Condensates and anomaly cascade in vector-like theories
PPrepared for submission to JHEP
Condensates and anomaly cascade in vector-liketheories
Mohamed M. Anber
Department of Physics, Lewis & Clark College, Portland, OR 97219, USA
E-mail: [email protected]
Abstract:
We study the bilinear and higher-order fermion condensates in 4-dimensional SU ( N ) gauge theories with a single Dirac fermion in a general representation. Augmentedwith a mixed anomaly between the 0-form discrete chiral, 1-form center, and 0-form baryonnumber symmetries (BC anomaly), we sort out theories that admit higher-order condensatesand vanishing fermion bilinears. Then, the BC anomaly is utilized to prove, in the absenceof a topological quantum field theory, that nonvanishing fermion bilinears are inevitable ininfrared-gapped theories with 2-index (anti)symmetric fermions. We also contrast the BCanomaly with the 0-form anomalies and show that it is the former anomaly that determinesthe infrared physics; we argue that the BC anomaly lurks deep to the infrared while the0-form anomalies are just variations of local terms. We provide evidence of this assertion bystudying the BC anomaly in vector-like theories compactified on a small spacial circle. Thesetheories are weakly-coupled, under analytical control, and they admit a dual description interms of abelian photons that determine the deep infrared dynamics. We show that the dualphotons talk directly to the 1-form center symmetry in order to match the BC anomaly,while the 0-form anomalies are variations of local terms and are matched by fiat. Finally, westudy the fate of the BC anomaly in the compactified theories when they are held at a finitetemperature. The effective field theory that describes the low-energy physics is 2-dimensional.We show that the BC anomaly cascades from 4 to 2 dimensions. a r X i v : . [ h e p - t h ] J a n ontents R R × S L R × S L R × S L × S β R × S L : the -index (anti)symmetric fermions 19 SU (4 k ) with 2-index symmetric fermions 204.2 SU (4 k ) with 2-index antisymmetric fermions 214.3 The BC anomaly at finite temperature 22 ’t Hooft anomaly matching conditions is one of the very few handles on the nonperturbativephenomena in strongly-coupled theories [1]. The anomaly is an unremovable phase in thepartition function that needs to be matched between the ultraviolet (UV) and infrared (IR),which imposes constraints on the viable scenarios of the phases of a given asymptotically-freegauge theory that flows to strong coupling in the IR. Recently, it has been realized thatthe class of ’t Hooft anomalies is larger than what has been known since the 80s. It wasdiscovered in [2, 3] that Higher-form symmetries may also become anomalous, which can beused to impose further constraints on strongly-coupled theories. These original papers werefollowed by a plethora of other works that attempted to use the new anomalies to studyvarious aspects of quantum field theory, see [4–22] for a non-comprehensive list.One can understand the new development as an anomaly of a global transformation onthe field content in the background of a fractional topological charge, an ’t Hooft flux [23, 24],of the center symmetry of the gauge group. This anomaly was further enlarged in [25] byconsidering the most general fractional charges in the baryon number, color, and flavor (BCF)directions. This anomaly was dubbed the BCF anomaly (or only BC anomaly when we have– 1 – single flavor), and was also studied in [26] on nonspin manifolds. One of the profoundconsequences of the BCF anomaly is the deconfinement of quarks on axion domain walls,a phenomenon that is attributed to an intertwining between the light (axion) and heavy(hadron) degrees of freedom at the core of the domain wall. The intertwining between thedifferent degrees of freedom can also have an important effect on models of axion inflation[11].In this paper we consider a 4-dimensional asymptotically-free SU ( N ) gauge theory witha single Dirac flavor Ψ in a general representation R and strong-coupling scale Λ. The theoryadmits a U (1) B baryon and Z dχ T R discrete chiral symmetries, where T R is the Dynkin indexof the representation. As the theory flows to the IR and enters its strongly-coupled regime,we assume that it forms a nonvanishing bilinear fermion condensate (cid:104) ¯ΨΨ (cid:105) (cid:54) = 0. Then, thediscrete chiral symmetry breaks spontaneously, Z dχ T R → Z , leaving behind T R degeneratevacua. These vacua are separated by domain walls of width ∼ Λ − . If the bilinear fermioncondensate vanishes, then higher-order condensates may form, which, in general, break Z dχ T R down to a discrete subgroup. We ponder on several questions:1. A theory with an ’t Hooft anomaly precludes a unique gapped vacuum. What doanomalies inform us about the breaking of Z dχ T R ? Is there an anomaly that grants thefull breaking of Z dχ T R down to Z ? Is this anomaly unique or there are several anomaliesthat yield the same result? Is one of the anomalies more restricting than the others,and does this depend on R ?2. How do the domain walls respond to these anomalies?3. How are the anomalies matched at finite temperature?Indeed, it is well-known that a vector-like theory admits a mixed anomaly between Z dχ T R and U (1) B , we denote it by Z dχ T R [ U (1) B ] , which needs to be matched between the UV and IR.If the bilinear condensate forms, then the existence of T R degenerate vacua will automaticallymatch the anomaly. Sometimes, however, a T R degeneracy is an overkill in the sense that onlya subset of T R vacua are needed for the matching. This happens if the anomaly Z dχ T R [ U (1) B ] gives a phase valued in a proper subgroup of Z dχT R . In this case we might set (cid:104) ¯ΨΨ (cid:105) = 0 andargue that higher-order condensates break the chiral symmetry to a subgroup that gives theexact number of vacua needed to match the anomaly. For example, SU (4) with a Diracfermion in the 2-index symmetric representation has T R = 6 and we expect that the bilinearcondensate, if it forms, breaks Z dχ spontaneously resulting in 6 vacua. The Z dχ T R [ U (1) B ] anomaly, however, is valued in Z and can be matched by 3, instead of, 6 vacua. Then, it isa plausible scenario, in the light of the Z dχ T R [ U (1) B ] anomaly, that the bilinear condensatevanishes and the four-fermion condensate (cid:104) ¯ΨΨ ¯ΨΨ (cid:105) forms and yields 3 vacua.Another anomaly that gives the exact same conclusion is Z dχ T R [gravity] , which resultsfrom the action of Z dχ T R on the fermions in the gravitational background of a nonspin manifold.– 2 –iven this classical result, one wonders whether a yet-to-be-discovered anomaly mayimpose a stronger constraint on the number of the degenerate vacua and gives us a nonper-turbative exact statement about this number. We address this question in the light of theBC anomaly and show that it provides constraints stronger than or equal to the constraintsfrom the traditional Z dχ T R [ U (1) B ] and Z dχ T R [gravity] anomalies. In particular, we show,in the absence of a topological quantum field theory, that SU (4 k ) with fermions in the 2-index (anti)symmetric representation has to break its discrete chiral symmetry down to thefermion number Z and yields exactly T k ± vacua. Thus, the BC anomaly excludes the abovementioned four-fermi condensate scenario.In fact, we examined all SU ( N ), with 3 ≤ N ≤
9, asymptotically-free gauge theorieswith fermions in a general representation and concluded that there are only two types oftheories that have a stronger response to the BC anomaly than the traditional anomalies.These theories are: (i) SU (4 k ) with fermions in the 2-index symmetric representation and(ii) SU (4 k ), k >
1, with fermions in the 2-index antisymmetric representation . Nonetheless,we shall argue that it is the BC anomaly, in fact, that “orders” the breaking of the discretechiral symmetry. We show that a domain wall that separates two distinct vacua couples to a3-form field a (3) that transforms non-trivially under a 2-form symmetry, which is at the heartof the BC anomaly. a (3) , however, is inert under both Z dχ T R and U (1) B . This observationseems to suggest that Z dχ T R [ U (1) B ] anomaly is matched by “fiat”. In the rest of the paperwe provide a justification of this hypothesis.Because of the strong-coupling nature of the 4-dimensional theory, it is extremely hardto provide a detailed analysis of what really happens in its vacuum; there is no separation ofscales and all phenomena, e.g., confinement and chiral symmetry breaking, take places at thesame scale ∼ Λ. In order to test our hypothesis, we study the fate of anomalies in a semi-classical setup. We compactify the vector-like theories on a small circle S L of circumference L , such that Λ L (cid:28)
1, and give the fermions periodic boundary conditions on S L . This is nota thermal theory; the periodic boundary conditions turn the thermal partition function intoa graded-state sum. We say that the theory lives on R × S L . In addition, we add adjointmassive fermions or a double-trace deformation in order to force the theory into its weakly-coupled semi-classical regime, without spoiling the original global symmetry. Effectively, theIR theory lives in 3 dimensions, it abelianizes, and becomes amenable to analytical studies.We can also go to a dual (magnetic) description, where the “dual photons” play the main rolein determining the pattern of the discrete chiral symmetry breaking. We show that the dualphotons couple nontrivially to the higher-form symmetry, and therefore, the BC anomaly iscommunicated from the UV to the deep IR. The Z dχ T R [ U (1) B ] anomaly, on the other hand,shows up as a variation of a local action and does not talk to the photons. In this sense, wesay that Z dχ T R [ U (1) B ] anomaly is matched by fiat. This analysis provides evidence that itis the BC anomaly that talks to the IR degrees of freedom. Our work uses and generalizesthe observation that was first made by Poppitz and Wandler [27] that cubic- and mixed- U (1) anomalies are matched by local background-field-dependent topological terms instead– 3 –f chiral-Lagrangian Wess-Zumino-Witten terms, while the 1-form center symmetry talksdirectly to the dual photons. We further study in detail the SU (4 k ) theory on R × S L with2-index (anti)symmetric fermions and analyze the dynamics that leads to the full breakingof Z dχ k ± , the expected result in accordance with the BC anomaly. As a byproduct, weidentify new composite instantons that play a major role in the IR.We also examine the fate of the BC anomaly as we heat up the the theory. The strongcoupling nature of the 4-dimensional theory hinders our ability to answer this question. Wecircumvent this difficulty, again, by studying the compactified theory at a finite temperature.Now, in addition to the spacial circle S L , we also have a thermal circle S β , where β is theinverse temperature, and we say that the theory lives on R × S L × S β . Effectively, it canbe shown that the theory is dual to a 2-dimensional electric-magnetic Coulomb gas. We donot attempt to solve the effective 2-dimensional theory since the strong-coupling problemmight resurrect near the confinement/deconfinement transition. However, we trace the fateof the BC anomaly on R × S L × S β and show that this anomaly “cascades” from 4 down to2-dimensions. We also use renormalization group equations to argue that the theory admitsflat directions in the dual photon space as we heat it up, and eventually the long-range forceof the dual photons, which were responsible in the first place for the breaking of the chiralsymmetry, is tamed indicating that the chiral symmetry is restored. In this case we find thatthe BC anomaly becomes “confined”, or in other words local, and is matched by fiat.This paper is organized as follows. In Section 2 we review the symmetries and thecorresponding background fields in 4-dimensional vector-like theories with a single Diracfermion in a general representation. We also review the essence of the BC anomaly andcompare it to the traditional anomalies. Next, we study the condensates and the role of theBC anomaly. In Section 3 we work out the construction of the vector-like theories on a smallcircle; we consider both the perturbative and nonperturbative aspects and we introduce thedual theory. Then, we show in great details how the BC anomaly is reproduced in the dualpicture and argue that it lurks deep in the IR. This is in contradistinction with the traditionalanomalies, since they are realized as the variation of local actions that do not communicatewith the IR degrees of freedom. We also trace the fate of the BC anomaly at we heat up thedual theory. In Section 4 we work out the details of SU (4 k ) on the small circle with fermionsin the 2-index (anti)symmetric representation and identify the microscopic objects that areresponsible for the full breaking of the discrete chiral symmetry. Finally, we consider thesetheories at a finite temperature and use renormalization group equations to understand therealization of the BC anomaly as we heat up the system. R We consider SU ( N ) Yang-Mills theory endowed with a single left-handed massless Weylfermion ψ in a representation R along with another left-handed massless Weyl fermion ˜ ψ – 4 –ransforming in the complex conjugate representation. Collectively, we can also talk about asingle Dirac fermion in R . The 4-dimensional Lagrangian reads L = − g tr F (cid:2) F MN F MN (cid:3) + θ π tr F (cid:104) F MN ˜ F MN (cid:105) + i ¯ ψ ¯ σ M D M ψ + i ¯˜ ψ ¯ σ M D M ˜ ψ , (2.1)where M, N = 0 , , , Z is defined over a large closed mani-fold. The Dynkin index of the representation is denoted by T R (we use the normalizationtr F [ T a T b ] = δ ab , where T a are the generators of the Lie-algebra) and its dimension is dim R .Strictly speaking, since the fermions are massless, we could rotate the the θ angle away byapplying a chiral transformation on ψ and ˜ ψ . Keeping the topological term, however, willserve a later purpose. The theory admits the global symmetry: G Global = Z dχ T R × U (1) B Z N/p × Z × Z (1) p , (2.2)where Z is the fermion number (which is a subgroup of the Lorentz group, and hence, wemod it out), p = gcd( N, n ), and n is the N-ality of R . Notice that Z N/p , which is a subgroupof the center group Z N , acts faithfully on the fermions, and therefore, we needed to mod itout since it is part of the gauge group. Z dχ T R and U (1) B are respectively the 0-form discretechiral and baryon number symmetries acting on ψ and ˜ ψ : Z dχ T R : ψ → e i π T R ψ, ˜ ψ → e i π T R ˜ ψ , U (1) B : ψ → e iα ψ, ˜ ψ → e − iα ˜ ψ . (2.3)Finally, Z (1) p , provided that p >
1, is the 1-form symmetry that acts on the fundamentalWilson’s loops.When the representation is real, then we slightly modify the above procedure since inthis case it is enough to have a single fermion without the need to introduce another fermiontransforming in the would-be complex-conjugate representation. We use the symbol λ forthe real Weyl fermions. For example, a single adjoint Weyl fermion defines super Yang-Millstheory with T adj = 2 N , dim adj = N −
1, and global symmetry Z dχ N × Z (1) N .We also need to turn on background fields of G Global since they play a pivotal role indetermining ’t Hooft anomalies. Introducing a background field of U (1) B is straight forward;we just include it in the covariant derivative. Thus, we write D = d + iA − iV (1) , where A is the color gauge field and its field strength is F = dA + A ∧ A , and V (1) is the 1-form U (1) B gauge field with field strength F B (2) = dV (1) . Introducing background fields of discretesymmetries is more involved. In order to turn on a background field of the discrete chiralsymmetry Z dχ T R , we introduce a pair of 0-form and 1-form fields (cid:0) b (0) , B (1) (cid:1) that satisfy therelation 2 T R B (1) = db (0) and demand that the integral of the 1-form field db (0) over 1-cyclesis in Z , i.e., (cid:72) db (0) = 2 π Z , which in turn implies (cid:72) B (1) ∈ π T R Z , where the integral of B (1) is performed over 1-cycles. These fields are also invariant under the gauge transformation B (1) → B (1) + dω (0) and b (0) → b (0) + 2 T R ω (0) , and dω (0) has quantized periods over 1-cycles: The N-ality of a representation is the number of boxes in the Young tabulate mod N . – 5 – dω (0) ∈ π Z . One may think of b (0) as the phase of a charge-2 T R non-dynamical Higgsfield that acquires a vacuum expectation value and breaks a U (1) gauge field down to the Z T R discrete field B (1) . Under the transformation ψ → e i b (0)2 T R ψ and ˜ ψ → e i b (0)2 T R ˜ ψ the measureacquires a phase e i (cid:82) b (0)32 π tr F [ F MN ˜ F MN ]. Therefore, following the analysis of [27], one can thinkof b (0) as a background θ angle, and we shall use the former instead of the latter in thefollowing discussion.Next, we turn to the Z N center group of SU ( N ). As we mentioned above, only a Z N/p , p = gcd( N, n ), subgroup of the center acts faithfully on the fermions, leaving behind a global Z p that we may choose to turn on a background field associated to it. Yet, one can excite abackground field of the full center Z N owning to the baryon symmetry. The simplest way tounderstand this assertion is by examining the transition functions G ij on the overlap betweentwo patches U i and U j that cover the 4-dimensional manifold. On the overlap U i ∩ U j we have ψ i = G ij ψ j , G ij = G Z N ij G U (1) B ij , (2.4)where G Z N ij and G U (1) B ij are respectively the transition functions of the center and baryonnumber symmetries. A similar transformation holds for ˜ ψ . The consistency of the gaugetheory requires that the transition functions satisfy the following cocycle condition G ij G jk G ki = 1 (2.5)on the triplet overlap U i ∩ U j ∩ U k . The most general solution of the cocycle condition isobtained by taking G Z N ij = e i π nN and G U (1) B ij = e − i π nN , where the additional factor of n that appears in the exponent in G Z N ij accounts for the fact that the fermions transform in arepresentation of N-ality n . This explains why one can always excite the full Z N background.Indeed, when p >
1, then one may not use U (1) B and instead choose to turn on a backgroundfield of Z p ⊂ Z N . As it turns out, exciting the full Z N will impose stronger constrains on thetheory by employing the related ’t Hooft anomalies.The background field of Z N is an ’t Hooft flux that carries a fractional topological charge.The modern way of thinking of ’t Hooft fluxes is via higher-form symmetries, as was done in[28]. From now on, we consider Z (1) N Z (1) N we use a pair of 1-form and 2-formfields (cid:0) B c (2) , B c (1) (cid:1) such that N B c (2) = dB c (1) , see [29]. The periods of B c (1) are quantizedin multiples of 2 π : (cid:72) dB c (1) ∈ π Z , where the integral is over 2-cycles. Now, owing to therelation N B c (2) = dB c (1) , we obtain (cid:72) B c (2) ∈ πN Z . Next, we define the U ( N ) connection˜ A ≡ A + B c (1) N I N × N with gauge field strength ˜ F = d ˜ A + ˜ A ∧ ˜ A . The field strength ˜ F satisfiesthe relation tr F ˜ F = dB c (1) = N B c (2) . Going from SU ( N ) to U ( N ) introduces a non-physicalextra degree of freedom. In order to eliminate this degree of freedom, we postulate thefollowing invariance ˜ A → ˜ A + λ (1) under the 1-form gauge field λ (1) . Subsequently, the pair (cid:0) B c (2) , B c (1) (cid:1) transforms as B c (2) → B c (2) + dλ (1) and B c (1) → B c (1) + N λ (1) , such thatthe relation
N B c (2) = dB c (1) remains intact. The covariant derivative of the matter field is– 6 –btained by replacing A with ˜ A , i.e., D = d + i ˜ A − iV (1) . The invariance of D under λ (1) enforces the baryon background field to transform as V → V + nλ (1) , where the factor of n isthe N-ality of the representation (recall the discussion after the cocycle condition (2.5)), andhence, we find that F B transforms as F B → F B + ndλ (1) . Turning on the baryon and the center background fields enables us to find the most generalperturbative ’t Hooft anomaly on a spin manifold. As was shown in [28], this is an ’t Hooftanomaly of the discrete chiral symmetry in the background of both Z (1) N and U (1) B fields, andhence, the name baryon-color (BC) ’t Hooft anomaly. Succinctly, we can compute the anomalyfrom the triangle diagrams with vertices sourced by the following 2-form combinations ˜ F − B c (2) and F B − nB c (2) , which are invariant under the 1-form gauge transformation withparameter λ (1) . The triangle diagrams yield the following color and baryon number topologicaldensities: q c = 18 π (cid:104) tr F (cid:16) ˜ F ∧ ˜ F (cid:17) − N B c (2) ∧ B c (2) (cid:105) , q B = 18 π (cid:104) F B − nB c (2) (cid:105) ∧ (cid:104) F B − nB c (2) (cid:105) . (2.6)Then, we perform a discrete chiral transformation in the background of the BC backgroundto find that the partition function Z acquires the phase: Z Z dχ T R −−−→ e i πT R ( T R Q c + dim R Q B ) Z , (2.7)where Q c = (cid:82) q c and Q B = (cid:82) q B and the integral is performed over a closed 4-dimensional spinmanifold. Owing to the facts: π (cid:82) ˜ F ∧ ˜ F ∈ Z , π (cid:82) F B ∧ F B ∈ Z , and N π (cid:82) B c (2) ∧ B c (2) ∈ N Z , we find Q c = 1 − N and Q B = ( (cid:96) + nN ) , (cid:96) ∈ Z . Since T R Q c + dim R Q B is the Dirac-index,which is always an integer, then the phase of the partition function in the BC background isvalued in Z T R or a subgroup of it:BC Anomaly = e i πT R ( T R Q c + dim R Q B ) ∈ Z T R . (2.8)At this stage one might think that the BC anomaly does not impose on the dynamics anyfurther constraints beyond the traditional anomalies Z dχ T R [ U (1) B ] and Z dχ T R [gravity] , sincethe latter are also valued in Z T R : Z dχ T R [ U (1) B ] = Z dχ T R [gravity] = e i π dim R T R ∈ Z T R . (2.9) The most refined phase of the Z dχ T R [gravity] anomaly comes from a calculation on a nonspin manifold.Fermions are ill-defined when the manifold is nonspin, e.g. CP . In order to render the fermions well-definedon CP , we turn on a monopole background of U (1) B with charge . The fractional monopole flux combineswith the fractional flux of the gravitational CP instanton and yields an integer Dirac index = 1. Hence, oneimmediately finds the anomaly in (2.9). – 7 –owever, as we will argue in the next section, unlike the Z dχ T R [ U (1) B ] and Z dχ T R [gravity] anomalies, the BC anomaly is more restrictive and communicates non-trivial informationto the low-energy confining phase deep in the IR. This will be evident in the semi-classicalanalysis that we will perform on the theory upon compactifying it on a small circle. It isalso worth mentioning that one may compute the BC anomaly in a nonspin background, aswas done in [26]. We checked, however, that the BC anomaly on a nonspin manifold doesnot impose more restrictions on the condensates compared to the same anomaly on a spinmanifold.Finally, let us note that when p = gcd( N, n ) >
1, then we can also turn on the backgroundof Z (1) p ⊂ Z (1) N without the need to employ U (1) B . This can be accomplished by constrainingthe quantization of B c (2) over 2-cycles to obey (cid:72) B c (2) = πp Z , and hence, Q c = N π (cid:82) B c (2) ∧ B c (2) ∈ Np Z . Then, we encounter a mixed ’t Hooft anomaly between Z dχ T R and Z (1) p , whichgives the phase Z Z dχ T R −−−→ e i πT R ( T R Q c ) Z = e i π Np Z , (2.10)which is less restrictive than the phase from the BC anomaly. As we flow to the IR, the theory may or may not break its discrete chiral symmetry. Inthe following, we assume that: (1) the theory generates a mass gap and the discrete chiralsymmetry breaks, which can be probed via the non-vanishing color-singlet bilinear condensate (cid:104) ψ ˜ ψ (cid:105) or higher-order condensates, (2) the is no topological quantum field theory accompanyingthe IR phase , and (3) the theory does not form massless composite fermions in the IR. Theformation of the condensates, then, implies that in general the full or partial breaking of Z dχ T R takes place, leading to T R or fewer distinct vacua. The conclusion about the full breaking of Z dχ T R cannot be guaranteed unless there is an anomaly that is valued in Z T R and not only ina proper subgroup of it. Only in this case the saturation of the anomaly in the IR, indeed,demands the full breaking of Z dχ T R .If gcd( T R , dim R ) >
1, then (2.9) implies that both Z dχ T R [ U (1) B ] and Z dχ T R [gravity] anomalies do not necessarily demand the full breaking of the chiral symmetry; the par-tial breaking Z dχ T R → Z gcd ( T R , dim R ) is sufficient to match the anomalies. Similarly, whengcd( T R , (cid:0) T R Q c + dim R Q B (cid:1) ) >
1, then the BC anomaly can be matched via the breaking Z dχ T R → Z gcd( T R , ( T R Q c + dim R Q B )).In Tables 1 and 2 we display the asymptotically free representations of SU ( N ), 3 ≤ N ≤ Z dχT R insteadof Z dχ T R . Also, in this case the BC anomaly is reduced to the phase given by (2.10). The possibility of IR topological quantum field theory was considered in [10, 30]. – 8 – roup R T R dim R Z dχ T R [ U (1) B ] BC Condensate SU (3) (2 ,
0) 5 6 Z Z (cid:68) ˜ ψψ (cid:69) (3 ,
0) 15 10 Z Z (cid:68) ( ˜ ψψ ) (cid:69) (1 ,
1) 6 8 – Z (cid:104) λλ (cid:105) (2 ,
1) 20 15 Z Z (cid:68) ( ˜ ψψ ) (cid:69) SU (4) (2 , ,
0) 6 10 Z Z (cid:68) ˜ ψψ (cid:69) (3 , ,
0) 21 20 Z Z (cid:68) ˜ ψψ (cid:69) (0 , ,
0) 2 6 – 1 No constraints(0 , ,
0) 16 20 – Z (cid:10) ( λλ ) (cid:11) (1 , ,
1) 8 15 – Z (cid:104) λλ (cid:105) (1 , ,
0) 13 20 Z Z (cid:68) ˜ ψψ (cid:69) (2 , ,
1) 33 36 Z Z (cid:68) ( ˜ ψψ ) (cid:69) SU (5) (2 , , ,
0) 7 15 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , ,
0) 3 10 Z Z (cid:68) ˜ ψψ (cid:69) (1 , , ,
1) 10 24 – Z (cid:104) λλ (cid:105) (1 , , ,
0) 22 40 Z Z (cid:68) ( ˜ ψψ ) (cid:69) (1 , , ,
0) 24 45 Z Z (cid:68) ( ˜ ψψ ) (cid:69) Table 1 . The asymptotically free representations of SU (3) to SU (5). We use the Dynkin labels todesignate the representation: R = ( n , n , ..., n N − ) ≡ (cid:80) N − a =1 n a w a , where w a are the fundamentalweights. A representation is said to be real if ( n , n , ..., n N − ) = ( n N − , n N − , ..., n ). For example,(1 , , , , , , ,
0) are all real representations. In this case, one needs to be more carefulsince U (1) B is enhanced to SU (2) f flavor symmetry. We avoid this extra complication by consideringa single Weyl fermion, λ , whenever the representation is real. Then, the discrete chiral symmetrybecomes Z dχT R and the baryon number symmetry as well as the anomaly Z dχ T R [ U (1) B ] disappear.Notice that we exclude the defining representation (1 , , , ...,
0) since theories with fundamentals donot have genuine discrete chiral symmetries. In the next to last column we list the phases of both Z dχ T R [ U (1) B ] (which is equal to Z dχ T R [gravity] anomaly) and BC anomalies. In the last column wedisplay the higher-order condensate that saturates the BC anomaly. For all complex representations, except two cases, we find that both Z dχ T R [ U (1) B ] andBC anomalies yield the same phase. The exceptions are: • SU (4 k ) theories with fermions in the 2-index symmetric representation: R = (2 , , ..., T R = 4 k + 2 and dim R = 2 k (4 k + 1). • SU (4 k ), k >
1, theories with fermions in the 2-index anti-symmetric representation: R = (0 , , , ...,
0) with T R = 4 k − R = 2 k (4 k − T R , dim R ) = 2, while (cid:0) T R Q c + dim R Q B (cid:1) (2 , ,..., = 4 k + 3, (cid:0) T R Q c + dim R Q B (cid:1) (0 , ,..., = 4 k −
1, and hence, gcd( T R , (cid:0) T R Q c + dim R Q B (cid:1) ) = 1, makingthe BC anomaly more restricting than Z dχ T R [ U (1) B ] and Z dχ T R [gravity] anomalies. Then,– 9 – roup R T R dim R Z dχ T R [ U (1) B ] BC Condensate SU (6) (2 , , , ,
0) 8 21 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , , ,
0) 4 15 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , , ,
0) 6 20 Z Z (cid:10) ( λλ ) (cid:11) (1 , , , ,
1) 12 35 – Z (cid:104) λλ (cid:105) (1 , , , ,
0) 33 70 Z Z (cid:68) ˜ ψψ (cid:69) SU (7) (2 , , , , ,
0) 9 28 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , , , ,
0) 5 21 Z Z (cid:68) ˜ ψψ (cid:69) (1 , , , , ,
1) 14 48 – Z (cid:104) λλ (cid:105) (0 , , , , ,
0) 10 35 Z Z (cid:68) ( ˜ ψψ ) (cid:69) SU (8) (2 , , , , , ,
0) 10 36 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , , , , ,
0) 6 28 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , , , , ,
0) 20 70 Z Z (cid:10) ( λλ ) (cid:11) (1 , , , , , ,
1) 16 63 – Z (cid:104) λλ (cid:105) (0 , , , , , ,
0) 15 56 Z Z (cid:68) ˜ ψψ (cid:69) SU (9) (2 , , , , , , ,
0) 11 45 Z Z (cid:68) ˜ ψψ (cid:69) (0 , , , , , , ,
0) 7 36 Z Z (cid:68) ˜ ψψ (cid:69) (1 , , , , , , ,
1) 18 80 – Z (cid:104) λλ (cid:105) Table 2 . The asymptotically free representations of SU (6) to SU (8). For details see the captionof Table 1. Notice that the first non-vanishing condensate in the representation (0 , , , ,
0) of SU (6)is a 4-fermion operator since the fermion bilinear vanishes identically for group theory reasons, see[9, 31]. the BC anomaly demands the full breaking of Z dχ k ± and the formation of 4 k ± Z (1)2 symmetry acting on Wilson’s loop and gauging it leads toa trivial phase, as can be easily seen from (2.10). We conclude, in the absence of a topologicalquantum field theory, that nonvanishing fermion bilinears are inevitable in infrared-gapped SU ( N ) gauge theories with 2-index (anti)symmetric fermions.We also observe that when the phase of the BC anomaly is in a prober subgroup of thediscrete chiral symmetry, then a plausible scenario is that the bilinear condensate vanishesand higher-order condensates form. In the last column of Tables 1 and 2 we display thepossible higher-order condensate that saturates the BC anomaly. For example, the discretechiral symmetry of SU (4) Yang-Mills theory with a single Dirac fermion in the (2 , ,
1) repre-sentation is Z dχ and the formation of the bilinear condensate suggests that the theory admits33 vacua in the IR. However, the BC anomaly can be matched via the breaking Z dχ → Z ,suggesting that an IR phase with only 11 vacua is enough to match the anomaly. Thus, aplausible scenario that matches the anomalies is the vanishing of both the bilinear and four-– 10 –ermion condensates (cid:104) ˜ ψψ (cid:105) = (cid:104) ˜ ψψ ˜ ψψ (cid:105) = 0 and the formation of the six-fermion condensate (cid:68) ( ˜ ψψ ) (cid:69) ≡ (cid:104) ˜ ψψ ˜ ψψ ˜ ψψ (cid:105) (cid:54) = 0.The exceptional cases discussed above give us an insight into the special role of the BCanomaly compared to the traditional anomalies Z dχ T R [ U (1) B ] and Z dχ T R [gravity] . We arguethat it is the BC anomaly that lurks deep in the IR and demands the existence of multiplevacua. In Section (3) we put this hypothesis into test by studying the same theory on asmall circle. This setup enables us to perform semi-classical calculations and examine variousphenomena that are rather difficult, if not impossible, to understand in the strong-couplingregime. In particular, we will show that it is the BC anomaly that influence the IR dynamics,while the Z dχ T R [ U (1) B ] anomaly is the variation of a local action and is matched by fiat, butotherwise does not influence the IR dynamics.Before delving into the analysis on the circle, let us show how the BC anomaly is matchedin 4-dimensions deep in the IR. As the condensate forms, domain walls will interpolate betweenthe degenerate vacua. Let a (3) be the 3-form field that couples to the domain wall such that (cid:72) a (3) ∈ π Z and the integral is over 3-cycles. Then, one can write down the following5-dimensional Wess-Zumino-Witten term that matches the anomaly in the IR: S W ZW = (cid:90) W dω (0) ∧ (cid:20) da (3) − N π B c (2) ∧ B c (2) + dim R π T R (cid:104) F B − nB c (2) (cid:105) ∧ (cid:104) F B − nB c (2) (cid:105)(cid:21) . (2.11)Under a Z dχ T R transformation we use (cid:72) dω (0) ∈ π Z and find e − iδS WZW ∈ Z T R . A closerexamination of the action (2.11) reveals some important information about the IR physics thatcannot be seen without the BC anomaly. As we discussed above, the 2-form field transformsunder the 1-form gauge field λ (1) as: B c (2) → B c (2) + dλ (1) . This, in turn, demands that the3-form field transforms as a (3) → a (3) + N π B c (2) ∧ λ c (1) + N π λ c (1) ∧ dλ (1) , (2.12)which indicates that the Z (1) N Z dχ T R [ U (1) B ] anomaly.Here, all we need to do is to turn off B c (2) in (2.11). Then, we still find that e − iδS WZW ∈ Z T R .However, the the 3-form field that couples to the domain wall does not transform under U (1) B or Z dχ T R ; the Z dχ T R [ U (1) B ] anomaly is matched trivially.Although our analysis in 4 dimensions might sound like an academic exercise due to thelack of any control on the strong dynamics, in the next section we show how our reasoningbecomes manifest once we push the theory into its weakly-coupled regime. R × S L In this section we study the vector-like theories by compactifying the x direction on a smallcircle S L with circumference L and demand that Λ, the strong coupling scale of the theory, is– 11 –aken such that L Λ (cid:28)
1. In addition, the fermions obey periodic boundary conditions on S L .This setup guarantees that the theory enters its semi-classical regime, and hence, we can usereliable analytical methods to analyze it. We say that the theory lives on R × S L . Further,the analysis of the theory simplifies considerably if we force it into its center-symmetric point(more on that will be discussed below). This can be achieved either by adding a double-tracedeformation L DT = (cid:88) j c j (cid:12)(cid:12)(cid:12) tr F (cid:16) e ij (cid:72) A (cid:17)(cid:12)(cid:12)(cid:12) , (3.1)with large positive coefficients c i , or by adding massive adjoint fermions with mass ∼ L − .Both of these two alterations to the theory neither change its global symmetries nor its ’tHooft anomalies. However, we note that, depending on R , adding adjoint fermions might notachieve the goal of stabilizing the theory at the center symmetric point, as was discussed indetails in [32].This construction was considered before in a plethora of works, and we refer the readerto the literature for more details, see [33] for a review. Here, it suffices to say that the theorycompletely abelianizes at the center-symmetric point: SU ( N ) breaks down spontaneouslyto the maximal abelian subgroup U (1) N − . Now, all fields that appear in the low-energyeffective Lagrangian are valued in the Cartan subalgebra space, which we label by bold facesymbols. At energy scales much smaller than the inverse circle radius we dimensionally reducethe theory to 3 dimensions with effective Lagrangian: L = − L g F µν F µν − b (0) π (cid:15) αµν ∂ α Φ · F µν + 12 g L ∂ µ Φ ∂ µ Φ + V ( Φ ) + L ,f , (3.2)where µ, ν = 0 , , Φ = ( φ , φ , ..., φ N − ). The field Φ isthe gauge field holonomy in the S L direction: L A ≡ Φ . The second term is the 4-dimensionaltopological term dimensionally reduced to 3 dimensions. As we promised above, we tradedthe θ angle for the background field b (0) of the discrete chiral symmetry. The potential V ( Φ )is the Gross-Pisarski-Yaffe (GPY) potential [34], which results from summing towers of theKaluza-Klein excitations of the gauge field, the R fermions, and the massive adjoint fermions.We always force V ( Φ ) to be minimized at the center-symmetric point either by adding massiveadjoint fermions or double-trace deformation. The center-symmetric value of Φ is Φ = 2 π ρ N , (3.3)where ρ = (cid:80) N − a =1 w a is the Weyl vector and w a are the fundamental weights. See the discus-sion immediately before (3.15) for more comments on the meaning of the center-symmetricvacuum. The holonomy fluctuations about Φ have masses of order ∼ gL , and thus, we canneglect them whenever we are interested in energies much smaller than gL . The U (1) N − gauge fields F µν are given, as usual, by F µν = ∂ µ A ν − ∂ ν A µ . Both d Φ and F satisfy thequantization conditions (cid:72) d Φ ∈ π α a Z and (cid:72) F ∈ π α a Z , where the integrals are taken– 12 –espectively over 1- and 2-cycles, for all simple roots α a , a = 1 , , ..., N −
1. The simple rootshave length α a = 2 and satisfy the relation α a · w b = δ ab .Finally, we comment on the fermion term in (3.2). The 3-dimensional fermion Lagrangianis given by (here we consider the Lagrangian of ψ . An identical Lagrangian holds for ˜ ψ ) L ,f = i (cid:88) µ ∈R (cid:88) p ∈ Z ¯ ψ µ p (cid:20) ¯ σ µ ( ∂ µ + i A µ · µ ) + i ¯ σ (cid:18) πpL + µ · Φ L (cid:19)(cid:21) ψ µ p , (3.4)where µ are the weights of R and p is the Kaluza-Klein index. The effective 3-dimensionalfermion mass can be readily found from (3.4): M p , µ = | πpL + µ · Φ L | . This mass has to benon-vanishing for every non-zero value of µ , otherwise the low-energy U (1) N − gauge theorybecomes strongly coupled, which in turn, invalidates any semi-classical treatment. Yet, incertain situations, depending on R , nonperturbative effects (these are monopole instantonsand/or their composites) can give the fermions a small 4-dimensional Dirac mass, renderingthe theory IR safe. Alternatively, we can also turn on a holonomy of U (1) B , which ensures thatall the fermions are massive with mass ∼ L − . To this end, we decompose the 4-dimensional U (1) B V (1) = V (1)3 D + (cid:16) κL + V (0) S L (cid:17) dx L , (3.5)where V (1)3 D is the 1-form background field in R , κL is the U (1) B holonomy in the S L direction,and V (0) S L are the holonomy fluctuations. Turning on κ modifies the 3-dimensional fermionmass to M p , µ = | πpL + µ · Φ − κL | , and now all the fermions are massive with mass ∼ L − . We shall investigate the realization of the symmetries as well as the BC anomaly on R × S L in the semi-classical regime deep in the IR. In order achieve this goal we need to utilize a dual(magnetic) description. To this end, we introduce the dual photon σ as a Lagrange multiplierthat enforces the Bianchi identity (cid:15) αµν ∂ α F µν = 0. We augment the Lagrangian (3.2) withthe term π (cid:15) αµν σ · ∂ α F µν and then vary the combination with respect to F µν to find: F µν = − g πL (cid:15) µνα (cid:34) ∂ α σ + b (0) π ∂ α Φ (cid:35) . (3.6)Next, we break Φ into two parts: the vacuum Φ and the fluctuations around it ϕ such that Φ = Φ + ϕ . Substituting (3.6) into (3.2) we then find L = g π L (cid:32) ∂ α σ + b (0) π ∂ α ϕ (cid:33) · (cid:32) ∂ α σ + b (0) π ∂ α ϕ (cid:33) + 12 g L ∂ α ϕ · ∂ α ϕ + V ( Φ ) + L ,f . (3.7)The domain of σ can be determined as follows. The integral of d σ over 1-cycles is equal tothe electric charge enclosed by the cycles. Since all the electric charges are allowed probe– 13 –harges when the group is SU ( N ), then the domain of σ is the finest lattice, which is theweight lattice: (cid:72) d σ ∈ π w a Z for all a = 1 , , ..., N − b (0) transforms as b (0) → b (0) + 2 π , Then, theinvariance of (3.7) under Z dχ T R demands that the dual photons shift as σ → σ − ϕ − C , (3.8)where C is a constant vector that belongs to the weight lattice, which is allowed owing tothe fact that it is the derivatives of σ and ϕ that appear in (3.7). The constant C can beunambiguously determined once we take the the nonperturbative effects into account. The theory also admits monopole-instantons. The action of the lowest Kaluza-Klein monopoles( p = 0 monopoles, where p is the Kaluza-Klein index) is S α a = 4 πg α a · Φ (3.9)for every simple root α a , a = 1 , , ..., N −
1. There is also one extra monopole instanton thatcorresponds to the affine root α N = − (cid:80) N − a =1 α a with an action S α N = π g + πg α N · Φ .Module O (1) normalization coefficients, the ’t Hooft vertex associated with each monopole,including the affine monopole a = N , is given by: M a = e (cid:16) − π g + ib (0) (cid:17) δ Na e − πg α a · Φ e i α a · (cid:18) σ + b (0)2 π ϕ (cid:19) ( ψ ˜ ψ ) I a , a = 1 , , ..., N . (3.10)The exponent I a is the Callias index that counts the number of the fermion zero modes inthe background of the monopole [35, 36]. The index of the lowest Kaluza-Klein monopole isgiven by [32, 37]: I a = (cid:88) µ (cid:22) Φ · µ π (cid:23) α a · µ , a = 1 , , ..., N − , I N = 2 T R − N − (cid:88) a =1 I a . (3.11)Each monopole vertex has to respect the global symmetries. First, it is evident that M a is invariant under U (1) B . Next, in order to respect the invariance under Z dχ T R we express theconstant C in (3.8) as a general vector in the weight lattice as C = 2 π N − (cid:88) a =1 K a w a . (3.12)Then, the invariance of each vertex under Z dχ T R fixes the values of K a : K a = I a T R . (3.13)– 14 –s we shall discuss below, in some cases the lowest Kaluza-Klein monopoles are insuffi-cient to construct the full low-energy effective potential V ( σ ). Thus, we need to turn into thefirst excited monopoles. Their actions can be obtained from (3.9) by replacing Φ → Φ + π α a [38]: S p =1 α a = 8 π g + 4 πg α a · Φ . (3.14)This action suggests that a p = 1 monopole can be thought of as a composite configurationof the original monopole plus a Belavin-Polyakov-Schwarz-Tyupkin (BPST) instanton . Thenumber of the fermion zero modes in the background of the excited monopoles can be readfrom (3.11) after adding T R extra zero modes of ψ and T R extra zero modes of ˜ ψ .The proliferation of monopoles or monopole-composites will lead to confinement andchiral symmetry breaking. Several examples that illustrate the important points of this paperwill be worked out in later sections. R × S L Next, we turn on a background field of the Z (1) N center symmetry and examine the BC ’tHooft anomaly on R × S L . This can be achieved by recalling the exact same procedure wefollowed in 4 dimensions. Here, however, we can entertain the fact that all fields are valuedin the Cartan subalgebra space, and at energies much smaller than L − we need to follow thedegrees of freedom that enter the semi-classical analysis. We adopt the exact same procedureused in [27] to study the center-symmetry in super Yang-Mills theory.To this end, we enlarge the abelian group U (1) N − to U (1) N by going to the R N basis[39]. The wights of the defining representation in the R N basis are ν A = e A − N (cid:80) NA =1 e A ,for A = 1 , , ..., N , and { e A } are basis vectors spanning the R N space, while the simple rootsare given by α A = e A − e A +1 , for A = 1 , , ..., N . Let ˜ F A be the U (1) N fields in this basis.Then, the periods of ˜ F A are given by (cid:72) ˜ F A = 2 π Z , where the integration is performed on2-cycles. In this basis we have one spurious degree of freedom, which can be eliminated byimposing the following constraint on the U (1) N fields: (cid:80) NA =1 ˜ F A = 2 πn , for some integer n . Everything we have said about ˜ F A also applies to ˜ ϕ A , the U (1) N gauge field componentalong S L .Upon compactifying the theory on S L , the 4-dimensional Z (1) N symmetry decomposes intoa 1-form symmetry that acts on Wilson’s loops on R (here we need to compactify R on alarge 3-torus) and a 0-form symmetry that acts on Polyakov loops wrapping S L . The lattervanish in a center-symmetric vacuum tr F (cid:2) e i Φ · H (cid:3) = 0, where H are the generators of theCartan subalgebra. Thus, the background fields of the Z (1) N symmetry decompose as: B c (2) = B c (2)3 D + B c (1) S L ∧ dx L , B c (1) = B c (1)3 D + B c (0) S L dx L , (3.15) The action of a BPST instanton is π g and can be thought of as the composite of all the monopoles thatare charged under the simple and affine roots. Therefore, a BPST instanton has a total number of 2 T R fermionzero modes. – 15 –uch that the conditions N B c (2)3 D = dB c (1)3 D and N B c (1) S L = dB c (0) S L are obeyed. The various 0-form and 1-form fields obey the quantization conditions (cid:72) − cycle dB c (1)3 D ∈ π Z , (cid:72) − cycles dB c (0) S L ∈ Z , (cid:72) − cycle B c (2)3 D ∈ πN Z , (cid:72) − cycles B c (1) S L ∈ πN Z .Next, we use the fact that the 4-dimensional combination ˜ F − B c (2) is invariant under the1-form gauge transformation via the 1-form field λ (1) . Thus, we can write a 3-dimensionaleffective field theory, which is invariant under the same λ (1) transformation, by replacing eachcomponent of F by ˜ F A − B c (2)3 D and each component of d ϕ by d ˜ ϕ A − B c (1) S L in (3.2). Thus, weobtain the bosonic part of the Lagrangian(we suppress V ( Φ )): L bosonic3 D = − L g N (cid:88) A =1 (cid:16) ˜ F Aµν − B c (2) µν, D (cid:17) (cid:16) ˜ F µν,A − B µν,c (2)3 D (cid:17) − b (0) π (cid:15) αµν N (cid:88) A =1 (cid:16) ∂ α ˜ ϕ A − B c (1) α, S L (cid:17) (cid:16) ˜ F Aµν − B c (2) µν, D (cid:17) + 12 g L N (cid:88) A =1 (cid:16) ∂ α ˜ ϕ A − B c (1) α, S L (cid:17) (cid:16) ∂ α ˜ ϕ A − B α,c (1) S L (cid:17) . (3.16)Next, we need to eliminate the spurious degrees of freedom contained in ˜ F A and d ˜ ϕ A , andin the same time use a duality transformation to write the effective action in terms of the U (1) N dual photons ˜ σ A . Both of these requirements can be implemented using the followingauxiliary Lagrangian: L auxilary = − π N (cid:88) A =1 (cid:15) λµν ∂ α ˜ σ A ˜ F Aµν + 14 π (cid:15) µνα u α N (cid:88) A =1 (cid:16) ˜ F Aµν − B c (2) µν, D (cid:17) , + 14 π v α N (cid:88) A =1 (cid:16) ∂ α ˜ ϕ A − B α,c (1) S L (cid:17) , (3.17)where u α and v α are the two Lagrange multipliers used to impose the two constraints: N (cid:88) A =1 (cid:16) ˜ F Aµν − B c (2) µν, D (cid:17) = 0 , N (cid:88) A =1 (cid:16) ∂ α ˜ ϕ A − B α,c (1) S L (cid:17) = 0 . (3.18)Then, we substitute (3.18) into (3.16) and vary L bosonic3 D + L auxilary with respect to ˜ F Aµν to find:˜ F Aµν = B c (2) µν, D − g πL (cid:15) µνα (cid:32) ∂ α ˜ σ A − u α + b (0) π (cid:32) ∂ α ˜ ϕ A − N N (cid:88) B =1 ∂ α ˜ ϕ B (cid:33)(cid:33) . (3.19)Finally, we substitute (3.19) into L bosonic3 D + L auxilary to obtain the dual Lagrangian: L bosonic, dual3 D = g π L N (cid:88) A =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ α ˜ σ A − N N (cid:88) B =1 ∂ α ˜ σ B + b (0) π (cid:32) ∂ α ˜ ϕ − N N (cid:88) B =1 ∂ α ˜ ϕ B (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 12 g L N (cid:88) A =1 (cid:12)(cid:12)(cid:12) ∂ α ˜ ϕ A − B c (1) α, S L (cid:12)(cid:12)(cid:12) − π N (cid:88) A =1 (cid:15) αµν ∂ α ˜ σ A B c (2) µν, D . (3.20)– 16 –his is the exact same Lagrangian that was obtained in [27] for super Yang-Mills theory. Aswe show below, this Lagrangian needs to be augmented with the fermionic part to match thefull BC anomaly.The last term in (3.20) is going to play the main role in what we do next. In terms ofdifferential forms, this term reads: L bosonic, dual3 D ⊃ − π N (cid:88) A =1 d ˜ σ A ∧ B c (2)3 D . (3.21)Under a Z dχ T R transformation d ˜ σ A and b (0) transform as d ˜ σ A → d ˜ σ A − d ˜ ϕ A , b (0) → b (0) +2 π (see(3.8)), and only the term − π (cid:80) NA =1 d ˜ σ A ∧ B c (2)3 D contributes to the variation of L bosonic, dual : e iδS bosonic, dual D = e − i π (cid:80) NA =1 (cid:82) d ˜ ϕ A ∧ B c (2)3 D . (3.22)Then using the second constraint in (3.18), (cid:80) NA =1 d ˜ ϕ A = N B c (1) S L , along with the quantizationconditions of B c (2)3 D and B c (1) S L we find e iδS bosonic, dual D = e − i πN . (3.23)The above manipulations show that the Z (1) N background lurks deep in the IR and thatit couples to the dual photons. This, however, does not capture the full BC anomaly; westill need to compute the variation of the fermions action in the Z (1) N and U (1) B backgrounds.This can be obtained from the U (1) B topological charge density, the second equation in (2.6).Substituting (3.5) and (3.15) into (2.6) and integrating by parts along the S L direction, weobtain the fermion contribution to the variation of the action: δS fermionic = 2 πT R dim R π (cid:90) (cid:16) dV (1)3 D − nB (2)3 D (cid:17) ∧ (cid:16) dV (0) S L − nB (1) S L (cid:17) . (3.24)Using the quantization condition π (cid:82) dV (1)3 D ∧ dV (0) S L ∈ Z along with the quantization condi-tions of B (2)3 D and B (1) S L we find: δS fermionic = 2 π dim R T R (cid:16) nN + (cid:96) (cid:17) , (cid:96) ∈ Z , (3.25)and finally we recover the BC anomaly on R × S L : e iδS bosonic, dual D + iδS fermionic = e i πT R (cid:16) T R ( − N ) + dim R ( nN + (cid:96) ) (cid:17) , (3.26)which is exactly (2.7), the BC anomaly computed directly on R .We conclude the following: – 17 – Our analysis shows hat the Z (1) N center acts non-trivially on the dual photons and, whenaccompanied with the contribution from U (1) B , it produces the correct BC anomalydeep in the IR. This suggests that the BC anomaly is seen and influence the dynamicsat all scales. • Unlike the BC anomaly, which makes use of the higher-form symmetries, the traditional’t Hooft anomalies are variations of local terms in the action when the theory is compact-ified on a small circle. This is clear from the treatment of δS fermionic above. Switchingoff the center background B (2)3 D and B (1) S L , we immediately lose the term (3.21) and find δS bosonic, dual = 0 and δS fermionic = dim R T R × integer. This is exactly the Z dχ T R [ U (1) B ] traditional ’t Hooft anomaly. We see right away that this variation of the action is aphase that does not talk to the photons; the dynamics on R × S L have to obey theBC anomaly, while it is transparent to the traditional 0-form anomaly. The latter isobeyed by fiat. This observation generalizes the observation that appeared first in [27]:the cubic- and mixed- U (1) anomalies are matched by local background-field-dependenttopological terms instead of chiral-Lagrangian Wess-Zumino-Witten terms and the 1-form center symmetry talks directly to the dual photons. • It is also important to emphasize, as is well known, that matching the BC anomalyon R × S L precludes a unique gapped vacuum. Such vacuum leaves δ ˜ σ A = 0, andhence, δS bosonic, dual = 0, a variations that does not match the anomaly. Therefore, theanomaly implies that either there exist massless dual photons in the spectrum and/or thediscrete chiral symmetry has to break spontaneously, which yields multiple degeneratevacua. We shall see examples of these two possibilities in the following sections. R × S L × S β In this section we continue our investigation of the BC anomaly as we heat the semi-classicaltheory that lives on R × S L . Turning on a finite temperature T is equivalent to compactifyingthe time direction x on a circle S β of circumference β = T and giving the fermions anti-periodic boundary conditions on S β . We say that the theory lives on R × S L × S β . In orderto follow the anomaly from R × S L to R × S L × S β , we decompose the background fields B c (2)3 D and B c (1)3 D into fields in the R and S β directions: B c (2)3 D = B c (2)2 D + B c (1) S β ∧ dx β , B c (1)3 D = B c (1)2 D + B c (0) S β dx β , (3.27)such that the constraints N B c (2)2 D = dB (1)2 D and N B c (1) S β = dB c (0) S β are obeyed. The backgroundfields obey the quantization conditions (cid:72) dB (1)2 D ∈ π Z , (cid:72) B c (2)2 D ∈ πN Z , (cid:72) dB c (0) S β ∈ π Z , and (cid:72) B c (1) S β ∈ πN Z . At finite temperature we may dimensionally reduce the 3-dimensional effectivefield theory down to 2 dimensions. In particular, using (3.27), the term (3.21), that contains– 18 –he anomaly, reduces to: L bosonic, dual2 D ⊃ − π N (cid:88) A =1 d ˜ σ A ∧ B c (1) S β , (3.28)where we have neglected the dual photons derivative in the time direction. Physically, thiscorresponds to keeping only the zeroth Kaluza-Klein mode of the dual photons and neglectingthe higher modes. Under a discrete chiral transformation the dual photons transform as d ˜ σ A → d ˜ σ A − d ˜ ϕ A and the variation of the 2 dimensional action becomes δ L bosonic, dual2 D = 12 π N (cid:88) A =1 d ˜ ϕ A ∧ B c (1) S β . (3.29)Further, we use the second constraint in (3.18), (cid:80) NA =1 d ˜ ϕ A = N B c (1) S L , to find the variationof the 2-dimensional action δS bosonic, dual2 D = − N π (cid:90) B c (1) S L ∧ B c (1) S β = − πN , (3.30)which is identical to the variation of the 3-dimensional dual action. This part of the anomalycombines with the contribution from the fermionic action (3.25) to reproduce the BC anomaly(3.26) at finite temperature.The important observation is that the 2-dimensional dual photons still couple to the Z (1) N center background field, and hence, we expect the anomaly to play a role even at finitetemperatures. Nonetheless, there is an extra layer of complication in 2 dimensions, thanksto the compact nature of σ . In 2 dimensions σ have both momentum modes, which areresponsible for the logarithmic Coulomb-like force between the monopole instantons, andwinding modes. The latter are monodromies of σ with a UV cutoff of order L . Thesemonodromies are the W-bosons and heavy fermions that were not captured by the the lowenergy effective field theory in 3 dimensions. As we crank up the temperature and approachthe critical temperature of the phase transition/crossover, the heavy excitations inevitablypop up from vacuum and participate in the dynamics alongside with monopoles and othercomposite instantons. Eventually, one needs to deal with an electric-magnetic Coulomb gas,which, in general, is a strongly-coupled problem.In this paper we avoid delving into the anomaly matching in the fully-fledged electric-magnetic Coulomb gas, leaving it for a future investigation. In the next section, however, wegive an example that illustrates the idea of the BC anomaly matching at a finite temperaturegiven that we stay well inside the semi-classical weakly-coupled regime. Then, we commenton the fate of this anomaly at very high temperatures. R × S L : the -index (anti)symmetric fermions In this section we consider several examples on R × S L and on R × S L × S β that illustratethe main points of this work: it is the BC anomaly that is responsible for communicating the– 19 – igure 1 . The molecular instantons in the theory on R × S L . From left to right they are thebions, triplets, and higher composites. The latter composite is an example of molecular instanton in SU (8) with fermions in the 2-index antisymmetric representation. It consist of a p = 1 Kaluza-Kleinmonopole attached to 6 lowest-order monopoles. The 12 zero modes of the central monopole aresoaked up by the orbital monopoles (moons). There is a repulsive Coulomb force between the centralmonopole and the its moons, which is balanced by the attractive force due to the exchange of the zeromodes. The moons, on the other hand, repel each other since they are all charged under the sameroot, and thus, they are stabilized under this repulsive force. UV information to the deep IR. In particular, we found from our analysis in Section 2.3 thatthe BC anomaly is stronger than the traditional Z dχ T R [ U (1) B ] anomaly. Then, we showed inthe previous sections that it is the BC anomaly that couples to the dual photon, and thus,one expects that it controls the breaking pattern of the chiral symmetry. SU (4 k ) with -index symmetric fermions We work in the center-symmetric vacuum Φ = π ρ k , which can be attained by using adouble-trace deformation. The monopole vortices are given by (see (3.10); here we neglectthe holonomy fluctuations ϕ and set b (0) = 0): M a = e − π kg e i α a · σ ψ ˜ ψ , a (cid:54) = 2 k or 4 k , M k = e − π kg e i α k · σ (cid:16) ψ ˜ ψ (cid:17) , M k = e − π kg e i α k · σ (cid:16) ψ ˜ ψ (cid:17) . (4.1)Since all the monopoles are dressed with fermion zero modes, they cannot lead to confinementor breaking of the chiral symmetry. Yet, molecular instantons that are composed of twomonopoles (bions) [40, 41] or three monopoles (triplets) [42] can form, see Figure 1. Thestability of these molecules is attributed to the fact that the total potential seen by the twoor three monopoles admits a stable equilibrium point. This is ascribed to the competition– 20 –etween the repulsive Coulomb force from the dual photons and the attractive force fromthe exchange of the fermion zero modes (we say that the fermions zero modes are soakedup). In particular, notice that α a · α b = 2 δ a,b − δ a,b +1 − δ a,b − , and therefore, only monopoleand anti-monopole that are charged under neighboring simple roots can feel the repulsiveCoulomb force. The bions and triplets with the lowest fugacities are: M a M a +1 , ≤ a < k − k + 1 ≤ a < k − , M k (cid:0) M k − (cid:1) , M k (cid:0) M k +1 (cid:1) , M k (cid:0) M k − (cid:1) (cid:0) M k +1 (cid:1) , M k (cid:0) M k − (cid:1) , M k (cid:0) M (cid:1) , M k (cid:0) M (cid:1) (cid:0) M k − (cid:1) , (4.2)as well as their anti-bions and anti-triplets. The proliferation of bions and triplets generatesa potential of σ : V ( σ ) = − e − π kg { cos ( α k − α k − ) · σ + cos ( α k − α k +1 ) · σ + cos ( α k − α k − ) · σ + cos ( α k − α ) · σ + cos ( α k − α k − − α k +1 ) · σ + cos ( α k − α k − − α ) · σ }− e − π kg (cid:88) { ≤ i< k − }∪{ k +1 ≤ i< k − } cos ( α i − α i +1 ) · σ . (4.3)The triplets fugacity is exponentially suppressed compared to the bions fugacity and onemight be tempted to ignore the triplets. This, however, leaves some flat directions, i.e.,massless photons , which are lifted once we take the triplets into account.One can easily check that the potential admits a global minimum at σ = 0, and then wecan use the chiral transformation σ → σ − C , where C is given by (3.12), to obtain the restof the vacua: σ = 2 πn k + 2 w k + k − (cid:88) a =1 ,a (cid:54) =2 k w a , n = 0 , , ..., k + 1 . (4.4)As promised, there are 4 k + 2 distinct vacua, which are required to match the BC anomaly. SU (4 k ) with -index antisymmetric fermions We also work in the center-symmetric vacuum. The monopole vertices are given by: M a = e − π kg e i α a · σ ψ ˜ ψ , a (cid:54) = 2 k or 4 k , M k = e − π kg e i α k · σ , M k = e − π kg e i α k · σ , while the bions are M a M a +1 , ≤ a < k − k + 1 ≤ a < k − . (4.5)The proliferation of the bions and the two monopoles M k and M k leaves flat directions,and in order to lift them one needs to take into account higher Kaluza-Klein monopoles. Note, however, that massless photons can still match the BC anomaly. – 21 –efore discussing these higher order corrections, one wonders about the possibility of theformation of bion-like compositions between not neighboring monopoles, e.g., bions of the form M a M a +2 that could lift the flat directions. The problem, though, is that such compositionsare unstable against the attractive force due to the exchange of fermions zero modes. Also,the absence of any kind of Coulomb interactions between the monopoles (remember that α a · α a +2 = 0) eleminates the possibility of analytically continuing the coupling constant g ,i.e., sending g → − g , that could generate a repulsive coulomb force to compete with thefermion attractive force. This is the famous Bogomolny Zin-Justin analytical continuationprescription that has been used in several works to stabilize bion-like objects, see, e.g., [43].In summary, we do not expect bions of the type M a M a +2 to form in vacuum.Now, we need to go to the next-to-next-to-leading order in fugacity and consider thehigher Kaluza-Klein monopoles (3.14). A typical example of a complex molecule that can liftthe flat directions is composed of a p = 1 Kaluza-Klein monopole charged under α k , whichhas a total of 8 k − k − − α : M p =14 k (cid:2) M (cid:3) k − , (4.6)see Figure 1. The proliferation of the monopoles, bions, and higher composites generatesmasses for all photons and leads to the full breaking Z dχ k − → Z . The theory admits 4 k − σ = 2 πn k − k − (cid:88) a =1 ,a (cid:54) =2 k w a , n = 0 , , ..., k − . (4.7) In this section we attempt to partially answer the question about the BC anomaly matchingat finite temperature. As we pointed out in Section 3.5, we can reduce the problem to 2dimensions by compactifying the time direction on a circle and keeping only the zero modeof the dual photons. Definitely, if the temperature is high enough, then the W-bosons andheavy fermions will be liberated and their effects, in addition to the monopoles and compositeinstantons, have to be taken care of. The problem, then, reduces to an electric-magneticCoulomb gas, which in general is a strongly-coupled system. This Coulomb gas was consideredbefore in the SU (2) and SU (3) cases with adjoint fermions, see [44–48]. Non of these works,however, addressed the issue of anomaly matching. Here, we do not provide a full solution tothe anomaly-matching problem at all temperatures, which will be pursued somewhere else.Let us, at least, show how the BC anomaly is being matched as we crank up the temperatureand stay in the weakly-coupled regime. We comment on the fate of the BC anomaly at veryhigh temperatures at the end of the section.From here on we work in 2 dimensions. The general structure of V ( σ ) takes the formof a collection of cosine terms, see e.g., (4.3), V ( σ ) = (cid:80) m y m cos( Q m · σ ), where y m is thefugacity of the instanton, Q m , is its charge, and the sum runs over the various instanton types:– 22 –onopoles, bions, etc. One, then, expands the cosine terms and write the grand canonicalpartition function as: Z = (cid:88) m ∞ (cid:88) N m =0 y N m m N m ! (cid:90) N m (cid:89) j m =1 d (cid:126)x j m e − (cid:82) L , (4.8)where L = g π LT ∂ i σ · ∂ i σ + (cid:88) m J m · σ . (4.9)The latin letter i = 1 , R space and J m = Q m δ (2) ( (cid:126)x − (cid:126)x m ) is the current source ofan instanton of charge Q m located at (cid:126)x m . Then, we can solve the Gaussian system, ignoringthe monodromies of σ since they correspond to heavy electric excitations not accessible atlow temperature, to find the potential energy between two sources: V ( (cid:126)x , (cid:126)x ) = − πLTg Q · Q log T | (cid:126)x − (cid:126)x | . (4.10)Next, we substitute this result into (4.9) to obtain the grand canonical partition function ofa magnetic Coulomb gas: Z = (cid:88) m ∞ (cid:88) N m =0 y N m m N m ! (cid:90) N m (cid:89) j m =1 d (cid:126)x j m e πLTg (cid:80) m,m (cid:48) (cid:80) a (cid:54) = b Q am · Q bm (cid:48) log T | (cid:126)x a − (cid:126)x b | , (4.11)and we need to impose a neutrality condition on the gas to avoid IR divergences. In orderto understand what happens as we increase the temperature, we need to follow the fugacitiesof the magnetic charges under the renormalization group flow. Let us consider a pair ofmagnetic charges Q m and − Q m located at (cid:126)x and (cid:126)x and separated by a distance L . Thepair’s contribution to the partition function is (cid:18) y m ( a ) a (cid:19) (cid:90) d (cid:126)x d (cid:126)x (cid:12)(cid:12)(cid:12)(cid:12) (cid:126)x − (cid:126)x a (cid:12)(cid:12)(cid:12)(cid:12) − πLTg Q m · Q m = y m ( a ) (cid:18) La (cid:19) − πLTg Q m · Q m , (4.12)where a is a UV cutoff. Demanding the invariance of the left hand side under the renormal-ization group flow means: y m ( a ) (cid:18) La (cid:19) − πLTg Q m · Q m = y m ( ae b ) (cid:18) Lae b (cid:19) − πLTg Q m · Q m . (4.13)Taking the derivative with respect to b and setting b = 0, we obtain the renormalizationgroup equations of the fugacities dy m db = (cid:18) − πLTg Q m · Q m (cid:19) y m . (4.14)– 23 – igure 2 . The critical temperatures of SU (8) with fermions in the 2-index symmetric representation.The directions of the arrows indicate the temperature range at which the fugacity of the correspondingcharge becomes relevant. The domination of magnetic (electric) charges is indicated by black (red)arrows. See the text for a detailed discussion. Equation (4.14) determines the critical temperature above which the fugacity of a certainmagnetic charge becomes irrelevant: T cm = g πL Q m · Q m . (4.15)Therefore, as we heat the system, magnetic charges with bigger Q m · Q m decouple first. Thisis the Berezinskii-Kosterlitz-Thouless (BKT) transition.In oder to make sure that T cm is well within the semi-classical regime—so that we canneglect the effect of the electric charges, hence, the renormalization group analysis we per-formed above is justified—we need to compute the critical temperatures at which the electricexcitations, the W-bosons and heavy fermions, dominate the plasma. An electric charge withmass M will have a fugacity given by the Boltzmann factor y e = e − MT , and the electricpotential between two charges is given by: V ( (cid:126)x , (cid:126)x ) = − g πLT Q · Q log T | (cid:126)x − (cid:126)x | . (4.16)Then, we can repeat the above steps to find the renormalization group equations of the electricfugacities: dy e db = (cid:18) − g πLT Q e · Q e (cid:19) y e , (4.17)from which we find the critical temperature above which the electric charges proliferate: T ce = g πL Q e · Q e . (4.18)As expected, the bigger the electric charge Q e · Q e , the higher the critical temperature abovewhich it dominates the plasma, which is the exact opposite of the magnetic critical tempera-ture. Staying inside the semi-classical, magnetically disordered, regime demands T mc < T ec .– 24 –s an example, let us apply this treatment to SU (8) with fermions in the 2-index symmet-ric representation. This theory contains two types of magnetic charge: the bions, that carrycharge Q = α a − α a +1 , a = 1 , , ,
6, and triplets. There are also two types of triplets: the firsttype, e.g., M (cid:0) M (cid:1) has charge Q = α − α , and the second type, e.g., M (cid:0) M (cid:1) (cid:0) M (cid:1) has charge Q = α − α − α . Using the renormalization group equation of the magneticfugacities (4.14), we find 3 distinct critical temperatures: T c (1)triplet = g πL , T c (2)triplet = g πL , T c bion = g πL , (4.19)which correspond, respectively, to the temperatures above which the first triplet, the secondtriplet, and then the bions become irrelevant. Similarly, we use the weights of the 2-indexsymmetric representation, the fact that the W-bosons carry charges valued in the root lattice,along with the renormalization group equations of the electric fugacities to find 3 distinctcritical temperatures: T c (1) µ = 3 g πL , T cW = g πL , T c (2) µ = 7 g πL , (4.20)which correspond, respectively, to the temperatures at which a first group of heavy fermions,the W-bosons, and then a second group of heavy fermions become relevant.The 6 critical temperatures and the corresponding relevant excitations are depicted inFigure 2. At temperatures smaller than T c (1)triplet the chiral symmetry is fully broken and allthe photons are massive. For temperatures in the range T c (1)triplet < T < T c (2)triplet the first typeof triplets decouple leaving behind a vacuum with one flat direction, i.e., a single masslessphoton. This can be envisaged by studying the effective potential (4.3) after neglecting thefirst type of triplets. 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