking us from v to −v and again to v in the time direction. Then we have (3.92)

We could have obtained this result also from the approximation v|P
The relative entropy in the intertwiner algebra is found to be where k includes subleading factors depending on the size of the region. From this we get an upper bound to the twist relative entropy. The best bound follows by enlarging A to cover most of the twist region. Together with the lower bound arising from (3.90) we get This computes exactly the exponent c in (3.85).
For computing an upper bound in the intertwiner order parameter we could as well have followed the calculation at the end of section (3.3), considering many regions of size V A small with respect to V . We have to insert the probability p = P A − /2 in formula (3.48). From (3.48) we find, for N such regions A inside the ball R, covering it, (3.95) We get a worse upper bound than (3.94) but still shows the volume law is obligatory from the existence of multiple uncorrelated intertwiners.

Summary
The entropic order parameters clearly distinguish between the phases of QFT in these scenarios. The symmetry broken phase has a constant law for the intertwiner relative entropy and specific exponential decay with the volume of the enclosed regions for the twist parameter. In contrast, in the symmetric phase, this behavior is not possible. In the symmetric phase, if the charged fields become gapped, the intertwiner parameter decays to zero exponentially wit the distance at large distances (an "area law"), and the twist parameter has a constant law. This is the dual behavior of the SSB phase. There are intermediate regimes (as in the conformal case) where none of the order parameters display an "area law" and none a constant law.
It is not so surprising that the physics of these phases are captured by the relative entropies because they are related to the expectation values of the associated order-disorder operators. What is interesting is that the entropic approach, due to the certainty relation, relates in a quantitative manner the characterization of the phases in terms of the order or the disorder parameters. The present approach shows they are dual to each other, the duality relation given by the certainty relation (3.50).
In this sense, it is quite clear that it would not be possible to have area-area behavior for the dual parameters. Such putative phase conflicts with the certainty relation. Indeed, we have seen that the area behavior of one parameter is always tied to the constant behavior of the other. This is because to fulfill the certainty relation where one parameter decays exponentially with the area, an area worth of the dual operators with independent and approximately constant expectation values is necessary. It would be interesting to prove these interrelations more rigorously.

Gauge symmetry
We consider the algebra of a simple ring R which contains non-contractible one dimensional circles. Its complement R contains non contractible S d−3 surfaces. The group of non-local operators is Abelian. For d = 4 these two complementary "rings" have the same topology (if we compactify the space at infinity).
Let us first consider d > 4, where the ring and its complement have different topology. In analogy with the case of global symmetries, we have two possible acceptable algebras for the same region R (of course, other algebras containing only subgroups of the non-local operators can be considered as well). Here the relevant algebra is the one containing the non-contractible Wilson loops as well as the additive operators. For simplicity in this section, we call A(R) to the additive algebra in a region A add (R). The full algebra containing the non-local Wilson loops is The elements of A W (R) can be decomposed as for a fixed set of Wilson loop operators going around the ring, where r are representations of the center Z (or the uncharged subgroup of it). The elements a r belong to the additive algebra A(R). We can now use the conditional expectation that kills the Wilson loops studied previously Then we have two natural states in A W (R). The first is the vacuum ω and the second is ω • E W . A "magnetic" order parameter (since it measures magnetic fluctuations) is then given by Analogously, adding the t'Hooft loops to R we form the algebra The expansion of operators in this algebra is now where the indices run over the center of the group algebra. We can again define a conditional expectation eliminating the non trivial t' Hooft loops The "electric" order parameter is then Both of these order parameters vanish for theories with only global symmetries. In the same way, the order parameters for global symmetries vanish for theories containing only gauge sectors. The order parameters detect only their associated symmetries.
Considering this scenario, the observations done until the moment can be condensed in the following complementarity diagram with an associated entropic certainty relation given by (3.105) Let us now consider the case of d = 4. In this case R and R have the same topologies. They are both conventional rings once we compactify space at infinity. In other words, 't Hooft loops are one dimensional loops. This implies that the algebra (A(R )) contains both Wilson loops and 't Hooft loops, all based on R. Therefore the maximal algebra of region R is (3.106) These two sets of loops can be chosen to commute in R, and we can expand a generic element as a = z,r a z,r T z W r . We can also define a new conditional expectation E W T (a) = a 1,1 . The complementarity diagram reads in this case The associated entropic certainty relation is As explained in section 2.2.5, charged fields can break the group {W T } to subgroups, which may not have this particular product structure. The above still applies to such a scenario just by taking the conditional expectations that kill all remaining non-local operators, and where 2 log |Z| is replaced by the order of the group of non-local operators. In the above scenario, or more generally, when the group of non-local operators has a subgroup, we can choose other non-local algebras and define other relative entropies. We describe these parameters in the case of a {W T } group.
Another possible algebra for R in this case contains only Wilson loops, which we call A W (R), and another one only 't Hooft loops, which we call A T (R). This leads to two other natural order parameters, S A W (R) (ω|ω•E W ) and S A T (R) (ω|ω•E T ). The associated complementarity diagrams are . (3.109) The certainty relations read in this case We can show these relative entropies are not all independent. We will adopt here an obvious simplified notation, and call, for example S W T,0 = S A W T (R) (ω|ω • E W T ), and S W T,0 for the same quantity in the complementary region. Using this notation, the certainty relations described above are This last inequality also follows from the commuting square property of the conditional expectations E W , E T , and E W T = E W • E T = E T • E W and their respective algebras [69].

Bounds from Wilson and 't Hooft operator expectation values
The algebras generated by Wilson and 't Hooft loops closing groups (and no other operators) are isomorphic to the ones of representations of an Abelian group and the group algebra itself. Therefore, the formulas for the relative entropies for these algebras are the ones obtained for twist and intertwiners, but specializing for Abelian groups.
In particular, the algebra of a fixed set of 't Hoof loops is where the z s are elements of the center of the Lie group G. All these loops are unitary operators. Again, it is convenient to define projectors labelled by irreducible representations The conditional expectation is analogous to the twist conditional expectation Any state ω in this Abelian algebra is determined by the probabilities of the different sectors The relative entropy becomes where S T is the entropy over the t' Hooft loop algebra. We have For the Wilson loops the situation is similar, but replacing representations by elements of the group and viceversa. The minimal projectors of the Algebra are The conditional expectation kills all non trivial Wilson loops, The relative entropy becomes where S W is the entropy over the Wilson loop algebra.
Reducing to the algebras of non-commuting Wilson and 't Hooft loops, using monotonicity and the previous expressions we conclude (3.127) z R L Figure 9: Ring formed by the revolution around the z axes of a disk D of radius R, such that the inner radius of the ring is L.

Ring order parameter for the Maxwell field
In this section, we compute upper and lower bounds for a ring order parameter for the free Maxwell field. We find the behavior of the order parameter to be surprisingly well determined by these bounds.
Let us take a ring formed by the revolution around the z axes of a disk D of radius R, such that the inner radius of the ring is L ( figure 9) and A the additive algebra of the electric and magnetic fields inside the ring. A bigger algebra is obtained by adding to A a closed group of Wilson loops corresponding to charges that are multiples of some fixed charge q. We will be interested in the order parameter for this choice of algebra.
A choice of the smeared Wilson loops can be written in cylindrical coordinates appropriate for the ring geometry as W qn = e i q n D dr dz α(r,z) dϕ r (φ·A(r,z,ϕ)) . (3.128) We have used smearing functions localized at t = 0, and have to impose such that for a magnetic flux Φ B piercing through the hole of the ring W qn gets multiplied by e iqnΦ B , independently of the precise smearing function.
These loop operators form a group Z, and with them, we can form the ring algebra This algebra is independent of the smearing functions α since changes in α can be produced by additive operators in the ring. We want to compute the relative entropy where E W eliminates the non-additive Wilson loops.
The non-local operators for a ring in the Maxwell field form an infinite group R 2 of the electric and magnetic charges, and the log |G| of the certainty relation is divergent. However, this relative entropy over a discrete subgroup is finite, though it can take arbitrarily large values depending on the geometry. The result is indeed divergent (as − log(q)) in the limit of a continuous group q → 0. The complementary relative entropy involves the conditional expectation eliminating the t' Hooft loops with continuous magnetic charges g ∈ (0, 2π/q), which is divergent, as expected by the certainty relation.

Lower bound
Let's start by computing a lower bound to this relative entropy. We evaluate the relative entropy in the subalgebra W q generated by the Wilson loops (3.128). By monotonicity of relative entropy we have The algebra of the Wilson loops is Abelian and can be represented as the multiplicative algebra of functions on k ∈ (−π, π) by the identification The probability density in the k space is given by the equation Then, using Poisson summation formula for inverting 3.134 we get where Θ 3 is the elliptic function.
This gives the probability distribution corresponding to the vacuum state. The state ω • E W in the algebra of the Wilson loops just gives zero expectation value to all Wilson loops except for n = 0, which is the identity. Then the probability distribution in k space corresponding to this other state is Q(k) = (2π) −1 . The relative entropy is This depends through (3.136) on the smearing function α and the charge q. Later, we will analyze this dependence in detail.
A simplification of these expressions can be obtained for the limits of large and small q 2 Φ 2 B . For small q 2 Φ 2 B we can convert the sum into an integral in (3.136) getting Note that in the limit of a non compact Wilson loop group q → 0, the relative entropy diverges logarithmically with the charge ∼ −(1/2) log q 2 , and the same happens for the ring order parameter S A Wq (ω|ω • E W ).
In the opposite limit q 2 Φ 2 B 1 only the first terms in the sum (3.136) give a non negligible contribution and we get The best lower bound is obtained for the largest relative entropy for the subalgebra. This corresponds to a smearing function such that Φ 2 B is minimal, producing the largest difference between the vacuum and the ω • E W expectation values. These last are zero for non trivial Wilson loops. To solve this problem, we first express Φ 2 B in terms of α. From (3.128), writing the (r, z) coordinates as a vector u, we have where in (3.143) we have used the correlator of the vector potential in Feynmann gauge (3.144) Now, from (3.143) we see that finding α such that Φ 2 B is minimal, corresponds to minimizing α · K · α subject to the constraint α · 1 = 1, where 1 is the function that is identically 1 on the disk. The solution is These depend on the ring parameters L and R through the cross ratio determining the geometry of the ring (see appendix D). Consequently, the lower bound S Wq (ω|ω• E W ) will also be a function of the cross ratio (and q). We have computed numerically the smearing function (3.145) in a square lattice with site labels (i, j) ↔ (r, z). In these coordinates, the ring (R, L) is given by the set of points (i, j) such that (i − L − R) 2 + j 2 ≤ R 2 with L ≤ i ≤ L + 2R and −R ≤ j ≤ R. Alternatively, for the ring (R, η), we have (i − being rotationally invariant but mostly concentrated on the boundary for small η (thin ring), to a crescent moon concentrated on the inner left boundary of the disk for η ∼ 1.
The relative entropy can be solved analytically in terms of η for the opposite regimes L/R 1 and L/R 1. Let us start considering the limit of thin rings η 1. In this case, the expression of the kernel K is rotational and translational invariant, Moreover, in this regime, K · α is proportional to the Coulomb potential and therefore, the condition K · α = const means α is proportional to a charge density on a conductor disk. The solution to this problem can be obtained for squeezed ellipsoids using oblate spherical coordinates, and then taking the limit of the disk. The solution is wherer is the radial coordinate in the disk. This is in agreement with the profile shown in fig.(10) for η = 1/200. From ((3.148)) and (3.149), we see the flux that gives the best lower Wilson loop bound for large L/R and fixed q satisfies This gives an exponentially small relative entropy It remains now to study the opposite regime L/R 1. For this, it is convenient to consider the geometry corresponding to η ∼ 1 and focus on the complementary region of the ring: this is a thin ring withL/R 1. From our previous analysis, we know that for thin rings the smear function is mostly concentrated around the boundary. Then, in this limit, the original geometry of a ring with L/R 1 results equivalent to a much simpler one given by the complement of a large tube of radiusR. The smear function in this setup is translation invariant along the z direction, and the kernel K is the same as the one already found. Moreover, translation invariance implies that the equivalent problem cannot depend on z but only on the radial coordinate r α(r, z) =α (r) 2πL , We could not solve this limit analytically. However, by dimensional reasons, the flux giving the best lower bound in this limit has to be proportional toR/L where the constant c can be evaluated numerically by inverting a discretized version of the kernel K. We get The relation between new and original variables can be obtained from the cross ratio in this limit (see appendix D) Finally, from (3.139), (3.150) and (3.157), the best lower bound is From the above equation, the relative entropy increases logarithmically for wide rings R/L → ∞.

Upper bound
Having found the lower bound, let us now proceed with the upper bound. Such an upper bound can be found by considering the dual algebra of t' Hooft loops in the complement. We have (see appendix E) where S T is the entropy in the full algebra of t' Hooft loops, and S Tg is the one in the algebra of loops with magnetic charges multiple of g = 2π q , (3.160) due to Dirac quantization condition. Note that the upper bound (3.159) is not a relative entropy but a difference of entropies. This is the result of applying the entropic certainty and uncertainty relations restricted to the case of a subgroup (we are only considering discrete charges multiple of q) of the total symmetry gauge group (see appendix E). To calculate the entropy in T , we note the algebra is represented as the one of functions e ixk , where x ∈ R. The probability density then has a formula analogous to (3.138). Substituting the sum by an integral in (3.136) we get giving an entropy S T Regarding the entropy S Tg , the calculation follows the same line as the lower bound one. Note the loops in T g are represented as e igkn , k ∈ g −1 (−π.π). A calculation analogous to the one in the previous section gives the probability density The entropy is The upper bound is then given by S T − S Tg using equations (3.162) and (3.163). This is a function of g = (2π)/q and Φ 2 E . Again we have to use the best loop smearing with the smallest Φ 2 E to get the lowest entropy difference and the best upper bound. By electromagnetic duality, this is given by the same function used for the magnetic flux but evaluated in a complementary cross ratio Let us then compute the limits of wide and thin loops. The limit g 2 Φ 2 E 1 allows us again to convert the sums in integrals, and to integrate over the real line in (3.164), and the leading order in S Tg exactly cancels S T . The entropy difference is given by the following integral Replacing Φ 2 E ∼ cR/L and g = 2π/q we get ∆S = 1 2 which is compatible with (3.151) because c ≥ (2π) −1 . This confirms in this regime the relative entropy has a perimeter law For g 2 Φ 2 E 1 we get Q(k) ∼ |g|/(2π) up to exponentially small terms. Then we get ∆S = S T + log(g/(2π)) = 1 2 1 + log(g 2 Φ 2 E /(2π)) .  This is compatible with (3.158).
The upper and lower bounds give a surprisingly precise determination of the ring order parameter in this limit, Then, while the relative entropy in the ring can be very large for wide rings, it always differs from the one on the best Gaussian Wilson loop algebra by less than half a bit.
The numerical calculation of the lower and upper bounds on intermediary regimes is shown in figure (11) for different charges. The upper pair of curves corresponds to the electron charge.

Ring order parameters in CFT's
For a CFT there are no scales and the order parameters are dimensionless conformally invariant functions of the geometry. One can take advantage of the conformal symmetry and consider the toroidal rings previously described for the Maxwell field. The relative entropies must be a function of the cross-ratio η = R 2 /(R + L) 2 ∈ (0, 1), where R is the radius of a circle, and the torus is formed by rotating around an axis at a distance L from this circle. See figure 9. The complementary ring has cross-ratio 1 − η (see appendix D). All relative entropies are increasing functions of η due to monotonicity.
For small width R and large size L of the ring the relative entropies should go to zero exponentially in the perimeter S ∼ e −c L R , (3.173) matching the behavior of the loop operator expectation values. Indeed, the loop operators in this ring have a perimeter law, and this gives a lower bound. An upper bound follows from the certainty relation. Using a perimeter worth of small loops wrapped around the thin ring and well separated between themselves, we can show a perimeter law is an upper bound for thin loops. This follows from the ideas described in section (3.3) about improving bounds with uncorrelated dual operators. We cannot go further, however. This is because these larger dual loops must decay fast with the size and have non-trivial correlations in the conformal case, preventing the direct application of the ideas in (3.3).
Accordingly, for large width we expect the parameters to saturate the bound and approach log |Z| in the finite group case. From the certainty we expect For an infinite group U (1) n the order parameter involving the elements of the group is divergent for any η because the group is continuous. It has to be so to match the certainty relation too. However, the dual group is infinite and discrete, and we expect the corresponding relative entropy to behave as in the Maxwell field in the limit of wide rings The case of finite groups and conformal symmetry is quite special, given the tendency of non-Abelian groups to confine. It is achieved with a special balance of gauge and matter degrees of freedom. The matter should be in the adjoint representation to preserve the generalized symmetry. A famous case is N = 4 SU (N ) SYM theories.
Given the duality relation between η and 1 − η, there are some peculiar features in the conformal case for finite groups. The certainty relation gives S W T,0 (η) + S W T,0 (1 − η) = 2 log |Z| . (3.176) In particular, we have S W T,0 (1/2) = log |Z| . The first equation is easily understood if there is an analogous to electromagnetic duality symmetry. However, these equations do not seem to require some form of duality equating t' Hooft and Wilson loops.

Confinement and Higgs phases
The duality between the confinement and Higgs phases was transparently argued in 't Hooft's work [23]. In such work, the dual disorder parameters, the 't Hooft loops, were defined. They were defined by their simple commutation relations with the order parameters, the Wilson loops. It was argued that, although the confinement phase might be difficult to approach, given strong coupling issues, the associated physics is not that mysterious. We can study the dual Higgs phase at weak coupling.
In the confinement phase, the "electric" charges (the quarks) are confined, and this can be measured by the area law of the Wilson loop. In the Higgs phase, it is the magnetic charges the ones confined, and the 't Hooft loop the one displaying area law. The 't Hooft loop, a "disorder" parameter, is the natural order parameter for the Higgs phase. Although the physics of both phases is similar (or dual), the Higgs phase can be approached semi-classically.
In this section, we want to analyze what we expect for the relative entropy parameters in these phases. This provides a different perspective to the physics already known through the certainty relation. This relation is valid at any coupling and it relates the physics of order and disorder parameters. In the confinement-Higgs scenarios, it relates the physics of Wilson and 't Hooft loops.
As described for global symmetries, spontaneous symmetry breaking implies that the expectation value of the intertwiner should go to a constant, and the certainty relation then implies that the twist decays exponentially in the volume. In such a phase the intertwiner factorizes into the non-vanishing product of one point functions of the charged operators. A volume worth of constant intertwiners can be used to induce a volume law for the twist. This is very different from the conformal scenario, in which the intertwiner shows some decay typical of a conformal field theory, and the twist decays with the area of the boundary.
For gauge symmetries, we have a very similar picture. In both the confinement and Higgs phases the theory is expected to become gapped. We will focus on the Higgs phase where we have semi-classical control but the same (or dual) behavior is expected in the confinement phase. The gauge field then becomes massive. This has the consequence that the loop operators become uncorrelated and, further, as shown below, the Wilson loop displays a constant behavior. This implies through the certainty relation that the dual loop (the t' Hooft loop in the Higgs phase) is bound to display an area law. This is because we can place an area worth of constant uncorrelated Wilson loops crossing the sectional area of the t' Hooft loop, as shown in figure 12.
In the literature, this phenomenon is interpreted as a symmetry breaking of the ring generalized symmetry [8]. However, loops are almost always considered as line operators, without Figure 13: The vortex-instanton configuration giving the t' Hooft loop expectation value.
width. In this limit, the loop operator dual to the one displaying area law shows a perimeter law in general, as it happens in the conformal scenario, rather than a constant one. This constant behavior of loop operators is invoked in the literature to make a parallel to the case of spontaneous symmetry breaking of a global symmetry. It is argued that the loops can be dressed by local operators to convert the perimeter law into a constant behavior, while this cannot be done in the case of an area law. But by the very same means, a constant behavior can be induced as well for any loop operator having perimeter law, including the conformal case. However, for our purposes, this is unsatisfactory. The reason is that we need the loop operators to be real (smeared) operators that satisfy a group law. This tells us they cannot be dressed arbitrarily.
There is no way to dress a loop in the conformal regime to have constant behavior while keeping the group fusion rules. Otherwise, the corresponding relative entropy will not have the perimeter law dependence on R/ that it has. This is not the case for a massive field. The dressing is replaced here by looking at operators that are smeared in a ring of a certain width. That is, to get to a constant law we need to widen the ring size.
To test this behavior we study the Higgs phase. In this scenario, the gauge field appearing in the Wilson loop has become massive, and this should lie at the root of the expected constant scaling. We thus consider a Wilson loop for a massive vector field. This is of course the case of Wilson loops inside a superconductor, which is a specific simple scenario of the Higgs phase. In the massive case, the Wilson loop typically shows a perimeter law. We want to show that with transversal smearing in a size larger than the scale of the inverse mass we can do better and obtain a Wilson loop whose expectation value is (almost) constant, independent of the perimeter.
We have to compute the expectation value of a Wilson loop in the classical regime where the Wightman correlator of a massive vector field is Since we are interested for the moment in the perimeter term, we take an operator invariant in theẑ direction, with a current in that direction given by where we have written y ≡ (x 0 , x 1 , x 2 ). We also need to normalize the charge of the operator W dy α(y) = 1 , (3.182) and α(y) to have support in the causal development of a region of size R in the coordinates Plugging this into (3.179), and considering a tube of large length L, the leading linear L dependence of the exponent S becomes Writing the smearing functions in momentum space we obtain from (3.183) the perimeter law The condition (3.182) on α gives the constraintα(0) = 1. The mass shell in (3.185) is separated from the point p = 0 and the Fourier transform of the smearing function can be chosen to be exponentially small outside p ∼ R −1 . 20 Therefore we can device a smearing function such that the coefficient of the perimeter goes to zero exponentially fast for mR 1. Note that a pure spatial smearing α(y) = δ(t)β(x 1 , x 2 ) does not allow for this exponential suppression and we can get a power law suppression at most. The fields at t = 0 can still be written at t = 0 using the equations of motion. However, in that case our loop will also contain a term of the momentum of A (electric field) at t = 0 in the exponent. This is an instance of improvement produced by the locally generated operators on the expectation value of non locally generated one.
Therefore we can have a constant law for the Wilson loops that are wide enough. The gap then implies uncorrelated loops and an area law for the t' Hooft loop should arise from the certainty relation.
Let us see directly how this area law appears in the classical regime. The t' Hooft loop is a singular gauge transformation of the center of the group on a surface Σ of area A Σ and boundary in a ring R. For simplicity, let us think in a group Z 2 , where we have only one t' Hooft loop. This is the case of a spontaneously broken SU (2) gauge theory. To break it without converting the Wilson loops in the fundamental representation into local operators in the ring we need to couple the gauge fields with adjoint Higgs fields. More than one Higgs is necessary to break the symmetry completely. We will not enter into the details of the model building. Far from the surface Σ, and at both sides of it at t = 0, the Higgs fields stay into their vacuum values such that the filed is continuous at large distances outside the loop. So we expect a Higgs configuration that remains in its vacuum value at t = 0 and spatial infinity, deviating from it only near the Σ. We are interested in the area term and can take a large loop and neglect the boundary effects. Then the configuration of interest in only dependent on the coordinates x 0 , x 1 , where x 1 is the coordinate perpendicular to Σ. The t' Hooft loop sits at x 0 = 0, x 1 = 0 in this plane. The two Wilson loops W 1 , W 2 in the fundamental representation in figure 13 pass through the t' Hooft loop at x 0 = x 1 = 0, closing above and below t = 0. They get a factor −1 due to the t' Hooft loop insertion. Then the configuration is such that the circulation P e i x 1 <0 x 1 >0 dx µ Aµ of the 20 It cannot be zero because the Fourier transform of a function of compact support is an entire function. gauge field on the upper plane far from t = 0, is −1 and the same, in a time reflected manner, happens on the lower plane. The classical solution is a "vortex-instanton" that exists because of the insertion of the t' Hooft loop. The Higgs field as usual for these vortex configurations accompanies the rotation of the gauge field such as to minimize the action. It rotates from x 1 0 to x 1 0 an angle 2π in the group parameter but ends up at the same vacuum value φ 0 because 2π rotation is the identity on the adjoint representation. The full configuration has the same action as a 4π rotation vortex. The instanton action S I is a quantity with two dimensions of energy (it is action density over the surface). We get This gives a lower bound on the corresponding relative entropies 21 This also gives an upper bound to the relative entropy corresponding to Wilson loops in the wide complementary ring.
Now we compute an upper bound. For that, we need to understand how the sufficiently wide Wilson loop in the fundamental representation approaches maximal expectation value. We follow the same route as for the case of intertwiners in section 3.4.3. The Wilson loop can be decomposed into projectors P ± = (1 ± W )/2 (we are using W 2 = 1 for Z 2 ). We have P + ∼ 1 and P − = p 1. The expectation value of the projector P − follows again by inserting it into the path integral. The gauge field is then constrained to produce a 2π rotation in the gauge group at t = 0 along the path of the loop. This has to return for paths at negative times and paths at positive times. Therefore we have two vortices of 4π rotations, one after the other in time, which have the same classical action as the one previously discussed, see figure 14. They can be positioned anywhere along the path of the loop, but the contribution is quite concentrated around a cross-section of the loop of area A. We get The entropy on the algebra of W for small p is S {W } ∼ −p log(p). Therefore, taking the transversal area A ∼ A Σ to be as wide as possible for a loop interlocked with the t' Hooft loop, we get the leading exponential behavior of the upper bound This is consistent with (3.187) and gives the exact coefficient of the area in the exponent of the relative entropy order parameter. The same calculation shows that wide rings have a Wilson loop order parameter going exponentially fast to log(2).

Remarks on RG flows of order parameters
Having studied the order parameters in different phases, the main challenge becomes to understand the running of these parameters with the scale. A related objective would be to arrive at some conclusions about the possible realizations of a certain symmetry in the IR and UV. In this final section, we make a few comments towards these questions.
For any type of symmetry, the analysis of spontaneous symmetry breaking scenarios is typically phrased as follows. At low energies, the symmetry might be broken. This is signaled by some order parameter approaching a constant value and the dual order parameter decaying exponentially with some characteristic exponent. At high energies, where we approach some conformal fix point, the symmetry is restored. This is signaled again by the behavior of the order/disorder parameters. The question we want to comment concerns the transition between the different phases through the RG flow.
A natural route as we move from the UV to the IR is to consider scaling regions. For scaling regions, the terminology of phases and parameters is simplified considerably. Under scaling, the relative entropy corresponding to a region R of certain topology can either go to zero or tend to log |G| (or the logarithm of the order of a subgroup) as we scale R to infinity. This should happen independently of the precise shape of R. A second possibility is that the limit can be in the range (0, log |G|) and depend on the conformal geometry of R. In the first case, one of the symmetries (order vs disorder) is unbroken but the dual symmetry is broken. In the second case, which is the conformal one, we are forced to associate with it the idea that none of the symmetries is broken nor unbroken in the present sense.
In the same line, scaling non-local operators 22 leads, in the symmetry breaking scenario, to expectation values 1 or 0 in the large scaling limit. This is independent of the details of the shape. We just need to scale the characteristic length of the region to infinity. Operators with expectation value 1 correspond to the unbroken symmetry. They form a group because a 1 |0 ∼ a 2 |0 ∼ |0 implies 0|a 1 a 2 |0 = 1. In the conformal case, it leads to operators with intermediate expectation values, and this depends on the conformal geometry of the region.
These observations suggest that the breaking of a symmetry is tied to a gap. At least it seems tied to the absence of correlations between the relevant operators. This is simply because unitary operators where the expectation value is saturated to 1 have zero connected correlations between themselves. The same will happen for the operators with expectation value zero due to the commutation relations. If a 1 = a 2 = 0, for non local operators seated at spatially separated regions R 1 and R 2 , we have, using a b operator commuting with a 2 but not with a 1 , and such that b ∼ 1, The usual terminology in terms of area/perimeter laws for line operators is a bit more cumbersome especially because we can enlarge line operators in one direction only. For example, the improvement from perimeter to constant law for a loop occurs as we increase the width. It does hold only in certain cases and for loops that are wide enough. Further, this constant law does not persist without a perimeter term for exponentially large loops of constant width. The reason for these nuisances is that line operators are UV and IR operators at the same time, according to the two widely different scales involved in their geometry. 22 To associate unique operators to spacetime regions we can use the standard construction described in (2.2.3) From the present perspective of the RG flow, one would like to prove that some phase is realized in the IR by connecting this phase with the departure of conformal behavior in the UV. For example, to prove confinement, it may not be necessary to compute the area law of Wilson loops in the IR, but to address the question of which UV behavior leads inevitably to this area law if that were possible. This approach would of course be especially appealing, given that for asymptotically free gauge theories, the UV is under perturbative control.
In investigating the change of the order parameters with the scale it is not possible to use directly monotonicity of relative entropy for scaling regions. The reason is that none of the two scaled regions will be contained in the other for non-trivial topology. In the scaling limit, the order parameters are ordered by the inclusion ordering on conformally equivalent shapes (i.e. the cross-ratio in the special ring shapes used in the paper). This ordering is trivially realized in the case of broken symmetries.
Even if the simple monotonicity property is not enough to obtain a UV-IR connection, the heuristic ideas around entropic order parameters in this paper suggest that there is indeed a tendency for the increase of asymmetry between dual order parameters as we move to larger regions. If some non-local operators have more expectation values than the complementary ones, say a > b , this seems to seed still larger expectation values for the a operators in larger regions (and smaller ones for the b operators), through the certainty relation. Though it is not clear how to keep under control the effect of correlations, it suggests that the expectation of a connection between the UV and IR asymmetries may not be hopeless. With the purpose to illustrate further this point, in the next subsection we construct a, admittedly quite crude, toy model. But first, let us end this section discussing briefly some special entropic order parameter that exists only for global symmetries.
For the case of global symmetries, we can describe this tendency to symmetry breaking in more precise terms by using the theory F containing charged operators. In this case, there is a simpler relative entropy than the ones studied so far. This is the relative entropy in just one topologically trivial region introduced in [20], and mentioned in section 3.4, (4. 2) The reason this quantity has not been discussed above is that it is not easily generalizable to the case of gauge symmetries. In any case, it was studied at sufficient length in [20].
This quantity will be trivially zero if there is no SSB, just because the two compared states are identical in this scenario. However, if there is SSB, this relative entropy will be non zero at all scales. Further, by monotonicity, it is always an increasing function of the region R. This leads to the following conclusion. Even if the relative entropy goes to zero at the UV, however, small the deviation from zero, it will not go down again as we move to the IR. More interestingly, it cannot remain at a small value. To observe this we notice that for this relative entropy, the role of the intertwiners in the two-ball order parameter discussed above is played by the charged operators inside the ball. If there is a charged operator with non zero expectation value we can take many copies of this operator separated by large distances between each other, such that they are statistically independent. Using the results of section (3.3), we conclude that this will make the relative entropy to grow until log |G| is reached if the symmetry is completely broken. If the symmetry is only partially broken to a subgroup H ⊂ G, it will tend to log(|G|/|H|) instead. This argument exposes the general idea described above. Once a global symmetry is broken in the UV, notwithstanding the size of breaking, there is no way back. The symmetry will be completely broken in the IR. 23 The same could be said about an explicit breaking, driven by a small perturbative relevant operator in the UV.
From the operatorial point of view, detecting SSB in the UV is also easy in the F theory: it corresponds to a non zero expectation value of the charged operator. However, this expectation value is non-perturbative in the UV, and then not easy to understand in the perturbed UV theory. The same remark applies to the relative entropy (4.2). Even if this relative entropy shows the irreversibility of the SSB phenomenon, there is no clear indication on how to exploit it in the UV.
If we stay in the neutral model O a signal of SSB for small balls is difficult to separate from UV fluctuations. Correlators of a charged-anti-charged neutral operator (intertwiner) will be almost conformal, and concavity properties in a small ball do not tell if this will end up as a constant, a conformal, or a massive case, as we move to large distances (see appendix C.1). However, once it has set to a constant expectation value, by reflection positivity, the expectation value of the intertwiner will not start decaying again.

Toy model for the RG flow
This model is intended to illustrate more concretely the instability of the asymmetry between dual order parameters, and how it drives symmetry breaking. We think in a simple case of symmetry Z 2 . We have the non-local operators a and b with dimensions k and d − 2 − k respectively. To describe RG flows we fix a way of scaling a specific region R. Any such (sufficiently symmetric) region R can be characterized by two length scales r and . For example, for k = 1 we have loops, r is the radius of the loop as a one dimensional object, while is the width of the loop. We will fix to be sufficiently small and scale r from r to infinity. 24 Thus, we take generalized thin loops. Notice the dual region of a thin loop is not a thin loop.
Ideally we would want to find S A add ∨a and S B add ∨b as functions of r, for the algebras of the two types of thin loops. While this seems out of reach, the previous certainty principle, together with monotonicity of relative entropy, says that In these inequalities we have made explicit, by theS notation, that while we pursue the relative entropies in the left hand side as functions of a prescribed configuration scaling with r, the associated dual regions, albeit also defined by r and controlled by the dual non-local operators, are not in the same prescribed configuration. They are therefore not the same functions for r.
Given these inequalities, and the fact that for r/ 1 for the thin loops, we can let many dual thin loops cross the original thin loop, as in figure (12). We can then use the results in section (3.3) to estimate the upper bounds, under the simplifying assumption that we can neglect correlations between the thin loops. In order to obtain the best upper bound we need to locate as many thin loops as we can. The optimal configuration is then one where we put increasingly big thin loops to fill the original one (see figure (12). Writing the lengths in units of we can use formula (3.41) to obtain we attempt to extend this effective (local) symmetry at the IR to the UV it will probably get badly non-local, or highly broken, such as to make these relative entropies of regions either ill-defined or divergent. 24 This does not mean r needs to be in the IR nor in the UV scales.
where we define the function a(r), respectively b(r), to be the classical relative entropy between two probability distributions: the first is {1/2, 1/2}, and the second p a (r), respectively p b (r), associated to generalized thin loops of radius r. For thin loops the probabilities p a (r), p b (r) are near the distribution (1/2, 1/2) and we can replace This gives a pair of coupled inequalities involving only two unknowns, If a 0 > b 0 , the limit for r → ∞ gives f → a 0 − b 0 constant, and g → 0 exponentially fast (area law). If a 0 < b 0 , the opposite happens. There is a tendency of the RG flow to fall in one of the two possibilities and never come back, and the outcome only depends on data for small loops. The IR fate is controlled in this simplified toy model by the order between the dual entropic parameters. For other cases, to find the dual constant/area behaviour for each of the parameters we need to input from the start an asymmetry between the initial conditions. If that asymmetry is given, the outcome of the equations is a long period in which one of the functions remains constant and the other decays with the appropriate dual scaling.
The main drawback of this toy model lies in the fact that in the UV, the assumption that nonlocal correlators are uncorrelated, is invalid. Related to this comment, an interesting behavior in the previous case of d = 2, k = 0 arises if we take the limit (a 0 − b 0 ) a 0 . In this scenario there is a regime r pointing to a phase transition with some universal behavior when we cross the critical initial conditions a 0 = b 0 . The precise functional decay in this regime should again not be taken seriously, since in the conformal scenario we cannot use the approximation of uncorrelated thin loops.

Conclusions
In the description of a QFT in terms of algebras and regions, some basic relations are structurally natural. One of them is Haag duality, expressing that a region should contain all admissible operators allowed by causality. The other is additivity, expressing that operators in a region should be generated by local degrees of freedom in the same region. However, these properties are not required by the consistency of the theory. Only sufficiently complete theories should satisfy all these properties. In this paper, we have put forward the idea that the algebraic origin of symmetries in QFT is most naturally framed as the violation of these properties in regions with specific topology.
This point of view seems to be fruitful. To start with, considering algebras constructed additively, it is the case that different classes of symmetries correspond to violations of duality for regions with different topologies. We thus can see generalized symmetries as tied to violations of duality for regions with non-trivial homotopy groups π 0 , π 1 , π 2 , etc. These violations correspond respectively to global, local, and generalized symmetries, which are thus treated in the same footing. The focus on these simple properties of the net of algebras allows us to describe these symmetries without appealing to topological non-trivial spaces, excited states, or superselection sectors.
One key consequence that arises when taking duality as the fundamental starting point is that whenever duality is violated for a region with non-trivial π i , then duality is also violated for the complementary region, which has non-trivial π d−2−i . Besides, the operators that violate duality in the complementary region are in one to one correspondence with those in the original region, and the commutation relations between both algebras are completely fixed. This provides a unified perspective on order/disorder parameters. These can be just defined as the operators that violate duality. They are necessarily non locally generated in the region in question and everything else follows from this. For global symmetries, we have intertwiners and twists. For local symmetries, we have (unbreakable) Wilson and 't Hooft loops, and the commutation relations that arise in our construction are exactly those enforced by the original definition in 't Hooft's seminal work [23]. Generalized symmetries follow similar patterns. In this light, the Dirac quantization condition, together with its generalizations, follows when enforcing causality on a possible completion of a net of algebras showing violations of duality.
Regarding gauge symmetries, being not physical symmetries, its true meaning has become a recurring theme in QFT. From the present perspective, the breaking of duality for ringshaped regions is an unambiguous physical remnant of the gauge symmetry. As we have shown, it is also related to a good definition of confinement order parameters. Indeed, loop order parameters satisfying area law necessarily need to violate duality in a ring. In this precise sense, the conventional confinement order parameters imply a violation of duality. In turn, this means the inclusion of algebras A(R) ⊆ (A(R )) is not saturated and entropic order parameters immediately appear that measure the non-trivial inclusion.
The last part of the paper has been devoted to study the properties of entropic order parameters in several cases of interest. In this context, there are some aspects to highlight. The first is the use of the entropic certainty relation. This quantitatively relates the physics of order and disorder parameters. We have confirmed such a prediction in different cases. The certainty relation gives a useful and geometrical picture of the origin and relations between the different laws followed by dual order parameters. A constant law for one order parameter forces an area law for the dual one. Area law for both parameters, or even area and perimeter laws, would be forbidden: the fluctuations in both parameters are high enough to prevent saturation of the certainty relation.
However, it is fair to say that we feel we have not yet understood how to profit from these relations in full force. For example, though it is known that area-area laws for complementary parameters should be forbidden, and we see a compelling heuristic reason for this, we could not prove this in the present approach in a rigorous manner. Including some information on the correlations between loops would be important for further progress. In the same line, our approach shows the importance to understand the behavior of wide loops, which are dual to thin loops. These latter have been the focus of almost all past efforts. A simple heuristic reasoning suggests that the change of a loop with the size is seed by the behavior of the dual loop in a self-consistent manner. A further understanding of this self-consistency is important to have a clearer picture of the RG flow on the order parameters.
There are d r representations of type r (one for each value of the index j) in the decomposition of the regular representation in irreducible ones.
Since the operators T (g) are Hermitian we have for the conjugate representationr Now we consider the operators based on two disjoint balls R 1 , R 2 , It is immediate these operators are under global group transformations (acting on both R 1 , R 2 ), and they are formed by linear combinations of operators with charge r in R 1 andr in R 2 . Therefore they are intertwiners of representation r. Using eq. (A.3) and where n r 3 r 1 r 2 is the fusion matrix of the group representations, we get intertwiners which close an algebra I r I r = r n r rr I r . (A.10) We also have from (A.8) Ir = (I r ) † , It is clear from (2.14,A.3,A.4) that these intertwiners and the invariant twists belong to the finite dimensional algebra of invariant operators of the form where f (a, b, c) = f (ha, hb, hch −1 ) imposes invariance under the global group. This algebra has dimension |G| 2 , and the invariant twists and intertwiners form Abelian subalgebras.

B Gauge field on a lattice
We consider pure gauge fields on the lattice based on a compact group G. We take a square lattice and think in terms of a finite group for simplicity. The basic variables (at fixed time) are elements U (ab) ∈ G of the gauge group G assigned to each oriented link l = (ab) joining neighbour lattice vertices a, b. The linkl = (ba) with the reverse orientation is assigned the inverse group element Ul = U (ba) = U −1 (ab) = U −1 l , and hence does not correspond to a different independent variable. The variables g a of the gauge transformations are elements of the group G attached to the vertices a of the lattice. The gauge transformation law is Consider the vector space V of all complex wave functionals |Ψ ≡ Ψ[U ], where U = {U (ab) } is an assignation of group elements to all links. The scalar product is defined in V as where U l is the variable corresponding to the link l = 1, ..., N L , for a lattice with N L links.
The gauge transformation at a vertex a is implemented by a unitary operator in V with the gauge transformed variables U given by (B.1) with g x = g δ a,x . Gauge transformations based at different points commute to each other.
The physical Hilbert space is the subspace H ⊂ V of gauge invariant functionals, The subspace H is also a Hilbert space with the scalar product (B. A maps H in itself. This is formalized with a conditional expectation This conditional expectation acts locally in the lattice. The gauge invariant gauge transformation operatorsC where [g] is the conjugacy class of g, and n [g] is its number of elements, form a set of gauge invariant constraint operators in A labelled by conjugacy classes [g]. These generators commute with all the elements of A and generate the center of this algebra. AllC ...Û r (a k a 1 ) = χ r (U (a 1 a 2 ) U (a 2 a 3 ) ...U (a k a 1 ) ) = χ r (U Γ ) , (B.10) where Γ = a 1 a 2 ...a k a 1 is an oriented closed path made by links in the lattice, the matrix indices are contracted, and χ r (g) is the character of the representation r. These magnetic operators are then labelled by closed lines and group representations. We have (W r Γ ) † = Wr Γ = W r Γ , whereΓ is Γ with the reversed orientation. W r Γ is unitary only if the representation is one dimensional. Wilson loops for elementary plaquettes will be called plaquette operators.
Wilson loops (at t = 0), together with the constraint operatorsC l · · ·Û r k l for the same link l appearing in Ψ[U ] into a linear combination of theÛ r l for different r using the Glebsch-Gordan decomposition. This shows the wave function is at most linear in each of theÛ r l . This gives place to the spin network representation [72].
Gauge invariant local Electric operators can be defined for each link and conjugacy class [g] of the group as E Linear combinations of these operators gives us an electric operator for any element c = g c g g, c g = c hgh −1 of the center of the group algebra, It is immediate that E c l commutes with all Wilson loops not passing through the link l, and with all other electric variables based on any link. However, it does not commute with Wilson loops passing through the link l. The minimal projectors of center of the group algebra P r = d r |G| g χ * r (g)g , P r P r = δ rr P r , are labelled by irreducible representations. To these operators it correspond the electric projectors Let us choose the representation of the algebra in the space of gauge-invariant vectors H. All gauge constraints are then set to the identity,C a link with respect to Γ. We take plaquette operators of representation r with boundaries in Γ andΓ and sew them in order to produce locally, after the final plaquette is added, the operator W r Γ Wr Γ . Sewing plaquettes to Wr Γ we can displace it laterally to finally obtain, by local operations in R, W r Γ Wr Γ based on the same loop Γ.
For two generic Wilson loops based on the same path we have the fusion rule where n r 3 r 1 r 2 = n r 3 r 2 r 1 are the fusion matrices of the group representations. Therefore we obtain that the following operator is locally generated We want to select only one Wilson loop in this linear combination. Notice that in the operation of sewing two loops along a link in eq. (B.18) if we started with two loops in different representations the result vanishes. Then we can sew the operator (B.21) with a plaquette operator in representation r sharing a link with Γ to obtain a Wilson loop in representation r along a curve deformed from Γ in a plaquette. We can move this back to the loop Γ sewing a plaquette again. Then we finally conclude that we can locally generate in the ring R any W r Γ such that n r rr = 0 for some representation r.
Loops winding n times with |n| > 1 along the S 1 direction of the region R we can deform with local operators to wind n times along the same line Γ. This corresponds to χ r (U n Γ ). As a function of U Γ this is a class invariant function, and as such, it can be linearly decomposed into a sum of characters with coefficients that depend on the group. Then it can be decomposed into elementary loops of different representations winding just once along Γ. If we have a product of two loops along curves winding once, it can be locally transformed into a product of loops for the same path Γ, which can be also decomposed into elementary loops of different representations. Then, in understanding the non locally generated operators, we only need to worry about the case of simple loops winding once.
The locally generated loops can fuse, and the result is also locally generated. Locally generated loops form a subalgebra of the fusion algebra. For an Abelian group for example, n r rr = 0 for any r = 1, and no non trivial loop can be formed additively in this way. To see the structure of the loops that are non locally generated, and to prove they are such, we have to discuss t' Hooft loops, that are certain combinations of electric operators.
For a z in the center Z of the group G, the class [z] consist of a single element z, and E z l = L z l is gauge invariant. For any (d − 2)-dimensional surface Σ in the dual lattice and each element z ∈ Z we define the t' Hooft operator This is analogous to the electric flux through Σ. Recalling that C z a ≡ 1 in the H representation, we can multiply these operators in a volume A of the lattice to get We have used here the fact that as z commutes with all elements of the group, the action of the gauge transformations on the interior links of A cancel. Therefore, the electric "flux" corresponding to z vanished on any closed surface. In consequence, the flux is the same operator for two surfaces with the same boundary. This means that the t' Hooft operator corresponding to z is independent of the precise surface Σ in the definition and depends only on the boundary ∂Σ. For later convenience, we call this d − 3-dimensional closed surface Γ . Then, we have the t' Hooft loop operators T z Γ , defined for any z ∈ Z. We have (T z Γ ) † = T z −1 Γ = (T z Γ ) −1 , and these operators are unitary.
We could have used the electric fields E [g] l for any [g] in (B.22) but if g / ∈ Z this operator does not commute with local operators along the surface, and therefore is not an operator that can be thought localized along the boundary of Σ.
For a one-dimensional loop Γ interlocked with Γ (winding number one) it is not difficult to see that This uses the irreducibility of the representation r which implies through Schur's lemma that z is represented inside the loop as a matrix proportional to the identity for z ∈ Z. Then χ r (U z) = φ r (z)χ r (U ). The value φ r (z) is a phase which corresponds to one of the (one dimensional) representations of Z, and we have φ r (z) = φ * r (z −1 ) = φ * r (z). A similar calculation shows that the t' Hooft loop commutes with all Wilson loops with trivial winding number with Γ . This is because they cross the same number of times the surface Σ in opposite directions giving factors φ r (z) and φ * r (z) an equal number of times. Another way to see this is that for zero winding between Γ and Γ we can deform Σ to lie outside the support of Γ.
Choosing a Γ in the complement of the region R and interlocked with Γ we conclude that Wilson loops along Γ for representation where φ r (z) = 1 for some z ∈ Z cannot be locally generated in the ring R. Any locally generated operator commutes with the t' Hooft loop and this operator does not commute with W r Γ if φ r (z) = 1.
Then, we have two sets of representations, both of them closed under fusion. One is formed by the representations generated by the fusion of rr for all r, which we have shown give locally generated WL. The other set of representations is the one that is trivial (proportional to the identity) in the center of the group. The complement of this set (i. e. representations which are non-trivial on the center) we have shown give WL that cannot be locally generated.
Any representation r restricted to the center of the group can be put into a diagonal form, where the diagonal elements are proportional to phases. The conjugate representationr is given by the conjugate matrices, and rr is proportional to the identity. Therefore the first set of representations is included in the second. We knew this already from the above reasoning about the locally generated representations. We will show how these two sets coincide.
We need to introduce the adjoint representation D a (g) of a group. This is a representation in the group algebra (that is of dimension |G|) given by the adjoint action, D a (g) ( h∈G b h h) = h∈G b h ghg −1 . The group algebra is isomorphic to a space of block-diagonal matrices r M dr×dr , where an element of the group is represented on the block r by the corresponding irreducible representation. The adjoint action reduces to the adjoint action of D r (g) on each block, which is equivalent to the tensor product representation rr. Therefore the character of the adjoint representation is χ a (g) = r χ r (g)χr(g), and it contains all irreducible representations that can be formed by fusion of rr. On the other hand, if we view the adjoint action on a basis |h given by the elements of the group, the adjoint action produces a permutation of the basis elements and we have D a (g) hh = δ h,gh g −1 . The character then has another representation as χ a (g) = h δ h,ghg −1 . This sum is the number of elements of the group that commute with g. The elements of the group that commute with g form a subgroup called the centralizer of g, which we write C(g). Then χ a (g) = |C(g)|. It is immediate that χ a (z) = |G| for z ∈ Z and χ a (g) < |G| for g / ∈ Z. In consequence lim n→∞ χa(g) n |G| n = θ Z (g), the characteristic function of the center, which equals 1 for elements on Z and 0 otherwise. Fusing the adjoint representation with itself n times we get the character χ a (g) n , and it follows that for n large enough this will contain any irreducible representation which is constant on the center of the group. This proves the statement that the fusion algebra generated by rr for all r coincides with one of the characters which are constant on the center. 26 We will call the loop operators corresponding to this set of characters Ξ 1 .
This proves that all Wilson loops which commute with the t' Hooft loops are locally generated in the ring and those that do not commute with at least one t' Hooft loop are not locally generated. Further, every character ϕ s of Z induces a representation in G whose character evaluated on Z is proportional to ϕ s . Therefore, for every t' Hooft loop there is a Wilson loop that does not commute with it. In other words, t' Hooft loops are not locally generated in the complement R of R.
The character χ r of each irreducible representation of G when evaluated on Z is proportional to the character ϕ s one irreducible character of Z. Then, the Wilson Loops W r Γ are divided into equivalence classes Ξ s according to the characters ϕ s , s = 1, · · · , |Z| of Z. The class of the identity is the class of locally generated loops Ξ 1 . Further, Ξ 1 Ξ s = Ξ s , and the different classes select Wilson loops that are locally transformable to each other. The dual sets of non locally generated operators in R and R are then labeled by Ξ s and T z , where z ∈ Z and s labels a character of Z.
As we mentioned before, in the continuum theory of a Lie group the adjoint Wilson loop can be broken into pieces by the Wilson lines (2.46) formed with the curvature as charged fields. In the lattice, this type of Wilson lines are represented by two plaquettes joined by a segment which is passed in the two opposite directions, see figure 15. The plane of the plaquettes represents the indices of the curvature in (2.46). The adjoint representation on the Lie algebra is the adjoint representation of the group discussed above when we look at elements near the identity.
In fact, in analogy to the case of finite groups, for a Lie group, the adjoint representation generates all the representations that arise from fusing rr. As an example, for the group SU (2), the adjoint representation of spin 1 generates all integer spin representations, and the same is true for the fusion j × j for any j. Only half-integer representations are not locally generated. The center of the group is Z 2 formed by the identity and the 2π rotation. These commute with the integer spin representations. For SO(3) we do not have any non locally generated representations, and the center is trivial.

B.3 Algebra and maximal nets
Summarizing, we have as dual non-local operators in a (pure) gauge theory the Wilson loops W s labeled by characters χ s of the center Z and t' Hooft loops labeled by elements of the center. For d = 4 the set of non-local operators in a ring is doubled since both types of loops live in topology S 1 . Hence in this case the non-local group is Z × Z * . for any pair of dyons. This is automatically consistent with (B.26). As in the Maxwell case, there could be several solutions for a maximal set satisfying these conditions, including taking all the electric charges (1, s) or all the magnetic monopoles (z, 1). Several examples are worked out for the centers of Lie groups in [49]. We do not know of a classification of solutions to this problem for general finite Abelian groups.

C Perimeter law for additive operators
We want to show that the vacuum expectation values W of operators in the additive algebra of the ring decrease at most with a perimeter law W ≥ c e −µR (C.1) for large radius R, where c, µ are constants that depend on the operator. This would prohibit an area law, the expected behavior of confinement order parameters, for additively generated operators. However, so that this statement makes sense, we have to further qualify the operator W living in a ring, and how it is supposed to depend on the size of the ring.
First, we have to construct the operator for different ring radius in such a way that is "the same type of operator" but for a different ring. Otherwise, any behavior as a function of R can be obtained by multiplying the operator by an arbitrary function of the radius. We also need to increase the size of the ring keeping the cross-section constant. It would also be convenient to use only operators with positive expectation values.
We first discuss a simpler case of correlators of operators localized in two balls which corresponds to the order parameters of global symmetries instead of gauge symmetries.

C.1 Two point correlators
We look at the behaviour of the correlator of a pair of operators localized in two balls as the distance R between the two balls goes to infinity. We can conveniently take operators However, in an inequality such as (C.1) we intend, on the contrary, for a lower bound. There is also a lower bound associated with the clustering of operators. To see this we have to select operators with positive expectation values such that f (R) could not be zero or change sign. This is easily done by taking O 1 and O 2 to be CRT conjugate operators and cannot vanish for a non-trivial operator. In a massive theory and provided that f (∞) = 0 this will decay exponentially fast. We will show it cannot decay faster than exponentially in any theory. The slope V (R) cannot be negative for any R = R 0 . Otherwise (C.7) will imply a slope always negative for R > R 0 , which will lead to increasingly negatives V (R), and a violation of the cluster property. Hence, V (R) is increasing and concave. It means there is a limit for the slope (C.8) 27 In addition, f (R) is a completely monotonic function [76].
If V ∞ > 0 is positive we have an "area law" for large radius If V ∞ = 0 we may have different situations. A perimeter law would be f (R) ∼ constant = | O | 2 . An intermediate case is the conformal case V (R) ∼ log(R). The potential cannot increase faster than the area, in analogy with the result for loops.
For an orbifold, the intertwiner, that is a non locally generated operator in the two balls, has a constant law if the symmetry is spontaneously broken and area law or some milder logarithmic law if the symmetry is not broken. Hence, it is a good order parameter for symmetry breaking. However, an operator locally generated (not an intertwiner), can also have both area and constant law. This is a difference with what we expect for operators on rings, where locally generated operators (hence non-good for order parameters) have a potential that increases at most as the perimeter. This is in part because an operator in the two balls may be locally generated for a particular orbifold but it might also be an intertwiner for other groups. A way to regain the expected behavior of locally generated operators is to take O as a CRT positive operator, O = QQ. This prevents O = 0 and leads to a constant law. In this case, O(− R)O( R) can never be an intertwiner for any orbifolding since it is the product of charge-less operators.

C.2 Constructing additive ring operators
In analogy with the previous discussion, we limit the order parameters to be CRT positive operators, that is operators of the form with λ ij positive definite and O i on the right wedge. We could also take the limits of these types of operators. This gives us in particular W > 0. We also impose W to be localized in a ring and invariant under the rotation symmetry of the ring. This in principle should allow us to select a "cross-section" for the operator with which to construct the sequence of operators for different ring radius. Both these properties can be also imposed on non locally generated loops. However, we consider the case of locally generated operators in this appendix.
To begin with, we can divide the angular span of the ring in N pieces, and consider operators O Now we impose CRT positivity which requires that O ab N = O ba 1 with respect to the plane separating the two, and is convenient to impose O ab 1 = O ba 1 with respect to the plane bisecting the first angular sector. This relations together with rotation symmetry imply that the operator W is CRT positive. We also have that all the partial operators W k 1 ,k 2 = O a 1 a 2 k 1 O a 2 a 3 k 1 +1 · · · O ana 1 k 2 (C.14) are also CRT positive with respect to the plane dividing them in two.
The same argument used in the section 3.1 shows these operators have concave potential. We have W −k,k W −k ,k ≤ W −k,k W −k ,k , (C. 15) with the convention of mod N classes for the position indices. Then writing W k 1 ,k 2 = e −V (k 2 −k 1 ) we have for k, k > 0, for some µ ≥ 0, and this value of µ depends only of the expectation value of two adjacent operators.
Then the idea is to use the same matrix operator as a seed for creating operators in larger circles using rotations in larger circles. When N and R are large, and R/N = c fixed, the two adjacent operators have to be readjusted for larger circles by a very tiny contrary rotation of the reflected operators. In the limit, the value of µ converges, and we get the lower bound W (R) ≥ e −2π (µ/c)R−V (0) . (C.19)

C.3 Wilson type operators
The CRT positive rotational invariant and additive operators form an algebraically closed convex cone. To get a result at this level of generality we should analyze how to take limits of Z N symmetric to rotational symmetric operators, etc. We do not pursue the mathematical details of this construction any further. Rather we analyze the case of Wilson type operators of the form W = trP e i ds A(s) , (C. 20) which are suggested both by the standard presentation of Wilson loops and by the preceding discussion. Here P is the path ordering, A(s) is the rotation of A(0), and A(0) is a matrix of fields smeared in the direction perpendicular to the loop. A(0) is the cross-section from which can obtain a sequence of operators for rings of different radius by using rotations for different circles. CRT positivity here is reduced to where α(s) is a smearing function equal to one inside the interval (s 1 , s 2 ) and zero outside (s 1 − , s 2 + ) and smooth everywhere. The shape of the smearing function between one and zero is the same for any interval (s 1 , s 2 ). The concavity holds for the expectation values of these operators. To get the perimeter law for W we then only have to take care of the step going from an almost closed loop to a closed one. Between the closed loop and loops with one gap and two gaps, there is also an inequality from CRT by reflecting in some plane that does not pass through the gap. Given that the loops with one and two gaps satisfy perimeter laws with the same perimeter coefficient it follows the perimeter law for the full closed loop.
If A(s) is a gauge field in some representation where the loop cannot be broken by charged fields the construction does not work. The partial operators are not gauge-invariant. If we fix the gauge several problems appear. If we do not, the partial operators have expectation value zero, nothing goes wrong, except that we cannot use the calculation to put a useful lower bound on the closed loop.

D Conformal transformations of the ring
Consider a ring R formed by rotating around the z axis a circle of radius R, centred around the point z = 0, x = r 0 , in the plane z, x. Written in cylindrical coordinates the surface of the ring is given by the equation The intersection with the plane x, y is the circular corona with inner radius L = r 0 − R. In Cartesian coordinates the surface of the ring corresponds to the quartic equation (L 2 + 2LR + x 2 + y 2 + z 2 ) 2 − 4(L + R) 2 (x 2 + y 2 ) = 0 . We have η ∈ (0, 1), with the limit 0 corresponding to thin rings and 1 for thick ones.

(D.7)
A natural way to parametrize it is by means of x 1 (s 1 ) = √ α cos(s 1 ) , (D.8) x 2 (s 1 ) = √ α sin(s 1 ) , (D.9) x 3 (s 2 ) = √ 1 − α cos(s 2 ) , (D.10) x 4 (s 2 ) = √ 1 − α sin(s 2 ) , (D. 11) where s 1 , s 2 ∈ (−π, π]. Applying the inverse of the stereographic projection (D.6) to the above parametrization, we can explicitily check that the transformed surface in R 3 satisfies the equation (D.2) for This torus has cross ratio η α = 1 − α. In the same way, we can check that the interior of the torus is the region on the sphere described by the inequality whereas the exterior region is then described by the opposite inequality. The advantage of describing the torus on the sphere is that we inmediately see that the exterior of a torus is also a torus because it is described by the region (D.14) In the same way, using the inverse of another stereographic projectionφ : R 3 ∪ {∞} ↔ S 3 ⊂ R 4 on the plane x 1 = 0, 30 the region (D.14) is mapped into a torus in R 3 with dimensions L 1−α and R 1−α , and cross ratio η 1−α = α.

(D.17)
Due to the conformal invariance of the problem, it is important to remark that the above transformation has to be considered as a transformation that maps equivalence class of torus interiors with cross ratio η onto the equivalence class of torus exteriors with cross ratio 1 − η. After this transformation, we obtain a torus in the same conformal class satisfying (D.12), and it is to this new torus that we have to apply (D. 15-D.17).

E Relative entropy of a subgroup of non local operators
Let G be an Abelian group of non local operators and H ⊆ G a subgroup. In this appendix we study an upper bound to the relative entropy S A∨H (ω|ω • E H ) which gives a comparison of the state ω with the one resulting from erasing the information on the expectation values of the operators in H. LetG andH be the dual groups. We have the diagram Here G/H is a subgroup ofG formed by all the characters which are the identity over H The algebra of the groupG is represented as the one of functions over G through the characters χg(g). The probabilities p(g) are obtained from the expectation values as p(g) = 1 |G| g χg(g) * ω(g) .

(E.5)
The entropy is SG(ω) = − g∈G p(g) log p(g) . The probabilities q(g) of this new state are then given by q(g) = 1 |G||H| g,h χg(g −1 h) ω(g) . (E.8) For each element g ∈ G there are |H| elements gH with equal probability q(g). Then for each element x ∈ G/H (which we identify with a representative in G) we can define a probabilitỹ q(x) = 1 |G|