Semi-Abelian gauge theories, non-invertible symmetries, and string tensions beyond N-ality
PPrepared for submission to JHEP
YITP-21-01
Semi-Abelian gauge theories, non-invertiblesymmetries, and string tensions beyond N -ality Mendel Nguyen, Yuya Tanizaki, Mithat Ünsal Department of Physics, North Carolina State University, Raleigh, NC 27607, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
E-mail: [email protected], [email protected],[email protected]
Abstract:
We study a d lattice gauge theory with gauge group U(1) N − (cid:111) S N , whichis obtained by gauging the S N global symmetry of a pure U(1) N − gauge theory, and wecall it the semi-Abelian gauge theory. We compute mass gaps and string tensions for boththeories using the monopole-gas description. We find that the effective potential receivesequal contributions at leading order from monopoles associated with the entire SU( N ) rootsystem. Even though the center symmetry of the semi-Abelian gauge theory is given by Z N ,we observe that the string tensions do not obey the N -ality rule and carry more detailedinformation on the representations of the gauge group. We find that this refinement is due tothe presence of non-invertible topological lines as a remnant of U(1) N − one-form symmetryin the original Abelian lattice theory. Upon adding charged particles corresponding to W -bosons, such non-invertible symmetries are explicitly broken so that the N -ality rule shouldemerge in the deep infrared regime. a r X i v : . [ h e p - t h ] J a n ontents U(1) N − lattice gauge theory with S N global symmetry 4 U(1) N − lattice gauge theory 42.1.1 Villain formulation 42.1.2 Wilson formulation 62.2 Mass gap and spectrum 62.2.1 Multi-component Coulomb gas representation of the Villain form 72.2.2 Long-distance effective theory 82.3 Wilson loops and string tensions 102.3.1 Explicit evaluation of string tensions 12 S N global symmetry 133.2 Z N center symmetry 153.3 String tensions beyond N -ality, and noninvertible topological lines 163.4 Effect of dynamical electric particles 18 A.1 Differential forms on the lattice 21A.2 3d compact QED on the lattice 24
B Wilson to Villain at weak coupling 26
In all calculable, confining
SU( N ) gauge theories in continuum, such as the Polyakov modelon R [1, 2], the Seiberg–Witten model on R [3], and deformed Yang–Mills and adjointQCD on R × S [4, 5], the gauge dynamics Abelianize to U(1) N − at long distances. Whilethese models have taught us much about confinement, they have several features that wedo not expect of the dynamics of non-Abelian confinement. One particularly salient featurecommon to all of these models is the complete Higgsing of the S N subgroup of the SU( N ) gauge group. This Higgsing of S N pervades the physics of these theories: it always givesrise to multiple masses for the dual photons and generically to multiple fundamental stringtensions [6–8] (see Ref. [9] for a case in which fundamental string tensions remain equal).– 1 –he characterization of string tensions is an especially important point of differencebetween Abelianizing and non-Abelianizing confining gauge theories. Indeed, it is well-known that at asymptotically large distances, the string tensions of confining gauge theoriesthat do not undergo Abelianization should be solely characterized by center symmetry. Thisis not the case in the Abelianizing theories mentioned above, at least within the low-energyeffective field theory, where string tensions are dictated by charges under U(1) N − ratherthan N -ality [6–8].Rather curiously, however, it has been observed in numerical experiments that thedynamics of non-Abelian confinement admit an intermediate distance scale where the stringtensions are not solely characterized by center symmetry either [10–17]: they carry moredetailed information on the representations of the gauge group. This naturally suggeststhat we should try to construct an intermediate theory between Abelian and non-Abelianworlds to leverage what we know about the former to learn more about the latter. Asmentioned above, an understanding of unbroken S N should be an important clue in thisdirection.Thus, the purpose of this work is to construct a 3-dimensional lattice model in whichthese considerations can be addressed quite explicitly. We call it the semi-Abelian gaugetheory . To define it, we begin with a pure Abelian lattice model (henceforth to be referredto as the ‘Abelian model’) with gauge group U(1) N − such that the permutation group S N is present as a global symmetry. The semi-Abelian theory is then obtained by gauging this S N , and its gauge group is given by G gauge = U(1) N − (cid:111) S N . (1.1)As pure gauge theories are not usually equipped with non-Abelian global symmetries, theglobal or local S N symmetry of these models has some rather interesting consequences.We first show that in both models, the mass gap is generated via Polyakov’s mecha-nism whereby the proliferation of lattice monopole-instantons results in Debye screening.Crucially, the unbroken (or un-Higgsed) S N symmetry implies that the effective potentialreceives equal contributions from the monopoles associated with the entire SU( N ) rootsystem, which in turn leads to exact degeneracy for the N − dual photon masses. (As thegauge groups of both theories can be realized as subgroups of SU( N ) , we find it useful touse the language of SU( N ) representations.) This feature sharply contrasts with the massgeneration in the Polyakov model on R or in deformed Yang–Mills on R × S , where theeffective potential is sourced at leading order only by the monopoles associated with the(affine) simple roots and the N − dual photon masses are not degenerate.After studying these properties of local operators, we move on to study propertiesof test electric particles, which can be described by the behavior of Wilson loops. In3 spacetime dimensions, the Coulomb potential is already log-confining, but due to themass gap generated by the monopole-instantons, the interparticle potential becomes linear-confining. Using the dual formulation of the Wilson loop, we give a semi-classical formulafor the string tensions within a reasonable ansatz. We then find that there are infinitelymany string tensions. In particular, the semi-Abelian theory furnishes a unique fundamentalstring tension. – 2 –his, however, raises a puzzle about the string tensions. In order to study the spectralproperties of the confining forces for test quarks, we would like to have some symmetrythat acts nontrivially on the Wilson loops. One is the well-known center symmetry, whichhas been recently axiomatized in the framework of higher-form symmetry [18, 19]. Beforegauging S N , our model has a U(1) N − -form symmetry, which provides sufficiently strongselection rules to support infinitely many string tensions. But after gauging S N , the centerof the gauge group becomes tiny, Z ( G gauge ) = Z N , (1.2)so the -form symmetry group becomes Z N , as it is in SU( N ) Yang–Mills. And as we knowfrom
SU( N ) Yang–Mills, the Z N -form symmetry can only explain the N -ality behaviorof the string tensions at asymptotically large distances. However, the list of string tensionsfor the semi-Abelian gauge theory turns out to be unchanged by the gauging of S N . Thus,as in the case of SU( N ) Yang–Mills at intermediate distances, the string tensions of thesemi-Abelian theory cannot be dictated by N -ality alone.We find a resolution to this puzzle in a not-so-obvious but important symmetry ofthe semi-Abelian theory, a non-invertible symmetry . Indeed, after the generalization ofsymmetry to higher-form symmetry, it has been recognized that the most essential feature ofa conservation law is the existence of topological defects, at least in the context of relativisticquantum field theories (QFTs). In other words, as long as one keeps intact the existenceof topological defect operators, one may make up a new kind of symmetry by replacing orweakening other features of the generalized global symmetry of Ref. [18] (e.g. higher-groupsymmetry [19–26]). Non-invertible symmetries are also generated by topological defects, buttheir fusion rules do not conform to the usual group multiplication; as the name suggests,a non-invertible symmetry transformation need not have an inverse (which of course neveroccurs if the transformations form a group). The notion of non-invertible symmetry is stillin its infancy, and it seems that its mathematical formulation has been so far establishedonly in -dimensional spacetimes. Nevertheless, the utility of such topological operators inprobing quantum systems has been elucidated in several recent studies, as the notion ofsymmetry itself tends to be broadened [27–35]. In that context, the new symmetry goesby various names, such as non-invertible symmetry, categorical symmetry, etc. Here, wewould like to emphasize that the non-invertible symmetry clarifies an important feature ofour -dimensional semi-Abelian gauge theory. Thanks to the simplicity of the model, thesymmetry considerations we propose can be checked against concrete calculation.We construct a generator of a continuous non-invertible symmetry, and compute its ac-tion on several Wilson loops. By looking at its eigenvalues, we show that we can distinguishdifferent string tensions even if they correspond to representations of the same N -ality. Wealso discuss conditions where such extra selection rules by noninvertible symmetry are lostby the addition of dynamical electric particles, and we compare them with the standardstring-breaking arguments to check that they are consistent. Finally, as an application, wediscuss an example where the non-invertible symmetry is explicitly broken to a discretesub-symmetry, so that even though the number of string tensions becomes finite, there stillremain some string tensions beyond N -ality.– 3 – U(1) N − lattice gauge theory with S N global symmetry There are two basic models that we study in this paper:• U (1) N − Abelian gauge theory with discrete non-Abelian global symmetry S N • U (1) N − (cid:111) S N semi-Abelian gauge theoryThe second one can be obtained by gauging the S N U(1) N − lattice gauge theory The U (1) N − lattice gauge model with the S N global symmetry can be realized either bya standard Wilson-type formulation [36] or by a Villain-type formulation [37]. Since theseprovide somewhat complementary perspectives, we end up working with both. To give the Villain formulation, we consider a link field A (cid:96) valued in R N − and a plaquettefield n p valued in the root lattice Γ r ⊂ R N − of SU( N ). We take for the action S = 14 πe (cid:88) p ( F p + 2 π n p ) (2.1)where F p = ( d A ) p is the field-strength and e is the gauge coupling. The partition functionis given by Z = (cid:88) { n p ∈ Γ r } (cid:90) R N − [ d A (cid:96) ] e − S . (2.2)This theory is invariant under the -form gauge symmetry A (cid:96) → A (cid:96) + ( d λ ) (cid:96) , λ s ∈ R N − , (2.3)and the -form gauge symmetry A (cid:96) → A (cid:96) + 2 π β (cid:96) , n p → n p − ( d β ) p , β (cid:96) ∈ Γ r . (2.4)In view of the fact that R N − / (2 π Γ r ) (cid:39) U(1) N − , we see that this indeed defines a U(1) N − gauge model.One way to understand this Villain-type formulation [1, 38] is to imagine that we hadbegun with pure R N − gauge theory, Z = (cid:90) R N − [ d A (cid:96) ] exp (cid:32) − πe (cid:88) p F p (cid:33) , (2.5)and then considered gauging the discrete subgroup π Γ r of the R N − A (cid:96) (cid:55)→ A (cid:96) + θ (cid:96) , θ (cid:96) ∈ R N − , ( d θ ) p = 0 . (2.6)The simplest way to do that is to introduce the discrete Γ r -valued plaquette field n p , andthen demand that the local transformations (2.4) be gauge redundancies. Minimal couplingto the field n p would then produce the action (2.1) and the partition function (2.2).– 4 – lobal symmetries: Let us now discuss the global symmetries of this model. First, asalready noted above, there is a
U(1) N − U(1) N − rather than R N − thanks to the 1-form gauge structure (2.4).Importantly, the theory has a discrete non-Abelian 0-form global symmetry, A (cid:96) (cid:55)→ Π A (cid:96) , n p (cid:55)→ Π n p , (2.7)under O( N − ) transformations Π that preserve the root lattice Γ r . Such transformationsconstitute the automorphism group of the SU( N ) root system, and therefore the symmetrygroup here is G [0]global = (cid:40) S N (cid:111) Z ( N > (cid:39) Z ( N = 2). (2.8)The S N corresponds to the Weyl group of SU( N ), which is generated by the reflectionsin the hyperplanes orthogonal to the roots A (cid:96) (cid:55)→ A (cid:96) − α ( α · A (cid:96) ) , n p (cid:55)→ n p − α ( α · n p ) , α ∈ Φ, (2.9)where Φ is the set of roots for SU( N ). The pair ( A (cid:96) , n p ) thus transforms in the standardrepresentation D std of S N , which is the ( N − -dimensional irreducible representation.Meanwhile, the Z is simply generated by the reflection A (cid:96) (cid:55)→ − A (cid:96) , n p (cid:55)→ − n p , (2.10)which we may think of as charge conjugation. We note that, for N = 2 , these two operationsare identical.Note that the existence of the non-Abelian global symmetry (2.8) is somewhat unusualfor a pure gauge theory. In general, pure gauge theories without matter fields, eitherAbelian or non-Abelian, do not possess non-Abelian global symmetries. In the U(1) N − gauge theory we are considering, this symmetry is present. The gauging of the permutationgroup S N will generate a genuinely non-Abelian gauge theory, which we shall investigate.The basic observables we are concerned with are the Wilson loops, which are here givenby W w ( C ) = exp (cid:18) i (cid:90) C w · A (cid:19) , (2.11)with w in the weight lattice Γ w of SU( N ) . Note that it is invariance under the 1-formgauge transformations (2.4) that requires the electric charge to be a weight. The Wilsonlines transform under the 0-form discrete symmetry (2.7) as W w ( C ) (cid:55)→ W Π − w ( C ), (2.12)and under the 1-form center symmetry (2.6) as W w ( C ) (cid:55)→ W w ( C ) exp (cid:18) i (cid:90) C w · θ (cid:19) . (2.13)– 5 – .1.2 Wilson formulation To construct the U (1) N − lattice gauge theory in the Wilson formulation, we consider N gauge fields a (cid:96) , . . . , a N(cid:96) and a Lagrange multiplier v (cid:96) which is an integer-valued link-field.The dynamics is determined by the action S W = β (cid:88) p N (cid:88) i =1 (1 − cos f ip ) − i (cid:88) (cid:96) N (cid:88) i =1 v (cid:96) a i(cid:96) , (2.14)where the f ip = ( d a i ) p are the field-strengths. In the partition function, we integrate over a i(cid:96) ∈ [0, 2 π ] and sum over v (cid:96) ∈ Z : Z = (cid:88) { v (cid:96) ∈ Z } (cid:90) π [ d a i(cid:96) ] e − S W . (2.15)In particular, summation over v (cid:96) in the partition function produces the constraint N (cid:88) i =1 a i(cid:96) = 0 mod 2 π , (2.16)so that only N − of the photons are physical. One nice thing about this formulation is that the S N symmetry is manifest; it actssimply by permuting the N photons: ( a (cid:96) , . . . , a N(cid:96) ) (cid:55)→ ( a P (1) (cid:96) , . . . , a P ( N ) (cid:96) ) , P ∈ S N . (2.18)For now, we shall prefer to work with the Villain form over the Wilson one, becausethe former enjoys exact dualities which allow us to analyze the dynamics most simply.Nevertheless, the two formulations are equivalent at weak coupling, as we demonstrate inAppendix B. Later on, in Section 3 where we gauge the S N global symmetry, we will findthe Wilson form more convenient. In this subsection, we discuss the mass gap of the lattice Abelian gauge theory with S N global symmetry.First, as we shall review in Section 2.2.1, we note that the Villain form is exactly dualto a multi-component Coulomb gas; that is, the partition function can rewritten in the form Z = (cid:88) { q (˜ x ) ∈ Γ r } exp − πe (cid:88) ˜ x ,˜ x (cid:48) v (˜ x − ˜ x (cid:48) ) q (˜ x ) · q (˜ x (cid:48) ) , (2.19) We could integrate out v (cid:96) and any one of the photon fields. Then after some simple field redefinitions,we would obtain the action S W = β (cid:88) p N (cid:88) i =1 (1 − cos( ν i · f p )), (2.17)where the ν i are the weights of the defining representation of SU( N ) and f p is the field-strength of an( N − )-component Abelian gauge field a (cid:96) . – 6 –here v (˜ x ) is the lattice Coulomb potential, and q (˜ x ) is a Γ r -valued scalar field on thedual lattice. Here, one can interpret { q (˜ x ) } as a configuration of magnetic monopoles; q (˜ x ) is the magnetic charge of the magnetic monopole at ˜ x . As is familiar, the proliferationof monopoles in the Euclidean description of the vacuum results in Debye screening, andhence, the correlation length remains finite for any nonzero value of the coupling [1]. Whilethis is more or less self-evident, we can go further and obtain the long-distance effectivefield theory: Z = (cid:90) D σ exp − e π (cid:90) d x | d σ | + M (cid:88) α ∈ Φ + (cid:0) − cos( α · σ ) (cid:1) , (2.20)where the dual photon field σ is a π Γ w -periodic scalar, Φ + is a set of positive roots forSU( N ), and M ∝ e − const. /e /e . This effective description shows very clearly the presenceof a nonzero mass gap. It will be derived in Section 2.2.2.We can immediately observe that the N − dual photons must have exactly the samemass. The degeneracy is a consequence of the S N global symmetry inherited from themicroscopic theory. To see this, note that the dual photons σ transform in the standardrepresentation D std of S N : σ (cid:55)→ D std ( P ) σ , P ∈ S N . (2.21)The mass matrix for the dual photons, ( M σ ) ij = M (cid:88) α ∈ Φ + α i α j , (2.22)is also invariant under the S N transformation, D std ( P ) M σ D − ( P ) = M σ , P ∈ S N . (2.23)Since D std is irreducible, it follows from Schur’s lemma that M σ must be proportional tothe identity matrix. The mass gap is thus the ( N − -fold degenerate eigenvalue of M σ .By taking the trace of M σ and using α = 2 , one easily finds the mass gap to be M gap = √ N M ∝ √ Ne e − const. /e . (2.24) Here we show that the Villain form (2.2) of our theory is exactly dual to multi-componentCoulomb gas (2.19), using standard techniques in Abelian lattice gauge theory [2, 39–41].We derive the equivalence very briefly here, but the detailed derivation for the single-component
U(1) gauge theory is reviewed in Appendix A.2.We first note that the Poisson summation formula can be generalized on the weightand root lattices to give (cid:88) n p ∈ Γ r exp (cid:18) − πe ( F p + 2 π n p ) (cid:19) = (cid:88) k p ∈ Γ w exp (cid:0) − πe k p + i k p · F p (cid:1) (2.25)– 7 –p to an overall coefficient. By performing the A (cid:96) integration exactly, we obtain theconstraint (d † k ) (cid:96) = 0, which can be easily solved by setting ∗ k = d m , (2.26)where m (˜ x ) is a Γ w -valued scalar field on the dual lattice. After this replacement, thepartition function becomes Z = (cid:88) { m (˜ x ) ∈ Γ w } exp − πe (cid:88) ˜ (cid:96) (d m ) (cid:96) . (2.27)We now wish to replace m (˜ x ) by a continuous field; it can be done with the help of thePoisson summation formula again, this time in the form (cid:88) m (˜ x ) ∈ Γ w δ ( σ (˜ x ) − π m (˜ x )) = (cid:88) q (˜ x ) ∈ Γ r exp(i q (˜ x ) · σ (˜ x )), (2.28)which introduces the dual photon field σ (˜ x ) . The result is Z = (cid:90) [ d σ (˜ x )] (cid:88) { q (˜ x ) ∈ Γ r } exp (cid:32) − e π (cid:88) ˜ x ( ∂ − µ σ (˜ x )) + i (cid:88) ˜ x q (˜ x ) · σ (˜ x ) (cid:33) . (2.29)After performing the Gaussian integration over σ , we arrive at the multi-component Coulombgas representation (2.19). We now wish to pass to the long-distance effective description (2.20) [2, 39–41]. To this end,we first split the Green function ∆ − in (2.19) into two parts by adding and subtracting (∆ + M ) − : ∆ − = ∆ − (1 + ∆ /M ) − + (∆ + M ) − = u M PV + w M PV . (2.30)Here, u M PV (˜ x ) is the Green function of the Pauli–Villars regulated Laplacian ∆ M PV ≡ ∆(1+∆ /M ) , and w M PV (˜ x ) is the Yukawa Green function. Since w M PV (˜ x ) decays exponentiallyfast, we can take w M PV (˜ x ) = w M PV (0) δ ˜ x ,0 . Furthermore, it is straightforward to show that w M PV (0) = v (0) − O (1 /M PV ) ≈ − O (1 /M PV ) [42].With the above decomposition, we rewrite the Coulomb gas partition function as Z = (cid:88) { q (˜ x ) ∈ Γ r } exp − πe (cid:88) ˜ x ,˜ x (cid:48) u M PV (˜ x − ˜ x (cid:48) ) q (˜ x ) · q (˜ x (cid:48) ) − I (cid:88) ˜ x q (˜ x ) , (2.31) For clarity, we ignore the effect of nontrivial spacetime topology. This representation may be thought of as a ‘ Γ w -ferromagnet’ by analogy with the corresponding ex-pression with Z in place of Γ w . The Γ w -ferromagnet representation is exactly dual to the Γ r -componentCoulomb gas representation (2.19). While the latter converges rapidly at weak coupling e → , the formerconverges rapidly at strong coupling e → ∞ . – 8 –here I ≡ πv (0) /e , and then reintroduce the dual photon field σ to get Z = (cid:90) [ d σ (˜ x )] e − e π (cid:80) ˜ x σ (˜ x )∆ M PV σ (˜ x ) (cid:88) { q (˜ x ) ∈ Γ r } e i (cid:80) ˜ x q (˜ x ) σ (˜ x ) e − I (cid:80) ˜ x q (˜ x ) . (2.32)At this point, we want to perform a cluster expansion of the partition function. Forweak coupling, I is large, and so e − I is exponentially small. Thus, at leading order insemi-classics, we can restrict the summation over q (˜ x ) to { } ∪ Φ . Indeed, α = 2 for each α ∈ Φ , so all the monopoles whose charges are roots have the same minimal action I , andthere are N ( N − degenerate saddles at leading order in semi-classics. Performing thesummation over q (˜ x ) with this restriction then yields (cid:88) q (˜ x ) ∈{ }∪ Φ e i q (˜ x ) σ (˜ x ) e − I q (˜ x ) ≈ exp e − I (cid:88) α ∈ Φ + cos( α · σ (˜ x )) . (2.33)Finally, inserting this into (2.32), we get Z = (cid:90) [ d σ (˜ x )] exp − e π (cid:88) ˜ x σ (˜ x )∆ M PV σ (˜ x ) + 2 e − I (cid:88) ˜ x (cid:88) α ∈ Φ + cos( α · σ (˜ x )) , (2.34)which, upon taking the continuum limit, coincides with (2.20).We note that, in this derivation, we have neglected the effect of the spacetime topology,and thus the periodicity of the dual photon field σ is undetermined. Had we taken it intoaccount, we would have identified σ as a π Γ w -periodic scalar. (We elaborate on thissubtlety in Appendix A.2.) Remarks:
The fact that the sum over monopoles goes over all roots α ∈ Φ and thatall monopoles associated with these roots have the same action distinguishes our U(1) N − lattice gauge theory with S N symmetry from Yang–Mills adjoint Higgs systems which ex-hibit dynamical Abelianization SU( N ) → U(1) N − . In the latter, if the adjoint Higgs arealgebra-valued, as in the Polyakov model [1], the sum over monopoles at leading order insemi-classics is restricted to the N − simple roots α ∈ ∆ , while if the adjoint Higgs aregroup-valued, as in deformed Yang–Mills [4], the sum over monopoles is restricted to the N affine simple roots. There are monopoles associated with non-simple roots as well, butthese are higher action and do not contribute at leading order; in general, the monopolessplit into Z N -orbits with hierarchical fugacities e − S (cid:29) e − S (cid:29) · · · (cid:29) e − ( N − S . In ourpresent construction, S N permutation symmetry guarantees that all N ( N − monopolesassociated with the roots have the same action. In theories like the Polyakov model andSeiberg–Witten theory, S N is part of the gauge structure of the microscopic theory, but itis spontaneously broken by the vacuum expectation value of the Higgs field which imposes It is also worth nothing that in N = 4 SU( N ) super Yang–Mills theory softly broken down to N = 1 ∗ on R × S as well, it is necessary to sum over monopoles associated with non-simple roots in order tocapture the ground state properties correctly [43]. This data is encoded in an elliptic superpotential, butthe S N symmetry is still Higgsed in generic vacua. – 9 – p px x-3 x+1x x+1+2x+2 C = @D Figure 1 . Left: The dual of the plaquette p is the link ∗ p on the dual lattice intersecting p asshown. Right: D is the shaded region bounded by the curve C . The Poincare dual [ D ] is a bump1-form on the dual lattice that is 1 on each of the red links shown, and 0 everywhere else. an ordering on the eigenvalues of the adjoint Higgs. These models therefore exhibit O ( N ) different types of fundamental string tensions. We will see how the string tensions behavein our U(1) N − Abelian model in the following subsection.
In this subsection, we show that the Abelian gauge model confines and we approximatelydetermine the string tensions.We begin by showing how a Wilson loop W w ( C ) = exp (cid:0) i (cid:82) C w · A (cid:1) with electric charge w ∈ Γ w is computed in the long-distance effective theory [2]. For our purposes, it will sufficeto take C to be a contractible loop, so that it is the boundary of a 2-dimensional surface D . We can then write W w ( C ) = exp (cid:18) i (cid:90) D w · F (cid:19) = exp (cid:32) i (cid:88) p [ D ] ∗ p ( w · F p ) (cid:33) . (2.35)Here we have introduced the Poincaré dual [ D ] of D ; it is a bump -form on the dual lattice(see Figure 1) given by [ D ] ∗ p = (cid:40) if p ⊂ D ,0, otherwise . (2.36)Now let us repeat the derivation of the dual theory, this time with the insertion of theWilson loop. Using the Poisson resummation (2.25), the path-integral weight becomes exp (cid:0) − πe k p + i k p · F p + i[ D ] ∗ p ( w · F p ) (cid:1) , (2.37)where the last term comes from the Wilson loop. The integration over A produces theconstraint, d † ( k + ∗ ( w [ D ])) = 0 , which can be solved by ∗ k = − w [ D ] + d m , (2.38)– 10 –nstead of (2.26). At this point, the rest of the derivation proceeds exactly as before, andwe obtain (cid:104) W w ( C ) (cid:105) = (cid:90) D σ exp − e π (cid:90) d x (cid:12)(cid:12)(cid:12) d σ − π w [ D ] (cid:12)(cid:12)(cid:12) + M (cid:88) α ∈ Φ + (cid:0) − cos( α · σ ) (cid:1) . (2.39)This expression shows that the Wilson loop is realized as a defect operator in the dualformulation. That is to say, it is evaluated by removing the loop C from the spacetime andrestricting the path integral to configurations satisfying (cid:72) S d σ = 2 π w for small loops S that link with the loop C . Taking the ratio with the unconstrained path integral, we obtainthe expectation value of the Wilson loop.We are now in a position to approximately determine the string tensions. It will sufficeto compute the functional integral in (2.39) in the classical approximation. For convenience,let us take the loop C as well as D to lie in the z ≡ x = 0 plane. If we take the loop C tobe so large that D essentially fills the z = 0 plane, then the action density localizes around z = 0 . As a result, the area law decay for the Wilson loop W w ( C ) is observed, (cid:104) W w ( C ) (cid:105) ∼ exp (cid:16) − T w Area( D ) (cid:17) , (2.40)and its string tension is given by the minimal action density T w = min σ ( z ) e π (cid:90) + ∞−∞ d z (cid:18) d σ d z (cid:19) + M (cid:88) α ∈ Φ + (cid:16) − cos( α · σ ) (cid:17) , (2.41)with the boundary condition σ ( −∞ ) = 0, σ (+ ∞ ) = 2 π w . (2.42)To go further, let us take as a plausible ansatz σ ( x ) = w σ ( z ), (2.43)and then proceed to evaluate the string tension analytically within this ansatz. Substituting(2.43) into (2.41), we obtain T w = min σ ( z ) e π (cid:90) + ∞−∞ d z w (cid:18) d σ d z (cid:19) + M (cid:88) α ∈ Φ + (cid:16) − cos( α · w σ ) (cid:17) = e Mπ (cid:90) π d σ (cid:118)(cid:117)(cid:117)(cid:116) w (cid:88) α ∈ Φ + (cid:16) − cos( α · w σ ) (cid:17) , (2.44)which is the Bogomol’nyi–Prasad–Sommerfield (BPS) bound [44, 45]. Although this is justan upper bound for the actual string tension, we assume that it gives a reasonable estimate.– 11 – .3.1 Explicit evaluation of string tensions Using the formula (2.44), we shall evaluate the string tensions explicitly for a few cases.Here, we take w = µ , 2 µ , µ , which correspond to the highest weights of the fundamen-tal, symmetric, and anti-symmetric representations of SU( N ) , respectively. We will alsocomment on the case w = α ∈ Γ r , corresponding to the adjoint representation of SU( N ) .Let us start with w = µ , which is the highest weight of the fundamental representationof SU( N ) . We obtain T µ = e Mπ N − √ N (cid:90) π d σ (cid:112) − cos( σ )= 4 e Mπ N − √ N . (2.45)We will use this quantity as a unit for the other string tensions.We next consider w = 2 µ , the highest weight of the SU( N ) two-index symmetricrepresentation. Since σ wraps S twice, we find that T µ = 2 T µ . (2.46)Thus, the symmetric string tension is twice the fundamental one, which suggests that thesymmetric string can be interpreted as the sum of two independent fundamental strings.The multi-string ansatz is also a candidate, which may give a reasonable approximation ofconfining strings [46], so we will compare it with (2.43) for other strings, too.For the two-index anti-symmetric string, w = µ , we find T µ = e Mπ √ N − √ N (cid:90) π d σ (cid:112) − cos( σ )= 8 e Mπ N − √ N = 2( N − N − T µ . (2.47)For N = 3 , µ gives the conjugate representation of µ , and in this case we indeed findthat T µ = T µ . For N > , we find T µ < T µ < T µ = 2 T µ , and T µ ≈ T µ for N (cid:29) . Since µ = µ + ( µ − α ) , we can understand this upper bound T µ as a sumof two-independent fundamental strings T µ = T µ + T µ − α . Our calculation shows that,for the anti-symmetric string, the ansatz (2.43) gives a more severe upper bound for T µ .Lastly, let us consider the adjoint string w = α ∈ Φ . Applying the formula (2.44)within the ansatz (2.43), we obtain it as T α = e Mπ (cid:90) π d σ (cid:112) (1 − cos(2 σ )) + 2( N − − cos( σ ))= 4 e Mπ (cid:32) √ N + ( N − √ − (cid:114) NN − (cid:33) = 1 N − (cid:32) N + ( N − √ N √ − (cid:114) NN − (cid:33) T µ . (2.48)– 12 –ccording to this formula, the adjoint string tension satisfies T α ≥ T µ , and turns out tobe only slightly larger than T µ . But here it turns out that we can do a little bit better.Let us explicitly take w = α , and consider a double-string ansatz, in which the adjointstring consists of two fundamental strings, µ and α − µ . Up to permutation, µ and α − µ are related by complex conjugation, and we thus find T α = T µ + T α − µ = 2 T µ . (2.49)Therefore, for the adjoint string, the two-independent-string ansatz is slightly better than(2.43), unlike the case of anti-symmetric string. In this section, we consider the
U(1) N − (cid:111) S N gauge theory obtained by gauging the S N global symmetry of the U(1) N − gauge theory considered in Section 2. We call it thesemi-Abelian gauge theory. S N global symmetry In relativistic quantum field theories, global symmetry is generated by a set of codim- defects, which are topological and obey the group-multiplication law [18]. When theglobal symmetry is discrete, we can gauge it by summing over all possible networks of suchcodim- defects. This procedure may be obstructed by anomalies, which are characterizedby a topological action in one higher dimension [47–49]. We also note that the gaugingprocedure admits the freedom to add a topological phase to each network configuration ofthe topological defects as long as it is consistent with locality and unitarity.In this section, we gauge the S N symmetry of the U(1) N − lattice gauge theory ofSection 2. Assuming that its low-energy description enjoys emergent Lorentz symmetrydue to the cubic lattice rotational invariance, the gauging procedure of S N should fit intothe above general discussion. Absence of the S N anomaly is guaranteed by the explicitconstruction of the lattice gauge theory. As extra topological terms, there are Dijkgraaf-Witten (DW) terms [50] characterized by H ( B S N , U(1)) , which are nontrivial for all N ≥ . In this paper, we limit ourselves to the case without the d S N DW term.We shall now construct the semi-Abelian gauge theory on a cubic lattice, and thesimplest way to proceed is to gauge S N in the Wilson formulation (2.14). For concreteness, Although detailed information is not relevant for us as we neglect the nontrivial DW twist, let us give itsfull information for completeness, which may be useful for possible extensions. By the universal coefficienttheorem, we obtain H d ( B S N , U(1)) (cid:39) H d +1 ( B S N , Z ) (cid:39) H d ( B S N , Z ) because they only have the torsionpart. We can find in literatures that the list of the 3d DW twist is given as N · · · H ( B S N , U(1)) Z Z Z ⊕ Z Z ⊕ Z ( Z ) ⊕ Z · · · (3.1)For N ≥ , this group cohomology stabilizes and H ( B S N , U(1)) (cid:39) ( Z ) ⊕ Z , i.e. we can add threedistinct DW terms, two of which give the ( ± phases and the another one gives the phases exp (cid:0) π i12 n (cid:1) , inthe path integral of S N gauge fields. – 13 –t is convenient to realize the semi-Abelian gauge symmetry by N × N matrices. We canrealize an element of U(1) N − (cid:111) S N inside SU( N ) as P · C ∈ SU( N ), (3.2)where C = diag(e i a , · · · , e i a N ) with det( C ) = 1 describes the Cartan components, and P ∈ S N is the N × N matrix representation of a Weyl reflection, which is realized as asigned permutation matrix. The group multiplication law is given as ( P · C )( P · C ) = ( P P ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ S N · (( P − C P ) C ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ U(1) N − , (3.3)and this expression elucidates the semi-direct structure in a concrete manner. Letting ( P (cid:96) · C (cid:96) ) ∈ SU( N ) denote the link variable, the gauge-invariant plaquettes for the Lagrangianconsist of two terms: Re β tr N − (cid:89) (cid:96) ⊂ ∂p ( P (cid:96) · C (cid:96) ) + β tr N − (cid:89) (cid:96) ⊂ ∂p P (cid:96) . (3.4)The second term is the gauge-invariant kinetic term only for S N . By sending β → + ∞ ,we can impose the flatness condition on the S N gauge field, (cid:89) (cid:96) ⊂ ∂p P (cid:96) = N ∈ S N . (3.5)As it is this limit that fits into the general discussion given above for the continuum de-scription, we shall work with this flatness condition (3.5) on the S N link variables. We canreadily check that the Lagrangian (3.4) is equal to the Wilson action (2.14) when the S N gauge fields are trivial, i.e. P (cid:96) = 1 for all the links (cid:96) , by putting β = β . In this manner,we obtain the U(1) N − (cid:111) S N gauge theory out of the U(1) N − pure gauge theory.Because of the flatness condition (3.5), the local dynamics should not be much affectedby the gauging of S N , and all the interesting things in the deep infrared have to do withthe global aspects of the theory. To be more precise, let us assume that we prepare a suffi-ciently large torus T for the spacetime and that we are interested in computing correlationfunctions inside an open ball B ⊂ T , which has a trivial topology. Using the flatnesscondition, we may perform a local S N gauge transformation so that the S N gauge fields P (cid:96) are fixed to equal 1 inside B . Hence, the correlation functions should be identical withthose of the U(1) N − theory of Section 2 as it has a nonzero mass gap, as long as we neglectexponentially small corrections that vanish in the thermodynamic limit. In this sense, thegauging of S N is locally trivial.More physically, by sending the parameter β → ∞ in the Lagrangian, we make themagnetic monopoles for the S N gauge group extremely heavy. As a result, the S N gaugefields are deconfined; i.e. the Wilson loops of S N gauge fields obey the perimeter law atany length scale. Now, assume that we wish to compute the correlation functions of localoperators. For the sake of exposition, consider a two-point function of the U(1) N − gaugetheory (cid:104) O ( x ) O ( x ) (cid:105) U(1) N − , (3.6)– 14 –hough it is straightforward to extend the discussion to general n -point functions. As the S N global symmetry is not spontaneously broken in the U(1) N − theory, this correlationfunction has a non-zero expectation value in the thermodynamic limit only if O ( x ) O ( x ) contains an S N -singlet component. Thus, we may assume that O ( x ) O ( x ) is S N singletwithout loss of generality. Here, we note that the operator O , O can be S N non-singlet,but they have to be mutually conjugate representations. By introducing an S N Wilsonline W (S N ) ( x , x ) connecting x and x , we can construct the S N gauge invariant op-erator O ( x ) W (S N ) ( x , x ) O ( x ) . Since the S N gauge field is deconfined, we find that (cid:10) O ( x ) W (S N ) ( x , x ) O ( x ) (cid:11) U(1) N − (cid:111) S N is invariant under any continuous deformation ofthe path connecting x and x . In particular, by choosing a trivial path which does not goaround any nontrivial cycle of the spacetime, we obtain (cid:68) O ( x ) W (S N ) ( x , x ) O ( x ) (cid:69) U(1) N − (cid:111) S N = (cid:104) O ( x ) O ( x ) (cid:105) U(1) N − (3.7)in the thermodynamic limit. Thus, any local correlation function of the U(1) N − theorycan be recovered in the semi-Abelian gauge theory. Z N center symmetry In this and the following subsections, we discuss properties of the (electric) Wilson loopsin order to identify the string tensions from the viewpoint of the symmetry. Here, we payespecial attention to the -form symmetry, or center symmetry, of the U(1) N − (cid:111) S N gaugetheory.The -form symmetry is generated by codim- topological defects, whose fusion ruleobeys group multiplication [18]. In general, when we consider a pure gauge theory witha gauge group G gauge , the theory enjoys a -form symmetry with the symmetry group Z ( G gauge ) , which is the center of G gauge . Since this acts as Z ( G gauge ) phase rotations onWilson loops, this has been historically called the center symmetry.In our case, the gauge group is G gauge = U(1) N − (cid:111) S N , and thus Z ( G gauge ) = Z N . (3.8)To see this, it is convenient to use the embedding of U(1) N − (cid:111) S N ⊂ SU( N ) used above,and to consider the defining representation of the latter. Using Schur’s lemma, one sees thatthe N × N matrix representation of center elements must be proportional to the identitymatrix. Such matrices are included only in U(1) N − , which is the same as the Cartan factorof SU( N ) , and thus the center elements of U(1) N − (cid:111) S N are the same as those of SU( N ) .Before gauging S N , the -form symmetry group is given by Z (U(1) N − ) = U(1) N − sincethe theory is an Abelian gauge theory without any electric matter fields. Therefore, in viewof the -form symmetry, one might be led to claim that the semi-Abelian gauge theoryshould be more similar to SU( N ) gauge theories than the U(1) N − theory from which itcame.This, however, raises the following puzzle about the string tensions. In the U(1) N − theory, there are infinitely many different string tensions depending on the representationsof Wilson loops, which are characterized by charges of the U(1) N − -form symmetry. As– 15 –e have seen in the previous subsection, the local dynamics is not affected by the gaugingof S N because we can locally set the S N gauge field to be zero by gauge transformations.As long as we measure string tensions using large and contractible Wilson loops, the samediscussion from before should apply here, and thus, there have to be infinitely many differentstring tensions also for the U(1) N − (cid:111) S N gauge theory. But this seems rather unnatural,because the Z N -form symmetry is too weak to give selection rules for these string tensions.To put the question another way: How is the presence of infinitely many different string tensions compatible withthe finite center symmetry?
To make things more concrete, let us construct the generator of Z [1] N out of (U(1) [1] ) N − generators of the U(1) N − gauge theory. In terms of the dual photon field σ ∈ R N − / π Γ w from the monopole gas description in Section 2.2, the generators of (U(1) [1] ) N − are givenby U ( k ) θ ( C ) = exp (cid:18) i θ π (cid:90) C α k · d σ (cid:19) , k = 1, . . . , N − (3.9)where the transformation parameter θ is π periodic due to the π Γ w -periodicity of σ . Aftergauging S N , these operators are no longer gauge invariant because the dual photon field σ transforms under the standard representation of the Weyl permutations S N . Nevertheless,the generators of Z [1] N can be constructed as U n ( C ) ≡ N − (cid:89) k =1 U ( k ) πN kn ( C ) = exp (cid:18) i nN (cid:90) C ( α + 2 α + · · · + ( N − α N − ) · d σ (cid:19) . (3.10)Thanks to the periodicity of σ , the U n ( C ) are invariant under S N transformations and thusremain good operators for the semi-Abelian gauge theory. Moreover, we have the groupmultiplication law U n ( C ) U m ( C ) = U n + m mod N ( C ) (3.11)so U ( C ) ≡ U ( C ) only generates the Z N subgroup of U(1) N − . Other generic combinations (cid:81) k U ( k ) θ k cannot satisfy the S N invariance, and they drop out from the possible generatorsof the -form symmetry. N -ality, and noninvertible topological lines Let us explicitly check whether or not the string tensions of the semi-Abelian gauge the-ory obey the standard N -ality rule. Using the embedding U(1) N − (cid:111) S N ⊂ SU( N ) , weconstruct the Wilson loops with the SU( N ) gauge field first, and then we restrict it to the U(1) N − (cid:111) S N gauge field. As we can locally eliminate the S N gauge field, we may restrictthe SU( N ) gauge field to its diagonal component in a naive way, as long as the Wilson loopis contractible. The N -ality of the obtained Wilson loop is the same as that of the Wilsonloop with SU( N ) gauge fields.Let us write W k ( C ) = exp i (cid:16) µ − k − (cid:88) j =1 α j (cid:17) · (cid:90) C A , k = 1, . . . , N , (3.12)– 16 –o that, for example, the fundamental Wilson loop is given by W fd ( C ) = W ( C ) + W ( C ) + · · · + W N ( C ). (3.13)Each Wilson loop W i in W fd has the same string tension, and for large loops C it obeys thearea law: (cid:68) W fd ( C ) (cid:69) ∼ exp( − T µ Area). (3.14)Under the -form symmetry, Z [1] N : W fd (cid:55)→ e π i /N W fd , (3.15)or, more precisely, (cid:104) U ( C ) W fd ( C ) (cid:105) = exp (cid:18) π i N Link( C , C ) (cid:19) (cid:104) W fd ( C ) (cid:105) . (3.16)In order to determine whether string tensions are controlled by the -form symmetry, letus consider the adjoint Wilson loop, W ad ( C ) = (cid:88) i (cid:54) = j W i ( C ) W ∗ j ( C ) = | W fd | − N . (3.17)This has trivial N -ality, but we can readily check that its string tension is not zero usingthe result of Section 2.3: (cid:68) W ad ( C ) (cid:69) ∼ exp( − T α Area), (3.18)with T α (cid:39) T µ . This example clearly tells us that the string tensions of the semi-Abeliangauge theory carry detailed data of its gauge-group representations, which cannot be cap-tured by the Z [1] N symmetry.Something new is needed to explain the failure of the N -ality rule, and this is where thenon-invertible topological lines [27–35] come in. We can easily construct such an operatorby summing over all the S N conjugates of U (1) θ ( C ) : T θ ( C ) ≡ N ! (cid:88) P ∈ S N P U (1) θ ( C ) P − = 1 N ( N − (cid:88) α ∈ Φ exp (cid:18) i θ π (cid:90) C α · d σ (cid:19) . (3.19)Since this operator is S N singlet, it can be a physical operator of the S N -gauged theory.Since each operator in the sum is topological, so too is T θ ( C ) . Therefore, this S N -invariantoperator shares important features with the -form symmetry generators. However, thegroup multiplication law is not satisfied for T θ ( C ) , as one can easily check: T θ ( C ) T θ (cid:48) ( C ) (cid:54) = T θ + θ (cid:48) ( C ). (3.20) Following the same logic, we can in fact construct many more continuous families of non-invertiblesymmetries. Indeed, we can average over all S N conjugates of an arbitrary product U ( k ) θ ( C ) · · · U ( k r ) θ r ( C ) of the operators (3.9) to get a non-invertible symmetry generator T ( k , ... , k r ) θ , ... , θ r ( C ) . In particular, the non-invertible symmetry generator T (1, ... , N − θ , ... , θ N − ( C ) with θ k = 2 πk/N will coincide with the Z [1] N center symmetrygenerator U ( C ) , as one can readily check from (3.11). Thus, the Z [1] N center symmetry is actually containedwithin a continuous family of non-invertible symmetries. – 17 –ecause of the violation of the group multiplication property, we cannot regard T θ ( C ) as agenerator of an ordinary -form symmetry in contrast with (3.11). Instead, it is a generatorof a non-invertible symmetry.Let us consider the component of the Wilson loop that corresponds to the weight w ∈ Γ w . Its eigenvalue for T θ is given by N ( N − (cid:88) α ∈ Φ exp (i θ α · w ) . (3.21)As a consequence, the fundamental Wilson loop transforms as W fd (cid:55)→ N (cid:0) N − θ ) (cid:1) W fd . (3.22)More importantly, the adjoint Wilson loop also transforms nontrivially as W ad (cid:55)→ ( N − N −
3) + 4( N −
2) cos( θ ) + 2 cos(2 θ ) N ( N − W ad . (3.23)This elucidates that we can detect the detailed information of the Wilson loop beyond N -ality by using the non-invertible topological line operator T θ .As another example, we can detect the difference between the symmetric and anti-symmetric two-index representations, W sym and W asym , of SU( N ) , whose highest weightsare given by µ and µ , respectively. We have to note, however, that W sym is not an eigen-operator of T θ , because the two-index symmetric representation of SU( N ) decomposes intotwo irreducible representations of U(1) N − (cid:111) S N . Since µ = 2 µ − α and (2 µ ) · α =2 , each charge in the anti-symmetric representation appears in the list of charges of thesymmetric representation exactly once, and thus the correct eigen-operator is W sym − W asym .Indeed, one can check that (cid:68) ( W sym − W asym )( C ) (cid:69) ∼ exp( − T µ Area), (cid:68) W asym ( C )) (cid:69) ∼ exp( − T µ Area), (3.24)with T µ < T µ , as we have discussed in Section 2.3. We find ( W sym − W asym ) (cid:55)→ N − θ ) N ( W sym − W asym ), (3.25)and W asym (cid:55)→ ( N − N −
3) + 2 + 4( N −
2) cos( θ ) N ( N − W asym . (3.26)This gives another explicit demonstration of the fact that one can distinguish different stringtensions for representations of the same N -ality with the help of the topological operator T θ . In the previous section, we discussed the behavior of string tensions for the pure semi-Abelian gauge theory. String tensions do not obey the N -ality rule, and the presence ofnon-invertible topological lines explain why they carry more detailed information. In this– 18 –ection, we discuss what will happen to the string tensions once dynamical electric chargesare added.Once electric charges are incorporated as dynamical excitations, their pair creation canbreak confining strings if it is energetically favorable. If the fundamental electric chargeis added, we expect that all the confining strings can be broken and all Wilson loops willobey the perimeter law. If the adjoint charge is added instead, we expect that the stringtensions should obey the N -ality rule, because the adjoint Wilson loop would then obey theperimeter law. Can we justify these expectations from the viewpoint of topological lines?For this purpose, we need to identify which line operators cease to be topological oncethe dynamical electric charges are included. If a line acts nontrivially on the Wilson loopcorresponding to the dynamical excitations, then it is no longer topological after introducingdynamical charges [33]. This is because the corresponding Wilson loop can end on chargedlocal operators, so that the linking number is no longer well-defined; in other words, thetopological invariance of the symmetry operator is lost.Let us add dynamical adjoint particles, and then determine whether or not T θ is topolog-ical. Since the eigenvalue of W adj has to be for any topological operator, T θ is topologicalonly if ( N − N −
3) + 4( N −
2) cos( θ ) + 2 cos(2 θ ) N ( N −
1) = 1. (3.27)This is solved only by θ = 0 mod π , and thus only the trivial one T θ =0 = 1 is topological.This implies that the non-invertible symmetry ceases to be an exact symmetry once anadjoint matter field is added. On the other hand, the Z N -form symmetry is kept intactbecause the generator acts trivially on W ad . In this case, the string tensions obey the N -ality rule at least if the Wilson loops are sufficiently large, which is consistent with theobservation for d SU( N ) Yang–Mills theory.As a nontrivial exercise, we can add dynamical particles corresponding to n Γ r with n > , instead of W ad . Then the non-invertible line T θ is topological if ( N − N −
3) + 4( N −
2) cos( nθ ) + 2 cos(2 nθ ) N ( N −
1) = 1, (3.28)and this has nontrivial solutions, θ ∈ (2 π/n ) Z . Therefore, the continuous part of thenon-invertible symmetry T θ is explicitly broken by dynamical electric charges n Γ r , but thediscrete part T θ =2 πk/n , k = 1, . . . , n still generates a good non-invertible symmetry. As aresult, Wilson lines distinguished by T θ =2 πk/n can have different string tensions even if theyshare the same N -ality. In this paper, we have studied the properties of the semi-Abelian gauge theory in spacetimedimensions, where the gauge group is G gauge = U(1) N − (cid:111) S N . As we have imposed theflatness condition on the S N gauge field, we can locally eliminate it completely, so thespectral properties of the mass gap and string tensions can be calculated as the U(1) N − theory. We have seen that the mass gap is generated via the Polyakov mechanism as a– 19 –onsequence of monopole-instanton proliferation. We can classify their magnetic chargesusing the SU( N ) representation, and all the monopoles for the roots give equally dominantcontributions to the effective potential. This point is very different from the Polyakov modelor QCD(adj) with an S compactification, where only the monopoles associated with thesimple roots play the dominant role, and it comes from the S N invariance of our model.Using the dual formulation, we also computed various string tensions, and we found thatthere are infinitely many different string tensions. When the S N symmetry is not gauged,this can be explained very naturally in the context of the -form symmetry because thecenter of U(1) N − is U(1) N − itself, and thus the -form symmetry group is large enoughto explain the selection rules between infinitely many confining strings.A puzzle arises, however, after gauging S N , because the center symmetry is just Z ( G gauge ) = Z N . This is because most of the elements of U(1) N − do not commute withthe permutations, and the permutation invariance requires that the center elements be pro-portional to the identity matrix. Thus, the -form symmetry of semi-Abelian gauge theoryis as small as that of SU( N ) Yang–Mills theory, where the string tensions are character-ized by N -ality alone. Therefore, for the semi-Abelian theory, there is a clear discrepancybetween the actual behavior of the string tensions and the natural expectation from Z N center symmetry.We find that the discrepancy is resolved by recognizing the presence of noninvertiblesymmetry. We constructed the topological line operator T θ out of the U(1) N − -formsymmetry generators, which remain well-defined and topological after gauging S N but donot satisfy the group multiplication law. Though this operator is noninvertible, its actionon the Wilson lines yield eigenvalues that are able to distinguish representations with thesame N -ality. Thus, we have demonstrated the utility of an extended notion of symmetryin a d toy example of a gauge theory.We should mention that the formal development of non-invertible symmetry is still animportant task. In the case of higher-form or higher-group symmetry, their formalizationnot only provided the rigorous definition and generalization of the center symmetry, butalso gave new tools to analyze interacting QFTs, such as generalizations of anomaly match-ing [51–63]. It would be very nice if this repertoire of useful tools could be enhanced toinclude non-invertible symmetry.Lastly, let us present some speculation. As we stated in the introduction, a similarbehavior regarding the N -ality rule has been observed in simulations of SU( N ) Yang–Millson the lattice: there is an intermediate distance scale where the quark-antiquark potentialexhibits linear confinement but its string tension depends on the details of the gauge-grouprepresentation. Though it is widely believed that the string tension becomes solely dictatedby N -ality once the quark-antiquark separation becomes sufficiently large, it is logicallypossible that ‘sufficiently large’ is parametrically larger than the strong length scale Λ − at which confinement sets in. For instance, viewing N as a parameter, it may very wellbe that the N -ality rule sets in at a distance scale h ( N )Λ − (cid:29) Λ − , where h ( N ) → ∞ as N → ∞ . We think it would be an intriguing possibility if, even in pure Yang–Mills, someapproximate notion of non-invertible symmetry could be used to explain the behavior ofstring tensions beyond N -ality at these intermediate distances.– 20 – more striking example may be QCD with fundamental or two-index matter fields,where the 1-form Z N center symmetry is either completely or partially lost, or Yang–Millstheories with simply-connected gauge groups without a center, such as G . Even in caseswhere 1-form symmetry is completely lost, we believe that an approximate non-invertiblesymmetry could potentially give a precise meaning to confinement of arbitrary test charges,and therefore provide the long sought definition of confinement in such theories. Acknowledgments
We thank Shailesh Chandrasekharan, Hanqing Liu, Hersh Singh, Misha Shifman, MikeCreutz, Tin Sulejmanpasic, and Aleksey Cherman for useful discussions. The authorsthank the YITP–RIKEN iTHEMS workshop “Potential Toolkit to Attack NonperturbativeAspects of QFT –Resurgence and related topics–” (YITP-T-20-03) for providing opportu-nities of useful discussions. The work of Y. T. was partially supported by JSPS KAKENHIGrant-in-Aid for Research Activity Start-up, 20K22350. M. Ü. is supported by the U.S.Department of Energy, Office of Science, Division of Nuclear Physics under Award DE-SC0013036.
A Review of Abelian duality on the lattice
A.1 Differential forms on the lattice
There is, on the lattice, a close analog of the notion of differential forms, and it is especiallyconvenient for treating Abelian lattice gauge theories. Here we give a somewhat informalintroduction to this formalism, which we use throughout this article. See [38] for a moresystematic discussion.We begin with a bit of lattice geometry. Consider a d -dimensional cubic lattice Λ d . Sucha lattice contains ‘ r -cells’ c ( r ) for each r = 0, 1, . . . , d . Thus, the -cells are the sites s , the -cells the links (cid:96) , the -cells the plaquettes p , the -cells the cubes c , etc., and everythingis oriented. For example, for a link (cid:96) = ( x ; (cid:98) µ ) , the oppositely oriented link is given by − (cid:96) ≡ ( x + (cid:98) µ ; − (cid:98) µ ) , and these are to be viewed as distinct objects despite correspondingto the same unoriented edge. By convention, whenever we write a sum or product over r -cells, we do not double count r -cells that only differ by orientation.The ‘boundary operator’ ∂ takes an r -cell into the (oriented) sum of the ( r − -cells thatconstitute its boundary. For example, the boundary operator on a plaquette p = ( x ; (cid:98) µ , (cid:98) ν ) yields ∂ ( x ; (cid:98) µ , (cid:98) ν ) = ( x ; (cid:98) µ ) + ( x + (cid:98) µ ; (cid:98) ν ) − ( x + (cid:98) ν ; (cid:98) µ ) − ( x ; (cid:98) ν ). (A.1)Importantly, the boundary operator is nilpotent, ∂ = 0 . By a slight abuse of notation, wewrite c ( r − ⊂ ∂c ( r ) (A.2) In the case of sites, an opportunity for confusion may arise. In this notation, the sites s and − s correspond to the same point x , say, but are equipped with opposite orientations. – 21 – pp pp
243 1
Figure 2 . Coboundary operator on a link in a 3d lattice: δ(cid:96) = p + p + p + p if the r -cell c ( r ) contains in its boundary the ( r − -cell c ( r − . We thus have (tautologically) ∂c ( r ) = (cid:88) c ( r − ⊂ ∂c ( r ) c ( r − . (A.3)We also have a kind of dual to the boundary operator, the ‘coboundary operator’ δ . Ittakes an r -cell into the sum of the ( r + 1) -cells that each contains c ( r ) in its boundary, δc ( r ) = (cid:88) ∂c ( r +1) ⊃ c ( r ) c ( r +1) . (A.4)For example, for a link (cid:96) = ( x ; (cid:98) in a three-dimensional lattice, we have δ ( x ; (cid:98)
1) = ( x ; (cid:98) (cid:98)
3) + ( x ; (cid:98) (cid:98)
2) + ( x − (cid:98) (cid:98) (cid:98)
1) + ( x − (cid:98) (cid:98) (cid:98) (A.5)(see Figure 2). It is easy to show that δ = 0 .There is another lattice ˜Λ d , the ‘dual lattice’, that is naturally associated with theprimary lattice Λ d . The points of ˜Λ d are given by ˜ x = x + ( (cid:98) · · · + (cid:98) d ) , with x any pointof Λ d . These lattices are connected by an operator ∗ , which takes r -cells in the primarylattice into ( d − r ) -cells in the dual lattice and vice versa; it is defined as follows: for an r -cell c ( r ) in the primary lattice, ∗ c ( r ) is the unique ( d − r ) -cell in the dual lattice suchthat c ( r ) and ∗ c ( r ) intersect transversally, and such that the orientation of the ordered pair ( c ( r ) , ∗ c ( r ) ) is positive. For example, for a plaquette p = ( x ; (cid:98) (cid:98) in a 3d lattice Λ , we have ∗ p = (˜ x − (cid:98) (cid:98) (see Figure 1). On r -cells, we have ∗ = ( − ) r ( d − r ) . (A.6)We can now define differential forms on the lattice. An r -form ω is simply a gadgetthat assigns a value ω c ( r ) to each r -cell c ( r ) , and it extends as a linear map. To comparewith more conventional lattice field theory notation, consider for example a 1-form θ . Wemay write its value on a link (cid:96) = ( x ; (cid:98) µ ) as θ (cid:96) ≡ θ µ ( x ). (A.7)We define the ‘exterior differential’ operator d to take r -forms to ( r + 1) -forms accordingto the formula ( d ω ) c ( r +1) ≡ (cid:88) c ( r ) ⊂ ∂c ( r +1) ω c ( r ) = ω ∂c ( r +1) . (A.8)– 22 –o again compare with more conventional notation, we note that the differential d θ of the1-form θ on a plaquette p = ( x ; (cid:98) µ , (cid:98) ν ) is given by ( d θ ) p = θ µ ( x ) + θ ν ( x + (cid:98) µ ) − θ µ ( x + (cid:98) ν ) − θ ν ( x ). (A.9)We also define the dual d † of the exterior differential, the ‘codifferential’, which takes r -forms to ( r − -forms, according to the formula ( d † ω ) c ( r − ≡ (cid:88) c ( r ) ⊂ δc ( r − ω c ( r ) = ω δc ( r − . (A.10)It is easy to see that d = ( d † ) = 0 . The star operator ∗ takes r -forms on the primarylattice to ( d − r ) -forms on the dual lattice, and vice versa, according to the formulae ( ∗ ω ) ˜ c ( d − r ) = ω ∗ ˜ c ( d − r ) , ( ∗ ˜ ω ) c ( d − r ) = ˜ ω ∗ c ( d − r ) . (A.11)It is easy to show that on r -forms, we have ∗ = ( − ) r ( d − r ) . (A.12)One of the more useful features of lattice form notation is that it enables us to ‘integrateby parts’ mindlessly. That is, we have the formula (cid:88) c ( r ) ( d ω ) c ( r ) τ c ( r ) = (cid:88) c ( r − ω c ( r − ( d † τ ) c ( r − . (A.13)Actually, it is this partial integration formula that justifies calling d † the dual of d. Toillustrate the utility of the notation, let us prove (A.13): L.H.S. ≡ (cid:88) c ( r ) (cid:88) c ( r − ⊂ ∂c ( r ) ω c ( r − τ c ( r ) = (cid:88) c ( r − (cid:88) c ( r ) ⊂ δc ( r − ω c ( r − τ c ( r ) ≡ R.H.S. (A.14)Finally, let us discuss the lattice analog of the ‘Hodge decomposition’. As in thecontinuum, we define the Laplacian on forms by ∆ = dd † + d † d. In particular, on 0-forms ϕ , we have (∆ ϕ )( x ) = d (cid:88) µ =1 [2 ϕ ( x ) − ϕ ( x + (cid:98) µ ) − ϕ ( x − (cid:98) µ )]. (A.15) Forms annihilated by ∆ are called ‘harmonic’. It is simple to show that harmonic forms areannihilated by both d and d † . The Hodge decomposition is the statement that any r -form ω can be written uniquely as ω = d α + d † β + η , (A.16)where η is harmonic. We will not prove this here.– 23 – .2 3d compact QED on the lattice Here we discuss the dual representation of 3d
U(1) lattice gauge theory following the pre-sentation of Ref. [41] (see also [39, 40]). We also give some attention to global issuesinvolving the spacetime topology. We note that, although we restrict our presentation tothree dimensions, many techniques used here are also applicable in four-dimensional space-time lattices, where interesting phase diagrams have been expected through electromagneticdualities [64–68].We start from the Wilson formulation of the
U(1) lattice gauge theory in d = 3 space-time dimensions: exp( − S ) = exp (cid:32) β (cid:88) p (cos f p − (cid:33) = (cid:89) p e β (cos f p − . (A.17)Since the action is periodic in f p , we can expand exp( − S ) as a Fourier series:e β (cos f p − = (cid:88) k p ∈ Z e i k p f p I k p ( β ) e − β , (A.18)where I k p ( β ) is the modified Bessel function of the first kind of order k p . This representationis useful because it allows us to integrate over the link fields in the Abelianized theory in astraightforward manner.The partition function can be rewritten as Z = (cid:90) π [ d a (cid:96) ] exp( − S ) = (cid:88) { k p ∈ Z } (cid:90) π [ d a (cid:96) ] exp (cid:32) i (cid:88) p k p f p (cid:33) (cid:89) p I k p ( β ) e − β . (A.19)From this expression, in the weak coupling limit β (cid:29) , we can obtain the Villain form.Using the asymptotic expansion, e − β I k p ( β ) ∼ √ πβ e − β k p , we can rewrite the summationover k p by Poisson summation formula, (cid:88) k p ∈ Z e i k p f p √ πβ e − β k p = (cid:88) n p ∈ Z e − β ( f p − πn p ) (A.20)Here, n p can be viewed as the flux passing through the corresponding surface p . The totalflux passing through the surface of the cube c centered at ˜ x is (cid:73) ∂c n = q (˜ x ), (A.21)which is just the magnetic charge located at ˜ x . In the following, we concentrate only onthis weak-coupling limit that is exactly equivalent to the Villain formulation. Dual formulation, from Λ to ˜Λ : In order to obtain the dual representation of theVillain form, we perform the exact integration over a (cid:96) before the summation over k p in(A.19). As (cid:80) p k p (d a ) p = (cid:80) (cid:96) (d † k ) (cid:96) a (cid:96) , the exact integration over a (cid:96) enforces the constraint, ( d † k ) (cid:96) = 0. (A.22)– 24 –s a result, the partition function can be written as a constrained sum over the k p : Z = (cid:88) { k p ∈ Z } (cid:40)(cid:89) (cid:96) δ ( d † k ) (cid:96) ,0 (cid:41) (cid:40)(cid:89) p e − β k p (cid:41) . (A.23)To construct the dual formulation of the theory, it is useful to turn the constrained suminto an unconstrained sum. To this end, we consider the decomposition of ( ∗ k ) ˜ (cid:96) as ( ∗ k ) ˜ (cid:96) = (d m ) ˜ (cid:96) + ˜ a ˜ (cid:96) , (A.24)where m ˜ s is an integer-valued scalar field, and ˜ a ˜ (cid:96) is an integer-valued link field on the duallattice. Since k p satisfies the constraint (A.22), ˜ a can be regarded as a flat connection. Incomputing the partition function, we may make the replacement, k p → ( ∗ d m ) p + ( ∗ ˜ a ) p , and the constrained sum over { k p ∈ Z } becomes an unconstrained sum over { m ˜ s ∈ Z } and [˜ a ] ∈ H ( ˜Λ, Z ) . As a result, the partition function in the weak-coupling limit takes thesimple form, Z = (cid:88) [˜ a ] ∈ H (cid:88) { m ˜ s ∈ Z } exp − β (cid:88) ˜ (cid:96) ( d m + ˜ a ) (cid:96) , (A.25)after using the duality relation (A.24). This model is sometimes called the Z -ferromagnetwhen ˜ a = 0 .Two more steps are needed to convert Z -ferromagnet representation to a continuumQFT. First, convert the sum into an integration over a continuous variable. Using thePoisson resummation identity repeatedly through the lattice ˜Λ , (cid:88) m (˜ x ) ∈ Z δ ( σ (˜ x ) − πm (˜ x )) = (cid:88) q (˜ x ) ∈ Z e i q (˜ x ) σ (˜ x ) , (A.26)we immediately obtain the partition function as an infinite dimensional integral, Z = (cid:88) [˜ a ] ∈ H (cid:90) [ d σ (˜ x )] (cid:88) { q (˜ x ) } exp (cid:32) − π β (cid:88) ˜ x ( ∂ − µ σ (˜ x ) + 2 π ˜ a µ (˜ x )) + i (cid:88) ˜ x q (˜ x ) σ (˜ x ) (cid:33) , (A.27)where q (˜ x ) ∈ Z has an interpretation as the magnetic charge of a monopole-instanton atposition ˜ x ∈ ˜Λ . The kinetic term of this expression clarifies that ˜ a plays the role of thegauge field for the discrete shift symmetry σ (cid:55)→ σ + 2 π , and thus the dual photon field σ is π -periodic scalar. Having made this point, for simplicity of notation, we shall neglect theeffect of nontrivial topology from now on, and set ˜ a = 0 .The Gaussian integration over σ can be done exactly to produce the Coulomb gasrepresentation for the magnetic monopoles: Z = (cid:88) { q (˜ x ) } exp β (cid:88) ˜ x ,˜ x (cid:48) ( − π β ) q (˜ x ) v (˜ x − ˜ x (cid:48) ) q (˜ x (cid:48) ) , (A.28)– 25 –here v (˜ x − ˜ x (cid:48) ) is the three dimensional Coulomb interaction formulated on the lattice(lattice Green function), formally given by v (˜ x ) = ∆ − . Let us split this Green functioninto two parts by adding and subtracting (∆ + M ) − , ∆ − = ∆ − − (∆ + M ) − + (∆ + M ) − = ∆ − (1 + ∆ /M ) − + (∆ + M ) − = u M PV (˜ x ) + w M PV (˜ x ), (A.29)where u M PV (˜ x ) is the Green function of the Pauli-Villars (PV) regulated Laplacian ∆ M PV ≡ ∆(1 + ∆ /M ) , and w M PV (˜ x ) is the Yukawa Green function.With this decomposition, we reintroduce the scalar field σ for the PV regulated prop-agator u M PV (˜ x ) , and then we obtain the partition function as Z = (cid:90) [ d σ (˜ x )] (cid:88) { q (˜ x ) } e (cid:80) ˜ x (cid:16) − π β σ (˜ x )∆ M PV σ (˜ x )+ i q (˜ x ) σ (˜ x ) (cid:17) − π β (cid:80) ˜ x ,˜ x (cid:48) q (˜ x ) w M PV (˜ x − ˜ x (cid:48) ) q (˜ x (cid:48) ) . (A.30)Since w M PV (˜ x − ˜ x (cid:48) ) is a massive propagator with the PV mass M PV , it is exponentiallydamping if | ˜ x − ˜ x (cid:48) | (cid:38) /M PV . Therefore, we can take w M PV (˜ x − ˜ x (cid:48) ) = w M PV (0) δ ˜ x − ˜ x (cid:48) ,0 ,where w M PV (0) = v (0) − O (1 /M PV ) ≈ − O (1 /M PV ) . In this limit, the partitionfunction simplifies into Z = (cid:90) [ d σ (˜ x )] e − π β (cid:80) ˜ x σ (˜ x )∆ M PV σ (˜ x ) (cid:88) { q (˜ x ) } (cid:89) ˜ x e − π βv (0)( q (˜ x )) + i q (˜ x ) σ (˜ x ) , (A.31)In (A.31), π βv (0)( q (˜ x )) has an interpretation as the action of the configurations withmagnetic charge q (˜ x ) . Let us denote the minimal action by S = 2 π βv (0) .We now perform the dilute-gas approximation as the leading-order semiclassical ap-proximation. We only take into account the minimal effect of the monopole-instantonscorresponding to q (˜ x ) = ± seriously, and regard higher-order effects in e − S as unimpor-tant. As a result, we may approximate the sum over q (˜ x ) by (cid:88) q (˜ x ) e − S q (˜ x ) + i q (˜ x ) σ (˜ x ) = exp (cid:0) − S cos( σ (˜ x )) (cid:1) + O (e − S ). (A.32)Substituting this expression into (A.31), we obtain the local Lagrangian for the dual photonfield: Z = (cid:90) [ d σ (˜ x )] exp (cid:32) − π β (cid:88) ˜ x σ (˜ x )∆ M PV σ (˜ x ) + 2e − S (cid:88) ˜ x cos( σ (˜ x )) (cid:33) . (A.33) B Wilson to Villain at weak coupling
As mentioned in Section 2.1, semi-Abelian
U(1) N − gauge theory may also be given in theWilson formulation by taking the action S W = β (cid:88) p N (cid:88) i =1 (1 − cos f ip ) − i (cid:88) (cid:96) N (cid:88) i =1 v (cid:96) a i(cid:96) , (B.1)– 26 –here the a i(cid:96) ∈ [0, 2 π ] are U(1) gauge fields, the f ip = ( d a i ) p are the corresponding fieldstrengths, and v (cid:96) ∈ Z is a Lagrange multiplier. This expression has manifest S N globalsymmetry. The purpose of this appendix is to demonstrate the weak-coupling equivalenceof this formulation and the Villain one (2.1).The first step is to use (A.18) for cos( f ip ) for i = 1, . . . , N with the weak-couplingapproximation, and then to apply (A.20) for i = 1, . . . , N − . We obtain a new action S = β (cid:88) p N − (cid:88) i =1 ( f ip + 2 πn ip ) − i (cid:88) (cid:96) N − (cid:88) i =1 v (cid:96) a i(cid:96) + 12 β (cid:88) p k p − i (cid:88) p k p f Np − i (cid:88) (cid:96) v l a N(cid:96) , (B.2)where we have introduced integer-valued plaquette-fields n ip ( i = 1, . . . , N − ) and k p , overwhich we must perform the summation in the partition function. Then exact integrationover a N(cid:96) gives the constraint v = − d † k , (B.3)and then the action becomes S = β (cid:88) p N − (cid:88) i =1 ( f ip + 2 πn ip ) + i (cid:88) (cid:96) N − (cid:88) i =1 ( d † k ) (cid:96) a i(cid:96) + 12 β (cid:88) p k p = β (cid:88) p N − (cid:88) i =1 ( f ip + 2 πn ip ) + i (cid:88) (cid:96) N − (cid:88) i =1 k p f ip + 12 β (cid:88) p k p . (B.4)Applying the Poisson summation formula in terms of k p , we get S = β (cid:88) p N − (cid:88) i =1 ( f ip + 2 πn ip ) + β (cid:88) p (cid:32) N − (cid:88) i =1 f ip + 2 πn p (cid:33) , (B.5)where a new integer-valued plaquette-field n p has taken the place of k p .For convenience, let us rewrite this last action in the form S = β N − (cid:88) i =1 ( f ip + b ip ) + β (cid:32) N − (cid:88) i =1 f ip + b p (cid:33) (B.6)by defining b ip ≡ πn ip , b p ≡ πn p . Completing the square then yields S = β (cid:88) p N − (cid:88) i , j =1 D ij ( f ip + b ip + w p /N )( f jp + b jp + w p /N ) + β (cid:88) p w p , (B.7)where we have defined w p ≡ b p − N − (cid:88) i =1 b ip , D ij ≡ δ i , j . (B.8)At this point, we realize that if we are only interested in the weak coupling regime, thensince the fluctuations in w p are gapped and discrete, we are entitled to set w p = 0 . Makingthis step leaves us with the action S = β (cid:88) p N − (cid:88) i , j =1 D ij ( f ip + b ip )( f jp + b jp ). (B.9)– 27 –ow we note that the unimodular matrix M ij = δ i , j − δ i +1, j (B.10)satisfies M t DM = C , (B.11)where C is the Cartan matrix of SU( N ) : C ij = α i · α j = 2 δ i , j − δ i , j +1 − δ i +1, j . (B.12)It follows that we can make the field redefinitions a i(cid:96) → N − (cid:88) j =1 M ij A j(cid:96) , b ip → N − (cid:88) j =1 M ij B jp (B.13)with A i(cid:96) ∈ [0, 2 π ] , B ip ∈ π Z , by the unimodularity of M . This yields S = β (cid:88) p N − (cid:88) i , j =1 C ij ( F ip + B ip )( F jp + B jp ) (B.14)which is equivalent to (2.1). References [1] A. M. Polyakov,
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