Remarks on QCD4 with fundamental and adjoint matter

We study 4-dimensional SU(N) gauge theory with one adjoint Weyl fermion and fundamental matter – either bosonic or fermionic. Symmetries, their ’t Hooft anomalies, and the Vafa-Witten-Weingarten theorems strongly constrain the possible bulk phases. The first part of the paper is dedicated to a single fundamental fermion or boson. As long as the adjoint Weyl fermion is massless, this theory always possesses a Zχ2N chiral symmetry, which breaks spontaneously, supporting N vacua and domain walls between them for a generic mass of the matter fields. We argue, however, that the domain walls generically undergo a phase transition, and we establish the corresponding 3d gauge theories on the walls. The rest of the paper is dedicated to studying the multi-flavor fundamental matter. Here, the phases crucially depend on the ratio of the number of colors and the number of fundamental flavors. We also discuss the limiting scenarios of heavy adjoint and fundamentals, which align neatly with our current understanding of QCD and N = 1 super Yang-Mills theory. ar X iv :2 30 6. 01 84 9v 1 [ he pth ] 2 J un 2 02 3


Introduction
Quantum field theories (QFTs), fundamental or emergent, are important in modern physics. They are the cornerstone of the Standard Model of particles. In statistical mechanics, they are an efficient way to capture universal features of phase transitions. At the same time, in condensed matter, they are useful to parametrize the important, low-energy features of various materials. Non-abelian gauge theories are the most infamous of QFTs. They describe Quantum Chromodynamics (QCD). Quarks, labeled by 3 colors, interact with non-abelian gauge fields to produce a theory that makes sense as a genuine quantum field theory (i.e., has a continuum limit) and, in addition, has highly nontrivial features in the infrared, such as chiral symmetry breaking and confinement which have yet to have a complete theoretical understanding. This is why understanding QCD-like theories has a special place in theoretical physics. Most recently, a popular method of analyzing such theories was pioneered by the discovery of novel generalized global symmetries and 't Hooft anomalies [1][2][3], leading to an avalanche of discoveries (see, e.g., [4][5][6][7][8][9][10][11][12][13][14][15][16][17] for an incomplete list). In this paper, we will be concerned with one such theory, namely the SU(N ) gauge theory with either fermionic or bosonic quarks (i.e., matter in the fundamental representation), supplemented by a single Weyl fermion in the adjoint representation of the SU(N ) gauge group. In some sense, the theory in question is halfway between QCD (the version of our theory without the adjoint Weyl fermion) and N = 1 Super Yang-Mills theory (SYM) (our theory without fundamental matter).
We will mostly be concerned with a massless adjoint Weyl fermion so that there is always a Z χ 2N discrete chiral symmetry 1 . The fundamental matter (bosonic or fermionic) can be massless or massive. Let us send the mass of the fundamental matter to infinity. The theory becomes super Yang-Mills, for which many things are known. The theory breaks its chiral Z χ 2N symmetry down to Z 2 fermion number symmetry, leading to N degenerate supersymmetric vacua separated by domain walls that support a topological quantum field theory (TQFT). In this paper, we use the existence of a novel anomaly involving the baryon symmetry and the methods of effective field theory to show that such a phase persists in the bulk at any finite fundamental mass.
In particular, consider the simplest case of one adjoint Weyl fermion and one massive fundamental Dirac fermion. The global symmetries are only the Z χ 2N chiral 1 We use the superscript χ in Z χ 2N to distinguish the discrete chiral symmetry from other discrete groups that appear in the text. As we shall discuss, if the fundamental matter is fermionic and massless, the discrete chiral symmetry enhances to U(1) χ chiral symmetry. symmetry, acting on the adjoint Weyl fermions, and the U(1) B baryon symmetry, acting on the fundamental fermion. We will show that there is a mixed anomaly between the two symmetries for any mass of the fundamentals, indicating that either U(1) B or Z χ 2N is spontaneously broken 2 . The Euclidean path integral measure of this system is positive definite, and one can apply the Vafa-Witten theorem [19] to conclude that U(1) B cannot be spontaneously broken. Hence, anomaly matching conditions demand that Z χ 2N be spontaneously broken. In addition, Z χ 2N has a mixed anomaly with gravity; thus, even if the Vafa-Witten theorem did not hold (e.g., Vafa-Witten is applied to a fermionic fundamental matter, but as we will see, the same anomalies apply to bosonic fundamental matter) the breaking of U(1) B would not saturate all the anomalies. Therefore, introducing massive fundamental fermions does not change the bulk phase of SYM: the theory still has N vacua connected via domain walls. While there is no phase transition in bulk as we vary the mass of the fundamentals, we shall argue that phase transitions will occur on the domain walls. At m = 0, the Z χ 2N symmetry enhances to U(1) χ , and the domain walls melt away, leaving a Goldstone boson for the spontaneously broken U(1) χ .
If we replace a fundamental fermion with a fundamental scalar, a similar conclusion can be drawn 3 : no phase transition in the bulk takes place as the fundamental mass squared is driven from large and positive (where the boson decouples) to large and negative (where the boson condenses). Yet, a transition on the domain wall is expected to occur.
We also discuss theories with more fundamental flavors and establish analogous anomalies. For N f fundamental fermions, matching the anomalies in the IR happen via one of two channels depending on the number of flavors: either the theory breaks its symmetries spontaneously, or it flows to a conform field theory (CFT). These scenarios are summarized in Figures 1 and 13 for the single flavor and multi flavors, respectively.
We briefly give an incomplete review of the literature on the mixed representation QCD. Of course, 4d super QCD is the most well-known mixed-representation QCD-like theory. Thanks to holomorphy, a lot is known about the IR phases of these theories, which, by now, is textbook material. The recent work [20] is relevant 2 The anomaly manifests itself as follows. An insertion of the minimal U(1) B flux will activate the color 't Hooft fluxes. This happens because U(1) B is the quotient of the U(1) q quark symmetry by the center of the gauge group Z N , i.e., U(1) B ∼ = U(1) q /Z N . Consistency of the cocycle condition in the presence of the U(1) B background requires we also activate the 't Hooft flux of the center of SU (N ). When this happens, however, the instanton number becomes fractionally quantized, and Z χ 2N is explicitly broken to Z 2 by the presence of the U(1) B flux. This is similar to [4,18]. 3 There is a slight caveat to this statement.
to our studies, which analyzed domain walls in super QCD. The mixed fundamental/adjoint representation was also analyzed in the context of the weakly coupled R 3 × S 1 and adiabatic continuity [21]. In [22], theories with adjoint and higher mixed-representations were studied on R 3 × S 1 , and in [14], SU (6) with fermions in the adjoint and the 3-index antisymmetric mixed representations were studied in 4d and on R 3 × S 1 . In [23], a systematic analysis of chiral perturbation theory was done for arbitrarily mixed representations, while in [24,25] lattice simulations with fermion fundamental and anti-symmetric sextet representation of SU(4) gauge theory was done. More relevant for our work is [26], where the mixed representation of fundamental/adjoint fermions was simulated.

A warmup: Higgs phase of SU(2) gauge theory with a single scalar
The various anomalies involving the chiral symmetry Z χ 2N , like the mixed Z χ 2Ngravitational anomaly, or the pure Z χ 2N anomaly, are purely due to the adjoint fermions. But there exists an anomaly involving Z χ 2N and U(1) B -the baryon symmetry carried by the fundamental matter. Such an anomaly exists regardless of whether the fundamental matter is bosonic or fermionic. This seems surprising at first, as even if the fundamental matter is fermionic, there are no triangles involving U(1) B and the chiral symmetry. When the fundamental matter is bosonic, the anomaly seems even stranger still. We will illustrate this anomaly by a simple example of an SU(2) gauge theory with one fundamental scalar Φ and one adjoint Weyl fermion λ. The theory has a Z χ 4 discrete chiral symmetry, whose Z 2 subgroup is the fermion number. It acts on λ as λ → iλ . (1.1) There is also a U(1) symmetry acting on the scalar φ → e iα φ. The Z 2 subgroup of U(1) is, however, the center of SU(2) gauge group, so one can view the global symmetry as U(1)/Z 2 ∼ = U(1). We distinguish between two normalizations. First, we define U(1) B as the baryon symmetry, i.e., a symmetry under which the smallest charge of the baryon is unity. This symmetry group is related to the quark symmetry U(1) q , under which the quark has the unit charge, as follows U(1) B = U(1) q /Z 2 (or for general SU(N ) as U(1) B = U(1) q /Z N ). The scalar then transforms as φ → e iα/2 φ under U(1) B , where now α ∼ α + 2π.
If we condense the scalars, they fully Higgs the SU(2) gauge group, leaving only three ungapped free adjoint Weyl fermions associated with the algebra of SU (2). We can write the Weyl fermion as λ = λ a τ a 2 , where summation over a = 1, 2, 3 is implied and where τ a are the Pauli matrices. The fermions λ a are, however, not gauge invariant, and the correct gauge invariant operators corresponding to the three Weyl fermions are where we explicitly wrote out fundamental indices A, B, C = 1, 2 of SU(2) ( AB being the totally anti-symmetric tensor). We enumerate the charges of the three Weyl fermions under the Z 4 chiral symmetry and U(1) B symmetry 4 in Table 1. The effective theory is hence a theory of three Weyl fermions ξ 1,2,3 with a given charge assignment under the global symmetries given in Table 1. It is now straightforward to see that there is a standard triangle U(1) B − Z χ 4 anomaly. Here we found an anomaly using the IR theory 5 , but since the anomaly is RG invariant, it implies the same anomaly exists in the UV theory. Later, we will see more formally how this occurs in a more general setting.
At first, this anomaly sounds odd, as the baryon-number carrying fundamental scalar Φ naively has nothing to do with the chirally charged λ. Further, it generalizes to any fundamental matter charged under the baryon number in the presence of a massless adjoint Weyl fermion. In addition the scalar charged under the baryon number can be made arbitrarily massive, which goes against the traditional lore that such fields can participate in the anomaly. However in recent years there are many examples of this type. In [5,6] a mixed anomaly between U(1)-topological symmetry and SO(3) (or more generally PSU(N )) flavor symmetry was utilized in Abelian-Higgs models. The scalars can be massive, and the anomaly persists. In [4] an anomaly between T and vector-like flavor symmetries was also used in QCD in 4 The model enjoys an enhanced flavor symmetry, called custodial symmetry, because of the pseudo-reality of the gauge group. The true flavor symmetry is actually SO(3) with U(1) B as its subgroup. Since we have a triplet of Weyl fermions, two of which are charged as ±1 under the U(1) B , the three naturally fit into the triplet of SO(3) flavor group. 5 Note however that the Z χ 4 enhances to U(1) χ symmetry because all mass terms involving ξ 1,2,3 either preserve all of U(1) χ or keep only Z 2 ⊂ Z χ 4 .
4d, again for any mass of the flavor multiplet. In all these cases (including the one we discuss here) the decoupling limit results in a theory with a 1-form symmetry, while the anomaly involving flavor transmutes to an anomaly involving the 1-form symmetry.
In our theory, a way to understand this anomaly from the UV point of view is to start with a massless adjoint fermion and a fundamental scalar coupled to the SU(N ) gauge fields. Let the scalar have a very large positive mass. Then, the theory has a mixed anomaly between the discrete Z χ 2N chiral symmetry and an emergent Z [1] N 1-form center symmetry (the 1-form symmetry becomes exact in the infinite mass limit) [2] (see also [3]). Now, lower the mass of the scalar or even take it to be negative. The presence of the scalar breaks the 1-form symmetry explicitly, but it introduces a U(1) B 0-form symmetry. As we will see the anomaly involving the Z [1] N 1-form symmetry transmutes into the anomaly involving the U(1) B baryon symmetry. Roughly speaking the background gauge fields for both the U(1) B as well as the 1-form Z [1] N symmetry activates 't Hooft fluxes, which causes the anomaly to manifest itself. We shall see the details of this anomaly in the bulk of the paper using different methods. To see that the anomaly is there also for the theory with fermions, we can introduce a fundamental Dirac fermion Ψ consisting of two Weyl fermions ψ andψ in the fundamental and the anti-fundamental representation respectively. Further we postulate a Yukawa coupling If we add a mass to the scalars, we decouple them and the model reduces to an SU(2) gauge theory with a single fundamental Dirac fermion. On the other hand condensing the scalar and Higgsing the theory will reveal a composite fermion phase with the required mixed Z χ 4 -U(1) B anomaly, as we will see in a moment. Since the symmetries in these two limits are unchanged, the anomaly structure has to be the same.
We can also add a mass term mψψ + c.c.. The symmetries of the model involve the U(1) B symmetry and, if m = 0, the U(1) χ chiral symmetry. When the mass m = 0 the chiral symmetry reduces back to Z χ 4 . Table 2 on the left summarizes the charges of the fundamental fields under these symmetries 6 . Now let us condense the scalar φ like before, higgsing the gauge fields entirely.
In the gauge where φ = v 0 , the fermions ψ A and vλ 1 A + mψ A , while fermions λ 2 1 and mλ 1 A − vψ A remain massless, 3 fermions in total. The massless fermions can be written in a gauge invariant form as We summarize the charges under the global symmetries of these fields in the right of Table 2. As we will see these composite fermions precisely correspond to the proposal. Further we can send m → ∞, and then we restore the single Higgs model where η 0,1,2 → ξ 2,3,1 7 . We will have more to say about this model in Sec. 2.3.
Finally let us briefly add that we could also see the anomaly in SU(N ) gauge theory with fermionic flavors. One would then couple N −1 fundamental Higgs fields, with appropriate Yukawas designed to preserve the global symmetries of the theory, and condense them in such a way to Higgs the gauge fields completely. The resulting phase is always a phase of free fermions, and the anomalies become manifest. We will not pursue this in details

Outline
The rest of the paper is organized as follows. In § 2, we study in detail the theory with a single fundamental Dirac fermion in the presence of a massless adjoint Weyl. We identify the faithful global symmetry and the associated 't Hooft anomalies constraining the IR phase and its domain wall theory. In § 3, we repeat the analysis with a single fundamental scalar that replaces the single fundamental fermion. We 6 Note that since there is no Yukawa coupling forψ, it seems that there is an additional baryon symmetry rotatingψ only. This symmetry however is anomalous. To make it anomaly-free we must either rotate the adjoint λ by an appropriate phase, or ψ by an appropriate phase or a combination of both. It is easy to see that this results in two U(1) symmetries, which we choose to label as U(1) B and U(1) χ with fields charged as in Table 2. 7 Note that in the model without the (anti-)fundamental fermions we defined the scalar field not to transform under the chiral symmetry U(1) χ , so the identification of charges of Table 1 and 2 should be made up to this redefinition generalize the story in § 4 to several fundamental Dirac fermions and a single adjoint Weyl fermion and elaborate on their anomalies. We generalize the analysis to multifundamental scalars in § 5. We conclude and outline some possible future directions in § 6. The appendices summarize some points used in different parts of the paper.
In Appendix A, we show that the fermionic measure is positive definite. In Appendix B, we review the relation between spectral flow and the index theorem. Finally, the 3-loop β-function used in some of our analyses in § 4 is displayed in Appendix C.
2 Theory with one fundamental fermion: symmetries, anomalies, and the phase diagram In this section, we study the symmetries and anomalies of 4d SU(N ) gauge theory coupled to one Dirac fermion in the fundamental representation and one Majorana fermion in the adjoint representation. Note that we can alternatively view the Majorana fermion in the adjoint representation as a Weyl fermion, but not both simultaneously. In the rest of the paper, we will take the adjoint fermion to be Weyl for convenience of the analysis.

The massive case
Denote the SU(N ) gauge field that couples to the fundamental fermions by a and its counterpart for the adjoint fermions by a adj . Then the action of the fermionic sector is given by Here, λ is a left-handed Weyl fermion in the adjoint representation of SU(N ). Ψ is a Dirac fermion in the fundamental representation of SU(N ). In the chiral basis, we decompose Ψ as where we use the notation that both ψ andψ are left-handed Weyl fermions in the fundamental and anti-fundamental representations of the gauge group. The theory has a classical U(1) χ chiral symmetry acting on the adjoint: which reduces to Z χ 2N discrete chiral symmetry by the ABJ anomaly. Further, there is a U(1) symmetry acting on the fundamental quark: We will call this symmetry U(1) q , where q stands for quark. Note that if β ∈ 2π N Z, we can absorb the symmetry in the SU(N ) gauge transformation so that the true global symmetry group is U(1) B ∼ = U(1) q /Z N -the Baryon number symmetry. Under U(1) B , the smallest charge of the baryon is unity. Thus, the faithful global symmetry The fermion content is summarised in terms of their representations under various symmetries in Table 3 below. Finally, the action of the Z 2 subgroup of the symmetry U(1) B × Z χ 2N : coincides with the action of the Z F 2 fermion number. Thus, there is a mixing between the Spin spacetime symmetry and the Z χ 2N into a Spin-Z χ 2N := (Spin × Z χ 2N )/Z F 2 structure. We can then use this structure to define the theory on some orientable manifolds that are non-spin by turning on a non-trivial Z N ∼ = Z χ 2N /Z F 2 bundle on such a manifold whose obstruction to lifting to a Z χ 2N bundle is precisely the second Stiefel-Whitney class of the manifold 9 .
To study the anomalies, let us couple a background gauge field A q to the U(1) q symmetry by promoting where A is a U(N ) gauge field whose traceless part is dynamical. We write F for the field strength of A and F q for the field strength of A q . Note that tr F = tr(F q 1 N ) = N F q is quantized in integer units of 2π. Hence it follows F q can be fractionally quantized in units 2π/N . This is because U(1) q is not the proper global symmetry (i.e. there are no gauge invariant operators with the unit charge under 8 We can also use the cocycle conditions as a systematic way to find the faithful symmetries. See § 4. 9 Note that this only works for orientable manifolds M with non-vanishing H 1 (M ; Z N ) because otherwise, there can be no non-trivial Z N bundle in the first place.
U(1) q ) but U(1) B is. We will also make use of the properly quantized baryon gauge field A B = N A q , and its curvature F B = N F q . Now, applying the Z χ 2N : λ → e i πk N λ symmetry transformation, the action changes as where the trace is taken in the fundamental representation. In the above result the traceless part is subtracted, because the adjoint field λ cannot see the trace of F . Now we write The first term on the RHS is integer-quantized on a spin manifold. So the partition function changes under the Z χ 2N transformation as Recall that F B is a properly quantized U(1) field strength; on a closed spin manifold, we have 1 2 We can then see that the above phase is nontrivial for k = 1, 2, . . . , N −1. Thus, there is a mixed anomaly between the Z χ 2N discrete chiral symmetry and the U(1) B baryon symmetry 10 . As we mentioned in § 1.1, the reader might not feel at ease about a mixed anomaly between Z χ 2N and U(1) B since none of the two symmetries couples to the two fermion species simultaneously, i.e., one does not see such an anomaly from triangle diagrams 11 . Another equivalent way to obtain the anomaly is to realize that a minimal flux for A B field induces a 't Hooft flux for the color fields, which makes the color topological charge fractional, Q c ∈ Z/N , and thus leads to a reduction of the chiral symmetry to Z χ 2 . Such anomaly mechanism, where putting background fields for a global flavor symmetry forces the instanton number to be fractional, has been observed before in [4,10,27], or more closely related to our setup, in [18,28] where they were dubbed baryon-color-flavor (BCF) and color-flavor-U(1) (CFU) anomalies. We will discuss this point of view in § 4 in more detail. 10 Notice that when k = N , the phase is trivial, meaning that the fermion number symmetry is not anomalous with the U(1) B baryon symmetry. 11 However, recall that this is precisely how we observed the anomaly in § 1.1, by going to a Higgs phase and establishing the fermion content in the IR.
The theory also exhibits a mixed Z χ 2N -gravitational anomaly. Under a Z χ 2N rotation, the partition function transforms as where p 1 ≡ − 1 8π 2 trR ∧ R is the first Pontryagin class of the tangent bundle, R is the curvature 2-form, and the integral is taken on a closed 4-manifold M . Notice that on a spin manifold M p 1 ∈ 48Z.
Finally, there is a nonperturbative discrete anomaly arising from the symmetry group Z χ 2N . Recall that, because of the quotient Z F 2 between the spin group and the Z χ 2N symmetry group, we can define a theory on a non-spin manifold as long as the manifold admits a Spin-Z χ 2N structure. The anomalies in this structure is classified by the cobordism group [29,30] 6 For a Weyl fermion of charge q mod 2N under Z χ 2N , the anomaly is given by a pair of indices (ν a , ν b ) ∈ Z a × Z b where ν a and ν b are explicitly given by [29,31] 12 For instance, when N = 3, 6 Spin Z χ 6 ∼ = Z a=9 . The anomaly from our adjoint fermion is then ν 9 = −1 mod 9 ∈ Z 9 .
As we shall see this anomaly can be saturated by Z χ 2N symmetry breaking (see How are all these anomalies matched in the IR? The most natural way is to spontaneously break the Z χ 2N symmetry for any mass of the fundamental matter. We know this is the right answer when the mass of the fundamentals is large enough, but one could speculate some sort of bulk transition for small enough fundamental fermion mass. What could this phase be? In [32,33], it was shown that unitary and symmetry-preserving TQFTs are excluded in 4d. Another option is to have massless composite fermions. We will in fact propose such fermions for massless fundamental matter which will match all the anomalies. However one can use the argument of Weingarten 13 [34] to show that the meson mass is always smaller than the baryon. Since massless composites must necessarily include baryons to saturate the mixed Z χ 2N − U(1) B anomaly, it follows that the meson must also be massless, which is not natural unless it is a Goldstone boson. So it seems that Z χ 2N chiral symmetry must be spontaneously broken. The order parameter of this breaking is the bilinear condensate Trλλ. A non-vanishing expectation value of Trλλ breaks Z χ 2N down to the fermion number Z 2 and results in N distinct vacua connected via domain walls. Finally let us briefly consider a scenario where Z χ 2N is broken, but U(1) B is 12 We thank Joe Davighi for working out the general expression for ν b with one of the authors. 13 Weingarten considered QCD -i.e. a theory fundamental fermions only. However the argument is unchanged as the adjoint fermion does not invalidate the positivity of the measure as we show in the Appendix A.
spontaneously broken as well. However this is prohibited by the Vafa-Witten [19] argument.

The massless case
When the mass of the fundamental Dirac fermion goes to zero, i.e., m = 0, the U(1) B baryon symmetry remains unchanged, and the theory has an enhanced chiral (axial) U(1) χ symmetry 14 . To see this, note that there are more classical chiral symmetries in the massless case, which now acts on the fundamental Dirac fermion. The chiral transformation λ → e iα λ, ψ → e iβ ψ,ψ → e iβψ (2.14) produces a change in the action where f is the SU(N ) 2-form field strength. If we set β = −N α the action is obviously invariant, hence the transformation transformation is a good symmetry even at the quantum level. Thus, the faithful global symmetry is The fermions transform under G Global in the representations given by Table 4. 14 The adjoint/fundamental mixed representation was also studied on M 4 = R 3 ×S 1 in [21], which was shown to have a massless goldstone boson associated with the U(1) χ symmetry. Therefore, it was conjectured that the theory with a single fundamental flavor is continuously connected to the theory on R 4 as we decompactify S 1 .
The U(1) χ symmetry carries a 't Hooft anomaly. Indeed, the anomaly is a consequence of the triangle diagrams containing λ and Ψ. The anomaly coefficient is given by where the first two factors come from the fundamental fermions ψ,ψ, and the third one comes from the adjoint fermion λ. Note that the fundamentals carry charge −N under the chiral symmetry, and there are N colors, while the adjoint carries a charge 1 and there are N 2 − 1 colors. In addition to this cubic anomaly of U(1) χ , we also have a mixed anomaly between U(1) χ and U(1) B . If we were not careful about the modding by Z N in (2.17), we would find that the coefficient of this anomaly comes from the triangle diagrams U(1) χ [U(1) q ] 2 , which yield −2N 2 . This is the traditional 't Hooft anomaly. However, the modding by Z N in (2.17) refines this anomaly. To compute it, we put background fields for the U(1) B symmetry and perform the chiral transformation (2.16). We then have where F B = N F q is the properly normalized baryon background field. Notice that this is the same as (2.7) and (2.8) we found previously for Z χ 2N chiral symmetry, which is just α = πk N , k ∈ Z. The theory also has a mixed U(1) χ -gravitational anomaly. All these are captured by a 5d action 15 where A χ and F χ are the background gauge field and its field strength for the U(1) χ chiral symmetry, and where integral over the first Pontryagin class obeys M p 1 ∈ 48Z.

The IR phases
In the infinite mass limit, the Dirac fermion completely decouples, leaving us with a pure N = 1 SYM with gauge group SU(N ). The discrete chiral symmetry remains. 15 From the descent formulas, we have that 1 3!(2π) 3 6d F χ ∧ F χ ∧ F χ ∈ Z, so that the 5d CS theory is k On the other hand 1 2!(2π) 3 6d F χ ∧ F B ∧ F B ∈ Z implies that the 5d mixed CS theory is given by the action k Moreover, there is now an emergent Z [1] N 1-form global symmetry. Therefore, the global symmetry is given by Of course, the super Yang-Mills and the above symmetry will emerge as long as m Λ, where Λ is the strong scale. In this case, there is a mixed anomaly between the 0-form and 1-form symmetries in G Global m→∞ [3,6] 16 . In addition, there is, of course, the mixed gravitational anomaly with Z χ 2N and the nonperturbative [Z χ 2N ] 3 anomaly we discussed in § 2.1. All the anomalies are saturated in the IR by the spontaneous symmetry breaking of Z χ 2N down to Z F 2 via the formation of the bilinear condensate 22) where the N vacua are labelled by k = 0, 1, . . . , N − 1.
In the opposite limit, when m → 0, a priori, we can saturate the anomalies in two ways. One can spontaneously break U(1) χ symmetry or have massless composite fermions. Weingarten's identities exclude the latter whenever the Euclidean action is positive definite. The fact that our action is positive definite is demonstrated in Appendix A. So the only alternative is that U(1) χ is spontaneously broken. The IR theory contains a Goldstone boson: a compact scalar ϕ ∼ ϕ + 2π whose effective action is given by up to an overall multiplicative constant that scales as N 3 in the large-N limit (see § 2.4). The scalar ϕ is associated with the tr λλ ∼ e iϕ , and therefore, the operator e iϕ carries a charge 2 under U(1) χ . 16 To see this, couple a background gauge field to Z [1] N . This is a closed 2-cochain B 2 with Z N coefficient satisfying the 1-form gauge transformation 17 (1) with λ (1) a 1-cochain. It, therefore, defines a class B ∈ H 2 (M 4 ; Z N ), where M 4 is the 4-dimensional spin manifold on which our theory lives. The theory now admits a non-trivial P SU(N ) gauge bundles that are not SU(N ) gauge bundles, with the obstruction given by w 2 (P SU(N )) = B. Consequently, the instanton number can now be fractional and quantized in units of 1/N . The theta angle is now 2πN -periodic. As the classical U(1) χ chiral transformation with parameter α shifts θ → θ − 2N α, the non-anomalous subgroup is generated by α = π, instead of α = π/N as we had before when the periodicity was 2π. It is also simple to see that any local counter terms cannot restore this periodicity. Thus, the Z χ 2N chiral symmetry is anomalous, but its Z F 2 subgroup remains anomaly-free. We interpret this as a 't Hooft anomaly between the 1-form symmetry and the chiral symmetry.
The above IR action, however, does not reproduce the UV anomalies, in particular the [U(1) χ ] 3 , the [U(1) χ ][U(1) B ] 2 and the U(1) χ -gravitational anomalies. To fix this problem, we find a set of composite fermions that match anomalies. We assume that we have K massless fermions with charges q χ 1 , q χ 2 , . . . , q χ K under U(1) χ and charges q B 1 , q B 2 , . . . , q B K under U(1) B . Then they match the [U(1) χ ] 3 , mixed U(1) B and U(1) χ , and the U(1) χ -gravitational anomalies if and only if 18 and q χ k need not be distinct. In general, there are multiple ways of satisfying the above conditions for particular N . Here, we find a solution that works on all N , so it is a natural large N candidate.
First, note the identity Thus, we satisfy the [U(1) χ ] 3 anomaly by taking q χ 0 = −1 and q χ i = −2i + 1 for i = 1, . . . , N . On the other hand, if we take anomaly. Finally we also look at the mixed U(1) χ -gravitational anomaly, which is given by −1 + N i=0 (−2i + 1) = −(N 2 + 1) as it should be. Note that for particular values of N , we can match the anomaly with other choices, so the above formula should be interpreted as a natural choice reproducing the smooth large-N limit.
But we do not want the fermions to be gapless; as we pointed out, the Weingarten theorem [34] excludes this scenario. Instead, we want to supplement the Goldstone action (2.23) with fermions that are gapped in the bulk, but otherwise become massless on the vortex worldsheet. To this end, we write bulk fermion mass terms. Such 18 In fact, we could replace the second condition , without affecting any of our conclusions. It was proven in [18] that if a vector-like theory does not possess a genuine discrete chiral symmetry, massless composites that saturate the traditional 't Hooft anomalies will also match the CFU anomalies. This is easily seen by observing that the only difference between the traditional U(1) χ [U(1) B ] 2 and the anomaly (2.19) is the different normalizations of the U(1) B charges, which account for the multiplicative N 2 factor that appears in the traditional anomaly.
mass terms must respect U (1) B and can be made U(1) χ invariant by decorating them with the appropriate power of the operator e iϕ . With the charges q χ i and q B i we chose above, we can supplement the Goldstone action with We note that q χ i are odd, so q χ i + q χ j is even for any pair i, j. The above action should not be viewed as an effective theory of the bulk. Indeed the masses of the fermions above are expected to be of order Λ, which is the UV cutoff of the would-be effective theory 19 . Instead the purpose of including these fermions is two-fold. One purpose is to show explicitly that all anomalies can indeed be saturated by spontaneous breaking of U(1) χ , which is not immediately obvious. Indeed a symmetric mass terms for χ i are only possible if one assumes that its phase is a dynamical field, implying that a Goldstone mode is needed.
The second benefit of the above action is that unlike other modes with mass of order Λ, the above fermions must have a vanishing mass at the center of the vortex of ϕ Goldstone. So the above action contains a proposal for the degrees of freedom on the vortex. In other words, even though the masses of the bulk fermions are at the scale of the cutoff, they go down to zero as one approaches the vortex 20 .
Next, we introduce a small mass of the fundamentals. This explicitly breaks U(1) χ → Z χ 2N . The most relevant term that achieves the breaking is cos(N ϕ), and thus, the Goldstone action needs to be modified to (2.27) where g(0, N ) and f (0, N ) are finite numbers scaling as ∼ N 3 and ∼ 1 respectively in the large-N limit (see § 2.4). This gives mass to the Goldstone, as well as lifting the S 1 vacuum manifold to N vacua. We hence again have N vacua for small m. The domain walls have width ∼ 1 √ mΛ , and thus, they are thick compared to the strong scale. 19 The masses of these fermions can however be made arbitrarily small by coupling scalar fields with the appropriate Yukawa terms and going towards the Higgs phase. The Higgs phase is characterized by massless composites, so it is natural to assume the composites become light near the transition (provided that the transition is a continuous one). We will do this explicitly for the SU(2) example we introduced in § 1.1 at the end of this section. 20 It is perhaps interesting to note that coupling scalars in the fundamental representation of the gauge group will in general reduce the masses of the composite fermions, and it may even push the model into S-confinement. This model is also interesting for making connections with the Super QCD. We leave the study of this for the future.
We conclude that in both the limit of small and large m, the discrete chiral symmetry Z χ 2N is spontaneously broken to Z 2 . The simplest assumption is that the intermediate regime has no phase transition.
One wonders if there could be a different phase opening in the intermediate regime. Symmetry-preserving TQFTs are excluded in 4d [32]. Spontaneous breaking of U(1) B neither saturates the nonperturbative Z χ 2N anomaly, nor the mixed Z χ 2Ngravity anomaly. Further U(1) B cannot break by the Vafa-Witten theorem, which holds because of the positivity of the fermionic weight (see Appendix A). The massless composites are also excluded by the (Vafa-Witten-)Weingarten theorem [34]. Therefore any intermediate bulk transition seems inconsistent with the anomalies and Vafa-Witten-Weingarten theorems. We conclude there is no bulk phase transition as we decrease m all the way down to m = 0, at which point N vacua melt away and turn into a (pseudo-scalar) Goldstone boson. This behavior is summarized in Figure 1. For any non-zero m, the discrete chiral symmetry Z χ 2N is spontaneously broken to Z 2 , leaving us with N vacua. At m = 0, Z χ 2N is enhanced to U(1) χ , which also breaks spontaneously, giving rise to a massless Goldstone boson.
Finally, we introduce the adjoint fermion mass m adj . In this case, the theory has a physical θ parameter that can be removed by shifting the phases of m adj and m. Further, recall that there is a combined shift of the adjoint and fundamental phases, which does not affect the θ term. This means we can always make the fundamental mass m real and positive, putting all the θ dependence into the m adj phase. In the following, we use a normalization where the bare θ parameter is zero and m adj is taken to be complex m adj = e i θ N |m adj |. Since the theory is invariant under θ → θ + 2π, the phase diagram in the space of the real m and complex m adj will be symmetric up to a e i 2π N phase shift of m adj . When |m adj | is large, the theory is 1-flavor QCD at θ angle When m adj = 0, the phase diagram reduces to that of Figure 1, which we already discussed. When both m and |m adj | are large, the theory reduces to super Yang-Mills with a θ term given by the phase of m N adj , which we use as appropriate labeling of the parameter space. When θ = π, such a theory is expected to have T -broken vacuum. This phase is labeled by blue sheets. When m becomes small enough, however, the vacuum undergoes a phase transition into the trivially gapped phase. The line on which this happens contains a massless pseudo-scalar. The massless pseudo-scalar phase connects continuously to the massless Goldstone boson phase of (2.23).
given by the phase of (m adj ) N . At θ = π, the theory was shown to have a massless η particle at some particular m = m 0 ∼ Λ/N in the large-N limit [4]. The argument for this is as follows. The mass m of a single fundamental flavor is complex, and its phase is associated with the θ term. Restricting to T -invariant theories, we set θ = 0 and take m real, but it can be both positive and negative. The negative mass theory is equivalent to the θ = π theory with a positive mass term. If |m| is large, the theory is pure Yang-Mills which is believed to break T -symmetry spontaneously at θ = π (large negative mass) [3], but is trivial at θ = 0. Hence, there should be a phase transition restoring the T symmetry at some finite m = m 0 , with a corresponding massless pseudo-scalar particle at that point. In [3], it was argued that this m 0 is negative, but this is not crucial. Since we have adjoint and fundamental fermion masses, we made m real and positive by a non-anomalous chiral transformation and put all the phases in m adj . Hence, when the modulus of m adj is large and the effective θ = π, we should have a massless pseudo-scalar at m = |m 0 |.
What happens as we reduce m adj from infinity to small values? As m adj becomes comparable to Λ, it is natural to assume that the value of m 0 will change. However, the massless phase cannot just disappear as long as we pick m adj in such a way to preserve the T -symmetry 21 , and instead, all that will happen is that m 0 -the fundamental mass at which the massless pseudo-scalar exists -will start moving as a function of |m adj |/Λ until at m adj = 0 it becomes m 0 = 0 where this massless pseudoscalar phase fuses with that of the Goldstone pseudo-scalar of the spontaneous U(1) χ symmetry breaking. All of this is summarized in Figure 2.
This picture is analytically controlled for |m adj | Λ at any 22 N . If a nonzero m adj is introduced, it will induce the following term in the Lagrangian (2.28) so that the potential for ϕ is approximately with some positive constants a ∝ m and b ∝ |m adj | which scale as N and N 2 respectively in the large N limit. We can redefine ϕ + θ/N → ϕ and obtain.
Now, restricting to θ = 0, π, we can capture both by setting θ = 0 and extending a to be also negative. So positive a corresponds to the θ = 0 regime, and negative a corresponds to the θ = π regime. Now notice that if a is positive, the global minimum of ϕ is at 0 mod 2π. When aN 2 = −b, the mass of ϕ vanishes. Taking into account the large N -scaling, we have that a ∼ mΛ 3 N and b ∼ |m adj |Λ 3 N 2 , so that at m = m 0 ∼ − |m adj | N we have a massless pseudo-scalar. If a is dialed to be even smaller, then two vacua emerge, breaking T -symmetry (because ϕ is a pseudo-scalar).

Domain walls
We have seen that the theory with massless adjoint Weyl fermion and one fundamental Dirac fermion with mass m has no phase transition all the way to m = 0. For 21 Naively, any choice of m adj which is not real breaks T symmetry, mapping m adj → m * adj . However, recall that we can always perform a Z χ 2N rotation, which will not induce a θ-term, to remove the Z N phase from m adj . The point is that the theory where the phase of m adj is the (2N )-th root of unity is T -symmetric. 22 This differs from the opposite limit, |m adj | Λ, where one needs to invoke the large-N limit to carry out the analysis, as was done in [3]. every finite m, it supports N discrete vacua and therefore has domain walls. Could there be a phase transition on the domain wall? In the limit of m → ∞, the domain wall has a non-trivial inflow due to the mixed anomaly between the Z χ 2N chiral symmetry and Z [1] N 1-form symmetry, which is saturated by a TQFT [35]. When a single fundamental fermion is introduced, the only remaining anomaly is the mixed U(1) B -Z χ 2N , and the inflow on the n-domain wall we denote as D n 23 is where F B is the field strength of the U(1) B symmetry. Note that the 4d space over which the above integral runs is not the physical 4d space but the domain-wall world volume extension into a 5th-dimensional bulk. The above anomaly polynomial has no inflow and corresponds to some anomalous conductivities on the domain-wall theory [36]. In other words, the domain wall theory has fractional Hall conductivity but can otherwise be gapped. Indeed, we know that in the limit m → ∞, the theory is N = 1 super Yang-Mills, and we understand the domain wall theory well.
The D n domain wall theory is conjectured to be the 3d SYM theory with Chern-Simons level n, where we will take n to have values n = ±1, ±2, . . . , ± N/2 . Such a theory breaks the supersymmetry spontaneously, resulting in a Majorana Goldstino and a TQFT [37][38][39] Since a generic domain wall will not enjoy a time-reversal symmetry, the presence of fundamental matter is expected to induce a time-reversal non-invariant 3d mass term for a Goldstino, but a TQFT must remain robust as long as the fundamental matter is heavy enough. The exception to this is a time-reversal-invariant domain wall which exists whenever N is even and is given by n = N/2, where the TQFT U(N/2) N/2,N is non-trivially time-reversal invariant due to the level/rank duality [40,41]. In this case, there is a non-vanishing mod 16 pure T -anomaly [42,43], which descends from the mod 16 anomaly of the Spin-Z 4 symmetry in the bulk [44]. Both the Goldstino and the TQFT contribute nontrivially to this anomaly [35,38,45], and hence a Goldstino is prohibited from acquiring a mass unless the TQFT is destroyed. Does this happen on the domain walls as the fundamental mass is reduced?
The effective theory is given by a pseudo-Goldstone boson ϕ and the domain walls are just kinks of (2.27). Kinks carry tension ∼ √ mΛΛ 2 which makes them light and within the realm of the effective theory. But one may worry that there are extra light degrees of freedom on the kinks from the anomaly-saturating massive fermions (2.26). That kinks do not have fermionic zero modes case is seen as follows.
Consider the domain wall that takes ϕ 0 → ϕ 1 = ϕ 0 + 2π/N , assuming a profile of ϕ(x 3 ) in the 3-direction which interpolates between two vacua. We first "diagonalize" the symmetric mass matrix 24 25 . Once diagonalized, the system becomes a set of N + 1 4d Majorana fermions, with masses given by the diagonal entries of M D . We now want to check whether the Dirac operator for each i = 0, 1, . . . , N , has a zero mode. If it does, it will be the same as for which is a Hermitian matrix by construction. It is well known that the zero modes of the above operators are in one-to-one correspondence with the spectral flow of the matrix A (see Appendix B). Now, M D cannot depend on the constant piece of ϕ, as a U(1) χ rotation can remove the constant. Hence, the fermion mass spectrum does not change as a function of slowly varying ϕ. We can conclude that no fermion zero modes exist on the domain walls. Further, in super Yang-Mills, the n-domain walls D n are stable and do not decay into n × D 1 domain walls. This is not the case for the domain walls of (2.27), as the domain walls are the usual sine-Gordon kinks that repel each other. The massive fermions cannot induce attraction between the domain walls as they do not have any zero modes and can be integrated out. A sine-Gordon kink, e.g., varying in the x 3 direction, will have ϕ vary slowly so that its derivative ∂ 3 ϕ ∼ √ mΛ. Since we assume m Λ we can integrate out the massive fermions and write an expansion in powers of the derivatives, i.e., in ∂ µ (...)/Λ. Since the derivatives of the kink vary as √ mΛ, and since no zero modes exist by the above reasoning, such a gradient expansion is valid, and the effective domain wall theory is empty. 24 This is an abuse of terminology, as the "diagonalization" is not performed by a matrix P and its inverse P −1 but by a unitary matrix U and its transpose U T which is not necessarily its inverse. The "diagonalization" we refer to here is more properly called singular value decomposition. 25 Note that the kinetic term is precisely invariant under χ → U χ. The above makes it clear that while the domain walls support a TQFT for large fundamental mass m, no trace of a TQFT remains at small m, and D n domain walls disintegrate into D 1 domain walls. So while there is no phase transition in the bulk, a phase transition on the domain wall must occur at m/Λ ∼ 1. The conservative scenario is that of a single transition, although we cannot exclude multiple transitions on the domain wall. Below, we elaborate on some possible scenarios of this sort. This discussion is continuously connected to that of a single-flavor QCD in [4] (see also a related discussion in [46]), which is the limiting case |m adj | Λ and the adjoint fermions decouple. We illustrate the phase diagram in Figure 3.
Note that this behavior is also expected from the general lore of the effective field theory (EFT). The potential of (2.27) is V (ϕ) = mΛ 3 (cos N ϕ − 1). EFT description is robust as long as ∆V (ϕ) Λ 4 . This is always true for small fluctuations in any of the vacua, which can be trivially checked by expanding about ϕ = 2π N to find ∆V ∼ mΛ 3 (∆ϕ) 2 , where ∆ϕ are the fluctuations near the vacuum. Any fluctuations are small, i.e., ∆ϕ 1. Thus, we trivially find ∆V Λ 4 , even in the limit m ∼ Λ. Now, consider large-field excursions as we traverse a domain wall and go from one vacuum to another. In this case ∆ϕ ∼ 1, and thus, we find ∆V ∼ mΛ 3 . Then, for m Λ, we still find ∆V Λ 4 , and the effective field theory description is still robust; no rearrangement of degrees of freedom is needed on the wall to correct for anything. On the other hand, taking m ∼ Λ, we find that a large-field excursion causes ∆V ∼ Λ 4 . Now, the EFT description fails, and one expects some additional degrees of freedom on the domain walls to correct for the failure of the potential. Presumably, these degrees of freedom give rise to the TQFT or perhaps massless fermions in some intermediate phase (see below).
We can apply our general analysis above to the particular case of N even, where there is a time-reversal preserving domain wall D N/2 , as D n → D N −n under timereversal. In the limit of the large mass of the fundamental matter, the domain wall theory was conjectured to be the T -preserving N = 1 3d Super Yang-Mills [39]. When the mass of the fundamental matter is reduced in the bulk, it is natural to assume that the domain wall theory will be deformed by a massive fundamental multiplet. Such theories were discussed in [47], where different scenarios of IR phases were considered. In particular, in the large-N limit, it was argued that these 3d theories break the T -symmetry spontaneously. This is precisely what happens to the T -preserving domain wall D N/2 of the effective Goldstone theory (2.27). Such a domain wall would correspond to a shift ϕ → ϕ + π as the domain wall is traversed. The domain wall D N/2 breaks time-reversal invariance spontaneously, as ϕ can wind forward or backwards 26 . Now for N > 2 and even, the n = N/2 domain wall is unstable for small m. Still, one could make it stable by explicitly breaking the Z χ 2N chiral symmetry down to Z 4 by, for example, adding a term tr λ 4 term 27 , and conclude that such a domain wall breaks time-reversal symmetry spontaneously.
As our analysis was done for small fundamental fermion masses, it does not a priori exclude an intermediate domain-wall phase. To gain insight, let us replace the model given by (2.26) and (2.27) with a model that incorporates a full order parameter of the U(1) χ breaking, namely a complex scalar φ. The complex phase of φ can be identified with ϕ in (2.26) and (2.27). To reproduce the anomaly, we must again couple fermions with terms like 28 ∼ χ i χ j (φ * ) q i +q j where q i is the U(1) χ charge of the fermion χ i , and with some sort of a Mexican hat potential V (|φ|), roughly of height Λ 4 and width Λ . For the finite mass of the fundamental fermion m, we must deform the model with a term ∼ mφ N + c.c.. The potential of the scalar for m = 0 looks approximately as Figure 4 (a), while a small mass theory m Λ looks like Figure 4 (b). 26 Recall that ϕ is a pseudo-scalar so it reverses sign mod 2π under T . 27 This operator is irrelevant. However, the fact that this deformation is relevant in the IR effective theory means that such a deformation is actually dangerously irrelevant in the UV. However, we cannot take the continuum limit with this term. Alternatively, we can consider a deformation by adding a real scalar φ and coupling it to the adjoint fermion as follows iφ(tr λλ + c.c.). Such a theory will retain the Z χ 4 ⊂ Z χ 2N chiral symmetry. 28 We assume that q i + q j > 0. If this is not the case, the coupling can instead be written as At the origin φ = 0, there are massless fermions, which are gapped everywhere else. Importantly the N vacua are separated by a very shallow barrier, controlled by the dimensionless parameter m/Λ. This parameter controls the tension of the elementary domain wall and can be made arbitrarily small. Further, the domainwall D n with n > 1 will not be stable, as the minimum energy configuration would prefer to go through elementary domain walls rather than through the high peak of order Λ 4 in the middle. In particular, the T -preserving domain wall of even N theory will always prefer to surf through the shallow rim of the potential in this regime, thereby breaking T -symmetry. However, for m ∼ Λ one can imagine a potential like in Figure 4c) where the neighboring vacua are separated by the barrier of height Λ 4 , which is of the same order as the barrier in the middle. Hence, it becomes a subtle issue whether the T -preserving domain wall prefers a direct, Tpreserving route, where the mass of the fermions becomes zero, or the T -breaking route where the massless fermion point φ = 0 is avoided. The case N = 2, however, is different as a natural domain wall would go through the φ = 0 point as depicted in Figure 4 (d).
Nonetheless, it is important to note that when m ∼ Λ, we have no reason to believe the effective description in terms of the would-be U(1) χ order parameter, as this is reliable only for m Λ. In other words, it could be that before the scenario of Figure 4  Before we conclude this section, let us consider a model which indeed has a scenario described by Figure 4d) and is under full analytical control. To do this we will deform the SU(2) model with an adjoint Weyl and a fundamental Dirac fermion, by adding to it a scalar and introducing the Yukawa coupling exactly like we did in the second model described in § 1.1.
Let us label the bare mass of the scalar as m s in this model. Then as scalars are condensed we saw in § 1.1 that three free fermion phase develops χ 0 , χ 1 and χ 2 , with charges exactly consistent with our proposed all-N formulas in the m → 0 limit (see discussion below (2.24)). When m = 0 the transition from m 2 s Λ 2 to m 2 s −Λ 2 is changing from a U(1) χ broken to a composite fermion phase. We can hence study the theory around m 2 s = M 2 c -the critical mass-squared. Introducing the parameter Λ is described by a complex scalar φ and three fermions χ 0 , χ 1 , χ 2 with the following interactions with some couplings g 1 and g 2 . The above Lagrangian is valid for small |t| and m. As t is increased from negative to positive values a restoration of the U(1) χ symmetry ensues. However if t is negative, depending on whether the |t| m or |t| m the effective potential for the complex scalar m can look as in Figure 5. So as m is increased for fixed t, the domain wall undergoes a transition from a twice degenerate, T -broken phase to the composite fermion phase. Moreover, the effective theory still remains valid when m is large 29 in the sense that the order parameter becomes a real scalar φ with the Z 2 symmetry φ → −φ, which is a subgroup of Z χ 4 . Since there are still many residual 't Hooft anomalies involving this chiral symmetry, one expects that the qualitative effective model remains, allowing us to study the transition in the bulk by demoting the field φ to a real scalar and dropping the last term of (2.35). However here it is easy to see that the chiral-broken phase supports domain-wall fermions 30 . This is nicely consistent with the study of an analogous 3d theory in [47]. We summarise this model's bulk phase diagram and the domain wall phase diagram in Figure 6. We are unable to determine whether, in the scalar decoupling limit, the composite fermion phase on the domain wall persists. This is indicated by the question marks in the figure.

Comments on the large-N scaling
We will here briefly comment on the large N scaling of the effective Goldstone theory (2.27). To take the large N limit, we normalize the UV Lagrangian in a standard way where λ t is the 't Hooft coupling. The current associated with the chiral symmetry is given by where the extra factor of N in the last two terms are because the fields ψ andψ are charged with charge N under the U(1) χ . The current current correlator then has three pieces, which are schematically given by where J µ λ = trλσ µ λ, J µ ψ =ψσ µ ψ +ψσ µψ and A, B, C are some order one coefficients. We have that J µ ψ J ν ψ , J µ ψ J ν λ ∼ 1/N , J µ λ J ν λ ∼ 1. So despite the correlator in the first term above being suppressed by 1/N , the enhancing factor of N 4 causes the first term to dominate, so To reproduce this, we have to take that in the large N limit the coefficient of the kinetic term (∂ µ ϕ) 2 of the effective Goldstone theory needs to scale as N 3 . Let us give another, more heuristic, way to understand this result. Namely if we note that there are two fermion-bilinear order parameters of U(1) χ symmetry breaking tr λλ ∼ e iϕ = U λ and ψψ ∼ e iN ϕ = U ψ , we then expect that the effective Lagrangian is given by In other words, the order parameter U λ will have an a priori dominant kinetic terms, while the U ψ is expected to be 1/N suppressed because it couples to fundamental matter. The second term however is not sub-leading and is of order N 3 because the order parameter U ψ carries charge N , i.e. |∂ µ U ψ | = N 2 (∂ µ ϕ) 2 . So the second term above is dominant and is given by ∼ N 3 (∂ µ ϕ) 2 in the large N limit. It is perhaps not surprising that the fundamental contribution is important even at leading order, because the U(1) χ symmetry crucially depends on the presence of massless fundamentals. Indeed the way that importance shows up in the large N limit is through the dominance of the charge of the fundamentals under the U(1) χ . Now if one inserts the mass terms for the adjoint fields and the fundamental fields, we expect them to scale as N 2 and N respectively, i.e. the effective theory schematically becomes Notice that if we set the fundamental mass to zero, i.e. m = 0, then the pseudo-Goldstone has a mass 1/N allowing an analysis to be carried out for large N at any m adj . This is exactly what one expects when the adjoint fermion decouples. On the other hand setting m adj = 0, the pseudo-Goldstone mode gets an order one mass in the large N limit, and the analysis breaks down when m is of order Λ-the strong scale.

Theory with one fundamental scalar
Now we replace the fundamental fermion with a fundamental scalar φ. The action for the matter content is given by where V (φ) is the potential for the scalar field, which we can take to be of the form where we will not be concerned with interaction terms much, as long as they are there to stabilize the potential when m 2 < 0.

Symmetry and anomalies
Similar to the massive fundamental fermion case, the faithful global symmetry group is where the representation of each field under G Global is given in Table 5. Here, we again denote the global baryon symmetry as U(1) B ∼ = U(1) q /Z N , in terms of the quark symmetry U(1) q . The lowest charge of a gauge invariant operator under U(1) B is unity, while under U(1) q it is necessarily a multiple of N so that U(1) B is the faithful global symmetry. The (gauge non-invariant) field φ, however, is fractionally charged Just like in the fermionic case, there is a mixed anomaly between Z χ 2N and U(1) B such that the Z 2 subgroup of Z χ 2N remains anomaly-free. This anomaly can also be seen as follows. We turn on a fractional instanton color flux Q c ∈ Z/N . One also must turn on the fractional baryon-number flux to render the theory well-defined in the presence of scalars, which see both the color and baryon-number fluxes. The adjoint fermions, however, are uncharged under U (1) B . Thus, in the background of the fractional color flux, the partition function transforms as under a discrete chiral rotation. The phase is the above-mentioned mixed anomaly. The theory also exhibits a mixed Z χ 2N -gravitational anomaly. Both Z χ 2N -U(1) B and Z χ 2N -gravitational anomalies are identical to the anomalies of the theory with a massive fundamental fermion. This is expected since massive fermions and scalars carry the same global charges under U(1) B .

The IR phases
We already discussed the phase structure of the N = 2 case in § 1.1, so here we restrict ourselves to N > 2.
(I) When m 2 is positive and much larger than Λ 2 , the scalar decouples, and the theory is pure super Yang-Mills with N degenerate vacua.
(II) When m 2 is negative and large, φ will condense and acquire a vacuum expectation value 31 . Assuming that this happens at a scale above the strong scale of SU(N ), the condensation will higgs the gauge group SU(N ) down to SU (N − 1), under which the adjoint fermion of SU(N ) decomposes as In other words, the Higgs regime is effectively described by an SU(N − 1) gauge theory with one Weyl fermionλ in the adjoint of SU(N − 1), two Weyl fermions ψ,ψ in the fundamental and anti-fundamental of SU(N − 1), and one neutral Weyl fermion ν. It will be beneficial to discuss the deep Higgs regime m 2 → −∞ separately from the Higgs regime where m 2 is merely large and negative. We will see that finite Higgs VEV will induce dangerously irrelevant terms, which will change the IR physics.
In the deep Higgs regime with m 2 → −∞, the Higgs field fluctuation is suppressed together with the massive gauge bosons which now have infinite masses. Concretely, by gauge-fixing so that the VEV of φ takes the form the decomposition of the adjoint fermion λ is given by Note that mass terms forλλ,ψψ, and ν 2 are forbidden asλ, ψ,ψ and ν are charged under the original global Z χ 2N symmetry. Thus, the IR theory is fully equivalent to the SU(N − 1) gauge theory with a massless fundamental Dirac fermion and a massless adjoint Weyl fermion discussed in § 2.2, together with a decoupled Weyl fermion ν. The global symmetry enhances in the IR to This is an abuse of notation since this consideration is not gauge invariant. Instead, one should talk about fixing a gauge to be more precise.
where U(1) B is also identified with a combination of U(1) B and SU(N ) that leaves the Higgs VEV invariant. Under U(1) q , ψ andψ have charges +1 and −1, whileλ and ν are neutral. There are many anomalies as discussed in § 2.2. Thus, the effective theory of the deep Higgs regime is just a Goldstone theory of a spontaneously broken U(1) χ , along with a decoupled ν, i.e., Now, let us discuss the theory when m 2 is large and negative but finite. When −∞ < m 2 < −Λ 2 so that the scalar VEV is v Λ. Naively not much changes, as the masses for ψ,ψ, ν andλ are still forbidden because mass terms are not invariant under the Z N subgroup of the global Z χ 2N . However, higher fermi interactions will generally be induced, although suppressed by powers of 1/v. Since higher fermion terms are naively irrelevant, one may erroneously conclude that they can be neglected. However, since a condensate ofλλ ∼ e iϕ forms, such higher-fermionic terms become dangerously irrelevant and will change the reasoning. Consider, for example, the termλ 2N . Such a term is invariant under the Z χ 2N symmetry and is therefore allowed. This operator in the IR looks like trλ 2N ∝ e iN ϕ , and hence a cos(N ϕ) term will be induced. Further, since under the action by the generator of the Z χ 2N symmetry ϕ → ϕ + 2π/N , a mass term for ν is allowed in a form ννe −iϕ , and so the effective theory becomes 32

+ (other massive fermion terms) . (3.10)
The constants A and B are suppressed by some power of Λ/v. To estimate them, note that cos N ϕ has to come from at least 2N fermi interaction terms when the theory flows from SU(N ) to SU(N − 1) gauge theory, and so on dimensional grounds they must be 33 On the other hand, the νν term can come from the fermi interaction νν trλ †λ † + c.c.. By the same dimensional analysis argument, we need B ∼ Λ(Λ/v) 2 . This shows that the phase diagram of the theory is very similar to the one in 1 fundamental fermion case, except that now we have an additional ν fermion which is very light in the Higgs regime. Further, the domain-wall tension is exponentially small in N . We plot the phase diagram in Figure 7. 32 The "other massive fermion terms" are of the kind we discussed in the purely fermionic theory which is meant to saturate the various anomalies, which we do not bother writing. They have a mass of order Λ. 33 A ψ 2N term has mass dimension 3N , so it must be multiplied by 1/v 3N −4 to give the correct dimension 4 of the Lagrangian density. Let's now give a non-zero mass m adj to the adjoint fermion λ. The now-physical theta angle can be absorbed as a phase of the Majorana mass: m adj = |m adj |e iθ/N . For |m adj | Λ at a generic θ, there is no T -symmetry and after we integrate out the massive adjoint fermion the theory is an SU(N ) gauge theory with the fundamental Higgs field and a θ-angle. When m 2 Λ 2 , we can also integrate out the Higgs, landing on a pure SU(N ) Yang-Mills with the θ-angle. Instead, if we tune m 2 −Λ 2 so that the Higgs condenses, the theory flows to a pure SU(N − 1) gauge theory with the θ-angle. Since both SU(N ) and SU(N − 1) gauge theories are believed to be in the same phase for any θ-angle 34 , it is natural to assume that there is no bulk phase transition as the Higgs condenses.
In the opposite limit when |m adj | Λ, we can analyse the theory perturbatively around the theory (3.10). The adjoint mass term m adj tr λλ + c.c. descends to the adjoint and the Dirac mass term m adj trλλ + m adjψ ψ + c.c.. which induce the term proportional to −|m adj |Λ 3 cos (ϕ + θ/N ) in the effective action (3.10). Shifting ϕ → ϕ + θ/N , the potential for ϕ takes the form There is a proposal, however that SU(2) gauge theory may be massless at θ = π. See [3] and [48].
for some positive numerical constant c. The relative sign between the two terms in V (ϕ) is fixed by the fact that there is a single global minimum when |m adj | > 0 at θ = 0 as we know the theory to be trivially gapped. This is also true for other θ = π. At θ = π, however, the local minima of V (ϕ) for |m adj | are instead at ϕ = ±(2k + 1)π/N for an integer k. As we turn on a non-vanishing |m adj | Λ, most of the degeneracy is lifted apart from exactly 2 degenerate vacua at ϕ = ±π/N due to the symmetry of the cosine function. As the T -symmetry acts on ϕ by ϕ → −ϕ, this double-well potential corresponds to the phase where T is broken spontaneously. As |m adj | increases, the potential barrier between the 2 vacua gets less steep. It eventually becomes flat at at which point there is a second order phase transition to the trivially gapped phase, accompanied by a massless pseudo-scalar. Note that the value of |m adj | where a massless pseudo-scalar exists is exponentially small in N deep in the Higgs phase. Assembling the IR phases from different corners of the parameter space back together in one piece results in a phase diagram that likely looks like the one shown in Figure 8. Observe the similarities and differences to the phase diagram of SU(N ) QCD (f/adj) with one fundamental Dirac fermion in Figure 2: here, unlike in the fundamental Dirac version, the IR is always in the T -broken phase as |m adj | → ∞ at θ = π regardless of m 2 . Moreover, the massless Goldstone boson phase in the deep Higgs regime is always accompanied by a massless Weyl fermion ν, unlike in Figure 2 at m = 0 where there is no additional massless particle.
Like what we found in the fundamental fermion case, even though there is no phase transition in the bulk over a range of parameters where domain walls exist, there can be a phase transition on the domain walls themselves. When the adjoint mass vanishes, there is a phase transition on the domain wall joining adjacent Z 2N → Z 2 vacua. As m 2 → ∞, we again recover the N = 1 SYM, and the domain wall must also be decorated with a TQFT. On the other hand, as m 2 −Λ 2 , we see that the domain wall can be effectively described by a chiral Lagrangian, indicating that the domain wall theory is trivial. Thus, there must be a phase transition somewhere in the middle as we vary the parameter m 2 between the two extremes. Now, take the adjoint fermion to be so massive it decouples from the theory at θ = π, where T -symmetry is spontaneously broken. As m 2 → ∞, the theory becomes pure SU(N ) Yang-Mills at θ = π, whose T -breaking domain wall is equipped with the SU(N ) 1 Chern-Simons TQFT [3]. In the opposite limit, the scalar field condenses and higgses the gauge group down to the pure SU(N −1) Yang-Mills at θ = π, whose domain wall theory is now the SU(N − 1) 1 Chern-Simons TQFT, which is different from the SU(N ) 1 theory. Again, we have an indication that there must be a phase transition between these two limits. Together with the trivial domain wall phase close to the m adj = 0 axis, these 3 possible phases are shown tentatively in Figure 9.

Theory with multiple fundamental fermions
We consider SU(N ) Yang-Mills theory endowed with a single adjoint Weyl λ and N f fundamental Dirac fermions Ψ i , i = 1, . . . , N f , all with the same mass m. The Lagrangian for the matter sector is given by

Symmetry and anomalies
The massive case When m = 0, the faithfully acting symmetry group is The only addition here compared to the symmetry group of § 2.1 is the SU(N f ) factor. The matter fields transform under G Global in the representations given in Table 6 , where we decompose the Dirac fermion into 2 left-handed Weyl fermions ψ andψ in the fundamental and anti-fundamental representations of SU(N ), respectively. The extra quotient by Z N f arises because U = e 2πik/N f 1 N f , k = 0, 1, . . . , N f − 1, in the centre of SU(N f ) is the same as transforming Ψ i by e 2πik/N f ∈ U(1) B . It is also worth pointing out that, just like in the one-flavor case, the Z 2 subgroup of Z χ 2N acts identically as Z F 2 , allowing us to define the theory on a non-spin manifold that admits a Spin-Z χ 2N structure. The anomaly structure is very similar to that described in §2: there is again a mixed anomaly between Z χ 2N and the flavor symmetry U(N f )/Z N (the CFU anomaly). To see this for a general background gauge field for U(N f )/Z N , we first turn on background gauge fields A f for SU(N f ) and A q for U(1) q . However, since the bundle is for U (N f )/Z N and not SU(N f ) × U(1), these fields do not have their proper normalization. Instead, the discrete quotient imposes the constraints that where A 1 and A 2 are properly normalized U(1) background gauge fields, and are properly normalized U(N ) and U(N f ) background gauge fields, respectively. Under a Z χ 2N transformation λ → e 2πik/2N λ, the action effectively shifts by where f is the field strength of SU(N ) and we used the same derivation as in Eq.
(2.7) to obtain the second line. Since A 1 is properly normalized, F 1 ∧ F 1 /8π 2 is an integer on a spin manifold. Therefore, such a Z χ 2N transformation is anomalous unless k = N , that is, the Z 2 subgroup is non-anomalous.

The massless case
Without the mass term, the theory now possesses SU(N f ) L × SU(N f ) R flavor as well as U(1) q quark and U(1) χ axial symmetries (with appropriately modded common discrete centers). The action of the gauge and global symmetries on the fermion content is displayed in the following table. We choose the minimal charge assignments for U(1) q and U(1) χ .
This section explains the origin behind the mixed anomalies discussed in the previous sections. In doing so, we give the details that link the above derivations to the CFU anomaly computations of [18]. Table 7: Action of gauge and global symmetries in the multi-flavor massless SU(N ) QCD(f/adj) theory.
The theory possesses the following traditional 't Hooft anomalies: More constraining anomalies can be found by utilizing the faithful global group. To find the latter, one first needs to determine a subgroup in the center of that acts trivially on all the fields. Consider the transformation The condition that it acts trivially on all the fields ψ,ψ, and λ are : where n c are integers mod N , n L,R are integers mod N f , and α, β are U(1) phases. The conditions (4.8) ensure that the transition functions of the gauge and global symmetry bundles satisfy the cocycle conditions. The set of solutions to these conditions forms a subgroup of G that must be quotiented out from G to determine the faithful global symmetry group. We find that this subgroup is N f ) is the charge of λ under the global U(1) χ . This subgroup is generated by Thus, the faithful global symmetry of the theory is (4.10) To fully study the anomalies we note that the symmetries allow us to define a Spin-G Global := Spin × G Global /Z F 2 structure on our manifold and not just a Spin structure. This is possible when there is a Z 2 subgroup of G Global that acts identically to  When both q and Q are odd, we can take α = π, β = n L = n R = 0 as a solution. When Q is odd and q is even, we can take α = β = π and n L = n R = 0 as a solution. Lastly, when Q is even and q is odd, we can take n L = 0, α = π/Q, β = π + qπ/Q, and n R = N as a solution. Thus, it is possible to put the theory on a non-spin manifold that admits a Spin-G Global structure. Since the quotient Z F 2 only involves U(1) symmetries (either U(1) q × U(1) χ or with a Cartan of SU(N f ) R ), we can always turn on only the Spin c structure. Since all orientable manifolds admit a Spin c structure, we can define our theory on all orientable manifolds, including e.g., CP 2 (unlike in the massive case).
Let us now analyze the anomalies in more detail. For this purpose, we turn on the background field strengths F L , F R , F q , F χ , for SU(N f ) L , SU(N f ) R , U(1) q , and U(1) χ , respectively. Then, the 6d anomaly polynomial for our theory is given by , (4.13) and p 1 ≡ − 1 8π trR ∧ R is the first Pontryagin class and R is the curvature 2-form. The terms inside the square bracket capture 't Hooft anomalies of G Global while the last term is the mixed [U(1) χ ]-gravitational anomaly. Through the anomaly descent equations, we can see that, under a transformation e iα ∈ U(1) χ , the partition function changes as . (4.14) Note that, because of the discrete quotient in G global , various instanton numbers that appear above can be fractional. These are dubbed the color-flavor-U (1) (CFU) fluxes in [18,28]. More precisely, we can write F q and F χ in terms of properly normalized U(1) field strengths F 1 , F 2 , F χ as Then, the fractional parts of the instanton numbers for SU(N f ) L and SU(N f ) R are related to the fluxes F 2 , F χ by 16) and F 2 2 , F 2 ∧ F χ , F 2 χ ∈ 8π 2 Z. Equivalently, one can give the fractional parts of these instantons in terms of the obstruction to lifting a G Global bundle to a SU(N f ) L × SU(N f ) R × U(1) B × U(1) χ bundle, which we will call the product bundle. One specifies G Global bundle by specifying P SU(N f ) L , P SU(N f ) R , U(1) B /Z qQ , and U(1) χ /Z Q bundles. The obstructions to lifting these bundles to the product bundle are given by the second "Stiefel-Whitney classes" , and w (Q) 2 are directly related to the U(1) fluxes by On the other hand, w (L) 2 and w (R) 2 , which obstruct lifting P SU(N f ) L,R bundles to SU(N f ) L,R bundles are related to the fractional instanton numbers by [49] where P is the Pontryagin square operation. When N f is even, it is defined to be the cohomology operation [50] P : The image is even when M is spin, so P(w 2 (L, R))/2 is a well-defined cohomology class in H 4 (M ; Z N f ). When N f is odd, we define P to be simply the cup product, following Ref. [51]. Division by 2 makes sense because it is invertible in Z N f when N f is odd. The structure of the quotient given by the generators in (4.9) relates w (L,R) 2 to the other Stiefel-Whitney classes by (4.20) These are well-defined modulo N f : as w mod N f . Substituting these relations back into Eq. (4.19), we recover the expression for the fractional instantons in Eq. (4.16).
To understand how these fractional instanton numbers could alter our anomalies, it is instructive to consider concrete examples. Let us define the topological charges associated with SU(N ), SU(N f ) L,R , U(1) B , and U(1) χ , respectively, by (4.21) One can also calculate the Dirac indices in these center fluxes: which are always integers in a consistent background: there is a one-to-one correspondence between the solutions of (4.8) and the integrality of the Dirac indices.
The finest fractional charges are reached when we put on the background fields with lowest, non-trivial,fluxes for F 1 , F 2 , and F χ . To achieve this, let's consider the theory on the product manifold S 2 × S 2 . Then, we can take F 1 , F 2 , and F χ to be a sum of charge-1 monopoles centered on each sphere. More precisely, we take the first Chern class for F 1 , F 2 , and F χ to have the form c 1 = α + β where α, β are the two generators of H 2 (S 2 × S 2 ; Z). In this configuration, the various topological charges (CFU fluxes) are given by where n 1 , n 2 , k c , k L,R are integers.
Let Z[Â f ] be the partition function in the background of the vector-like flavor symmetry U(N f )/Z N , which in general activates the CFU fluxes. Then, under a U(1) χ rotation we have that and it is easy to see that the part that multiplies Q c cancels out. This should be expected since the theory is not endowed with a genuine Z [1] N 1-form symmetry, thanks to the fundamentals. Thus, what we find is an anomaly of mixed type between U(1) χ , U(1) B , and SU(N f ). The anomaly is exactly the same one we obtain from (4.14) after using the properly normalized U(1) field strengths F 1 , F 2 , F χ defined via (4.15). As a special case, we can consider fractional fluxes of F χ turned off and set n 1 = n 2 = 0: The corresponding Dirac indices are Then, under a U(1) χ rotation we find and CFU anomaly is (4.29) This is exactly the A ∧ F B ∧ F B (the mixed U(1) χ -U(1) B ) anomaly in (2.19) when we set N f = 1.
Finally, we briefly discuss the anomalies when we put the theory on a non-spin manifold. For concreteness, we consider the theory on CP 2 and assume both q and Q are odd. In this case, the topological charges are given by [52] Q χ = 1 2 While the Dirac indices are which are always integers when both q and Q are odd. Then, the anomaly on CP 2 reads Similarly, one can work out the anomalies when q and Q are even or mixed even/odd.

The IR phases
In the infinite mass limit, we can again integrate out the Dirac fermions, leaving us with N = 1 SYM with gauge group SU(N ). As discussed in § 2.3, there are N degenerate vacua and domain walls connecting them. At the massless point, the Z χ 2N discrete chiral symmetry enhances to U(1) χ . All anomalies are now given in terms of the anomaly polynomial. Saturating the anomalies can be achieved in the IR in one of three ways: 1. Composite massless fermions charged under the global symmetry.
The first choice can be ruled out by Weingarten's theorem [34] because the measure of the theory is positive definite (see Appendix A). Let's now discuss the second option, where anomalies are saturated by breaking the global symmetries Goldstones and a U(1) vortex. The condensateψψ is charged under the full SU(N f ) L × SU(N f ) R × U(1) χ , and breaks the group down to SU(N f ) V × Z 2q . The Z 2q phase is the unbroken subgroup of U(1) χ underψψ. However, the unbroken phase is anomalous, as can be checked using [U(1) χ ] [gravity] and the CFU anomalies given by (4.27) by setting α = 2π 2q . To avoid this problem, another condensate has to form in order to break U(1) χ to a non-anomalous group. The minimal choice is λλ, which transforms under U(1) χ as: λλ U(1)χ − −− → e −i2Qα λλ, and thus, the formation of λλ breaks U(1) χ down to Z 2Q . The combined condensates λλ andψψ break U(1) χ down to Z 2gcd(q,Q) = Z 2 . it is easy to see that this Z 2 has no mixed anomaly with SU(N f ) V nor U(1) B by setting α = π and k L = k R in (4.27). Moreover, since Ω Spin 5 (BZ 2 ) = 0 [53], there are no global anomalies in Z 2 itself. So for sufficiently small number or flavors N f < N * f we expect that the theory flows to a Goldstone phase corresponding to N f Goldstone bosons.
Another scenario is that the theory flows to a phase that preserves all the global symmetries, e.g., a conformal window. One may wonder which scenario is preferred. The answer to this question comes from comparing the number of effective massless degrees of freedom (DOF) between the UV theory and the SSB scenario [54]. The, effective degrees of freedom A of n B massless real scalars and n f massless Weyl fermions are given in terms of the free energy density F as (we turn on a small temperature T Λ, where Λ is the strong scale) It is expected that the phase with lower free energy, i.e., a smaller number of DOF, is preferred. Define the difference between the number of DOF in the UV and the SSB scenario by (4.34) If ∆A > 0, the SSB phase is disfavored, and the theory should flow to the conformal phase in the IR, provided that it is asymptotically free. In Figure 10, we plot both ∆A and the β-function versus (N, N f ). The phase with ∆A > 0 is displayed in green, while the asymptotically-free region is in blue. There is only a small intersection window between the two regions. The intersection window, however, lies completely inside the conformal window, as is evident from computing the 2-loop Bank-Zacks fixed point. The latter region is displayed in orange.  Figure 10: The horizontal axis is the number of colors N , while the vertical axis is the number of flavors N f . We display the asymptotically free region, β 0 > 0 in blue, while the region that supports a Bank-Zack fixed point, with α ≡ g 2 * 4π < 0.5 at the fixed point, is displayed in orange. The phase with ∆A > 0 is displayed in green. Most of this region lies outside the asymptotically-free region. There is only a small window where the regions with ∆A > 0 and α ≡ g 2 4π < 0.5 intersect.
In order to make the last point more quantitative, we study the theory in the Veneziano limit. Thus, we take both N and N f infinite, keeping the ratio R = N f /N finite. The β-function is given by (see Appendix C) and in the Veneziano limit, we have The theory is asymptotically free for R < 4.5, while it develops a Banks-Zaks fixed point at , (4.37) provided that R > 1.125. Notice that the fixed point is well under control as N → ∞. Thus, the β-function analysis predicts that the conformal window lies in the range 1.125 R 4.5. At finite but large N , we should expect the conformal window to be in the range where the lower bound R * is 1.125 to leading order in large N . Let us compare this result with what we get from the constraint ∆A > 0. In the Veneziano limit, the conformal behavior is favored when 38) which is solved by requiring R 4.36. This implies that the theory is in the conformal window when R is in the range 4.36 R 4.5. This result is consistent with the β-function analysis but gives a more stringent constraint on the conformal window. On the other hand, when ∆A < 0, the inequality only tells us that the SSB phase is favored compared to the weakly coupled conformal phase (whose DOF are the same as the UV theory), but it does not exclude the strongly interacting conformal phase, whose DOF are not as easily computed. When N is large, R * should still remain close to the value computed using the β-function in the Veneziano limit, and so should be lower than 4.36. Thus, when R is in the range R * ≤ R ≤ 4.36, the IR phase could be an interacting CFT.
Finally, we searched for isolated points at finite N and N f with ∆A > 0 without a trustworthy IR fixed point. Such isolated points indicate that the theory prefers to flow to the conformal window, even without a trustworthy fixed point. To be conservative, we define a trusted IR fixed point as the one with g 2 * /4π < 0.05 at both two-and three-loops. We found no evidence of such isolated points in the N -N f plane.
To summarize, even though ∆A calculations put a more stringent constraint on the conformal window, it is consistent with the perturbative calculations from the Bank-Zacks fixed point. This is summarized in Figure 10.

Fermion masses and the phase diagrams
When N f is low enough that chiral symmetry breaking occurs, we can broaden the scope of our analysis by turning on the mass of the adjoint fermion in addition to the fundamental fermions' mass. When m adj Λ, it can be integrated out so that we are left with SU(N ) QCD with N f fundamental Dirac fermions. The IR phase structure of this theory has been analyzed in Ref. [4], which we briefly recount here. At non-zero m, the theory has no time-reversal symmetry at all theta-angle except at θ = 0, π and flows to the trivially gapped phase in the IR. At θ = 0, the Tsymmetry is unbroken and the theory is still in the trivially gapped phase. On the other hand, the T -symmetry is spontaneously broken at θ = π, resulting in a phase with 2 inequivalent vacua. This persists for all value of |m| down to m = 0 where the flavor symmetry SU(N f ) V enhances to SU(N f ) L × SU(N f ) R . This enhanced chiral symmetry breaks spontaneously down to SU(N f ), leaving us with non-Abelian Nambu-Goldstone bosons as the theory flows to the IR.
Nothing much can be said quantitatively in the intermediate regime, except when the masses are small, m, m adj Λ, where we can analyze the IR theory in more detail through the chiral Lagrangian. We start by writing down the chiral Lagrangian at the massless point. As previously discussed, both tr λλ and ψψ condense, inducing the spontaneous symmetry breaking pattern 35 The target space of the chiral Lagrangian is then the coset space which we parametrize by the pair with the identification e iϕ , U ∼ e 2πi/Q e iϕ , e 2πiN/N f U (4.41) enforcing the Z Q discrete quotient. Another way to see that we need this Z Q identification is by noting that such a Z Q transformation leaves the condensates tr λλ ∼ e iQϕ , ψψ ∼ e −iqϕ U (4. 42) invariant. Note also that our parametrization implies that ϕ has charge 2 under the original chiral symmetry U(1) χ like the one-flavor case. The lowest derivative terms are given by where f 2 π and f 2 ϕ :=f 2 ϕ + N f q 2 f 2 π are two different 'pion decay constants', scaling with N as f 2 π ∼ N 2 and f 2 ϕ ∼ N 3 . Additional terms (including the WZW term) are needed to match the 't Hooft anomalies of the UV theory.
Turning on the positive masses m adj for the adjoint fermion λ and m for the fundamental fermions ψ,ψ, as well as the SU(N ) theta-angle θ, induces a potential on M 0 : For the purpose of finding the vacua, we can focus on U of the form U = e 2πik/N f 1 N f so that the symmetry SU(N f ) V is preserved. The potential now reads To complete the chiral Lagrangian, additional terms (including the WZW term) are needed to match the 't Hooft anomalies of the UV theory, but these will not be necessary for what we want to discuss next.
m adj = 0 When we tune the adjoint mass to zero, the theta angle becomes unphysical as it can be rotated away by an anomalous chiral rotation. The potential for ϕ thus reduces to 46) with obvious minima at ϕ = 2πk/QN . However, the number of distinct vacua are smaller than QN due to the Z Q quotient. Denote the k th vacuum by Then the equivalence (4.41) implies that Thus, there are only N distinct vacua. See Figure 11 for the visualization of a specific case.
There are domain walls connecting neighbouring vacua |k and |k + 1 . The domain wall configurations cannot preserve the full global symmetry of each vacuum. Following [4], one can show that the non-Abelian symmetry SU(N f ) V is necessarily broken down to S[U(1) × U(N f − 1)] by domain wall configurations, resulting in a non-linear σ-model on the domain wall with the target space coupled to a topological term induced by the WZW term in the bulk. We can also conclude that there must be a phase transition on the domain wall as we crank up the fundamental mass m, just like in the one-flavor case, because in the large mass limit, the domain-wall theory is a TQFT with no massless degrees of freedom. m = 0 When we set m = 0 instead of m adj , the potential becomes independent of U . The potential has its minima at ϕ = 2πk/Q for any integer k, and any U ∈ SU(N f ). The vacuum manifold M 0 = ( U(1)χ Therefore, when m = 0 and 0 < m adj Λ, only the non-Abelian NGBs remain massless. The Abelian NGB ϕ becomes massive. m, m adj > 0 When both masses are non-zero, we need to look at the full potential (4.45): There is a Z 2 time-reversal symmetry at θ = 0, π, which is broken explicitly at other values of θ. The symmetry transformation is given by at θ = 0, and by at θ = π. Most of the vacuum degeneracy is lifted by the non-zero M term in the potential, due to the fact that Q and q are coprime. The new vacua are those closest to ϕ = 0. At θ = 0, there is a single vacuum at ϕ = 0, k = 0, invariant under the Z 2 time-reversal symmetry. At θ = π, however, there are two degenerate vacua related to each other by the time-reversal symmetry (see Figure 12). It is clear that there can be no second order phase transition to the trivially gapped phase, unlike what we saw earlier in the one-flavor cases for both the Dirac fermion and the scalar fields. There is a CP N f −1 non-linear sigma-model on the domain wall connecting the two vacua just like in the case with m adj = 0 consistent with the proposal [4] for QCD. To summarise, when the adjoint fermion is massless, we can combine our results from § § 2, 4 together to obtain a phase diagram in terms of the fundamental fermions' mass m and , the ratio of the number of flavors to the number of colors R = N f /N . The phase diagram is shown in Figure 13.
On the other hand, when N f is fixed such that there is chiral symmetry breaking, we can vary the adjoint mass m adj and obtain the phase diagram shown in as shown in Figure 14 by piecing together various limits explored earlier. When all masses vanish, there are both a massless Abelian Nambu-Goldstone boson as well as non-Abelian Nambu-Goldstone bosons; only the non-Abelian ones remain when we turn on the adjoint mass. Contrast this with the single fundamental fermion case in Figure 2. There, the existence of an Abelian Nambu-Goldstone boson persists for all value of the adjoint fermion mass, whereas in the multi-flavor case, it only appears when all fermions are massless. There are no phase transition in the bulk as we dial the mass m down to zero, but there are phase transitions on the domain walls, just like in

Theory with multiple fundamental scalars
To cap off our analysis, let us turn to multiple scalar flavors scenario in this section. Even though the anomaly story goes much the same way as before, we will see that the IR behaves qualitatively differently as we increase the number of the scalar fields.
The action for the matter fields reads with the potential We give the same mass to all N b scalar fields, just like what we did in § 4, to preserve as much global symmetry as possible. Terms with larger power in φ are also assumed to preserve the maximal symmetry. Figure 14: The T -invariant slice of the phase diagram for QCD (f/adj) with multiple fundamental fermion flavors. Note that when the fundamental fermions' mass vanishes, the theta-angle is not physical. We choose to represent the adjoint mass along the negative mass axis to emphasize that it is the end point of the Z 2 broken phase.

Symmetry and anomalies
The global symmetry group is under which the matter content transforms in the representations given by Table 8.
There is a mixed anomaly between U(N f )/Z N and Z χ 2N , exact as explained in §4.1.

The IR phases
When the mass parameter m 2 > 0, the scalar fields do not condense. At the scale below the mass scale, we can integrate them out and again obtain the same phase as the N = 1 SU(N ) SYM: there are N distinct vacua and there are domain walls connecting them. When m 2 < 0, we need to include the quartic terms in the potential for stability. Let us combine the scalar fields into an N × N b matrix Φ, which transform under the SU(N ) gauge group and the SU(N b ) global symmetry group as Then the most general potential invariant under the gauge and the global symmetry group is and we require κ 1 + N b κ 2 > 0 for stability. By adding appropriate constant terms to this potential, we can complete the square and write the potential as with v 2 = |m| 2 /(κ 1 + N b κ 2 ). Φ must now acquire vacuum expectation value to minimize the potential, which is achieved by the configuration Φ † Φ = v 2 1 N b . In this Higgs regime, the IR phases are sensitive to the number of scalar flavors N b . We will now consider each different scenario in turn.
where U 1 and The transformed VEV is We see that the transformation leaves the VEV invariant if and only if we take U 1 = V † , and U 2 = e iθŨ , where the phase θ is determined by V (because we need det U 1 det U 2 = 1), but are free to chooseŨ to be any SU(N − N b ) matrix. We can clearly see that the gauge group SU(N ) is higgsed down to SU (N − N b ), while the rest of the gauge group is locked with the flavor symmetry U(N b ), resulting in the color-flavor locked U(N b ) cf global symmetry that survives in the Higgs regime. The discrete chiral symmetry Z χ 2N does not act on the scalar fields, so it remains intact in this phase.
The SU(N ) adjoint fermion λ decomposes into an SU(N − N b ) adjoint fermioñ λ, N b Weyl fermions in the fundamental representation of SU (N − N b ), ψ, N b Weyl fermions in the anti-fundamental representation,ψ, a neutral fermions in the adjoint representation of U(N b ), η, and one neutral Weyl fermion ν. The matter fields transform under the IR gauge group SU(N − N b ) and the IR global symmetry in the representations given in Table 9. Again, even though the free massless fermion matter content has an enhanced global symmetry, as discussed in §3.1, there are irrelevant terms in the Lagrangian that reduce the symmetry down to the one we have in the UV. The enhancement that lifts Z χ 2N to a continuous chiral symmetry happens only at v = ∞. The difference in the discrete quotient between G Global IR in Eq. (5.9) and G Global in the UV is not a cause for concern. It simply reflects the fact that we assign the U(1) B charge ±1 to the fundamental and anti-fundamental fermions ψ,ψ. Note that the flavor symmetry that acts on the scalar fields in the UV now acts on the fermions in the IR through color-flavor locking, as we have already seen in the one-flavor scalar case. We learn about the IR dynamics by looking back at the dynamics of the fermionic theory that we studied in § 4, because it is the theory that emerges in the intermediate region, barring a few extra Weyl fermions neutral under the gauge group that only couple to the rest via higher-order terms. Thus, we expect to have domain walls connecting N vacua in the IR. The fermions that transform non-trivially under the SU(N − N b ) gauge group are gapped out by the gauge dynamics, while the neutral fermions are gapped out by the interaction with the would-be Goldstone boson for the U(1) chiral symmetry.
In the deep Higgs regime where v Λ, there are two options for the IR phases, depending on the ratio between the number of flavors and the number of colors. There exists a critical point N b /N = R b * , below which we have chiral symmetry breaking, and above which the phase enters a conformal window. Here we assume that N b /N < 18, always in a range that the theory is asymptotically free. The value of R b * cannot be ascertained in the generic case due to the strong dynamics involved. In the Veneziano limit, however, the Banks-Zaks computation can be trusted, and we can estimate R b * to be in the adjoint representation, the fundamental representation, and the anti-fundamental representation of U(N b ), respectively. All of them have unit charge under the discrete chiral symmetry Z χ 2N . We can dress these fermions with scalars to form gaugeinvariant composites, as we did in (1.2). The special case N = 2 was considered as a warmup exercise in Section 1.1. Here, the global symmetry U(1) B is enhanced to SO(3) custodial symmetry; see Footnote 4.
When the number of flavors N b is equal to the number of colors N , the gauge group is completely higgsed just like in the case when N b is one less than N . However, there is not enough room in the gauge group SU(N ) to fully preserve U(N b ) = U(N ) global symmetry through color-flavor locking when Φ = 0. The U(1) baryon symmetry must now be broken spontaneously by the VEV scalar fields, while the non-abelian flavor symmetry SU(N b ) remains unbroken. The IR phase consists of one U(1) Goldstone boson, as well as a massless composite Weyl fermion in the adjoint representation of the global symmetry SU(N b ).
Things get more complicated when N b > N . We see from the SVD that the equation still minimizes the potential. The gauge group is still fully higssed. Moreover, the global symmetry now breaks spontaneously to SU(N ) cf × U(N b − N ), which can be shown by the same argument around Eq. (5.8) but in the opposite direction. In the IR, there are Goldstone bosons described by the coset space apart from the composite fermions that we have previously. The 't Hooft anomalies in the UV can be matched by the composite fermions and the WZW term in the effective Lagrangian.
To summarise, there are many IR phases of the theory as the number N b of the fundamental scalars and the mass squared m 2 of the scalars varies. The ultraviolet (UV) theory has Z χ 2N symmetry that acts on the massless adjoint. In addition, baryon-number U(1) B and flavor SU(N b ) symmetries 37 act on the scalars. The theory admits mixed anomalies between Z χ 2N and U(1) B as well as between Z χ 2N and gravity. For m 2 > 0, we can integrate out the scalars ending up with N = 1 super Yang-Mills theory. When m 2 < 0, we need to distinguish between different scenarios depending on N and N b . (A) N b < N , the gauge group is higgsed down to SU(N − N b ). In the IR, the theory enjoys emergent continuous symmetries. Assuming that the Higgs vev is not much larger than Λ, the continuous symmetries are explicitly broken down to Z χ 2N by dangerously irrelevant operators. The IR theory breaks the Z χ 2N chiral symmetry spontaneously, leading to N vacua and domain walls. (B) N b = N − 1, the gauge group is fully higgsed, in which case composite free fermions match the anomalies. These are the UV fermions dressed by scalars in a fashion that preserves gauge invariance. (C) N b = N , the gauge group is fully higgsed, the flavor SU(N b ) is intact, and U(1) B is spontaneously broken. The IR phase contains one massless adjoint fermion in the global flavor group and one Goldstone boson. (D) N b > N , the gauge group is fully higgsed, and the continuous symmetry is broken, leading to many Goldstones. These various cases are neatly summarised in a phase diagram, shown in Fig. 16 below.

Conclusions and future prospects
In this work, we analyzed the SU(N ) gauge theory with one massless adjoint fermion and massive matter, either bosonic or fermionic. The special case of one fundamental flavor of varying mass was analyzed in detail. The abundance of 't Hooft anomalies involving discrete chiral symmetry Z χ 2N , combined with Vafa-Witten-Weingarten theorems, restricts the bulk phase quite strongly, which results in spontaneously broken discrete chiral symmetry for any mass of the fundamental fermion or boson, leaving N degenerate vacua in the bulk. Further, the bosonic theory is related to fermionic theory in the sense that when the boson condenses, the theory is higgsed down to SU(N − 1) with one fundamental fermion. So most conclusions about the theory with fundamental bosons can be drawn by studying the fermionic theory.
The massless fundamental fermion limit is particularly interesting, where the discrete chiral symmetry enhances to U(1) χ , and the domain walls melt into Goldstone bosons. Studying the small fundamental mass regime can be done systematically by perturbing the Goldstone theory. In particular, we studied the domain walls between the N vacua and, with the exception of the one T -preserving domain wall, found them all to be trivial. On the other hand, in the opposite limit of infinite fundamental fermion mass, the theory becomes Super Yang-Mills, of which many things are known. The domain walls of Super Yang-Mills are conjectured to hold a TQFT. As the fundamental fermion mass is dialed from small to large, no bulk phase transition occurs, but our analysis shows that a transition must occur on the domain wall. Further, we conjectured that the domain wall theory is IR dual to the corresponding SU(N ) gauge theory with both fundamental and adjoint matter in 3d. Of particular interest is the T -preserving domain wall, which exists if N is even. This domain wall cannot be made trivial as it carries a U(1) B − T mixed anomaly. The corresponding 3d theories were studied in [47], where it was proposed that anomalies can be saturated either by composite fermions or spontaneous T -breaking, with large N arguments favoring the latter. The 4d domain wall analysis indicates that T breaking is preferred, at least for a small enough mass of the fundamentals. We, however, speculate that there may be a composite fermion phase on the domain wall for some intermediate mass of the fundamentals for N = 2. We also discuss the decoupling limit of adjoints which results in the usual QCD.
We generalized the one fundamental flavor case to multi-flavors and discussed the bounds on the conformal window from the a-theorem and the Banks-Zacks 2-loop fixed point. For a sufficiently low number of flavors, we analyze the chiral Lagrangian and map out the phase diagram in the bulk and on the domain wall. As with one flavor, we discuss the decoupling limit of adjoints.
We end this section by discussing some future prospects. The conjectures about the bulk phase analyticity can be tested on the lattice. More interesting would be to study domain walls on the lattice, or the corresponding 3d gauge theories. The lattice studies of domain walls would require using twisted boundary conditions or spatially varying θ-term, which generically introduce a complex action problem that hinders numerical simulations. Studies of corresponding 3d theories directly also generically require bare Chern-Simons terms and complex fermionic measures, which again hinders lattice studies. Another interesting approach is to study soft SUSY deformations of super QCD setup, and see carefully what happens on the domain walls as supersymmetry is broken.
where D = σ µ D µ with σ µ = (I, τ 1 , τ 2 , τ 3 ) , (A. 2) and τ i are the Pauli matrices. We will take D µ to be the covariant derivative in a real representation of some gauge group, so that D * µ = D µ . We can write the complex Weyl fermion as a real fermion by writing λ = λ 1 +iλ 2 . Then we have that Then we have that λ 1 iσ µ D µ λ 1 and λ 2 iσ µ D µ λ 2 do not get a contribution from the anti-symmetric σ 2 matrix, while −λ 1 σ µ D µ λ 2 and λ 2 σ µ D µ λ 1 do not get a contribution from the symmetric matrices I, σ 1 , σ 3 . Now organize λ 1 and λ 2 into a column vector Ψ = λ 1 λ 2 . We can write the above action as Now let us set Notice that the matrix is real and anti-symmetric. It is also unitary because C † = C −1 . We want to set Γ µ = Cγ µ , so that γ µ = C −1 Γ µ . We then have that Note that all the Gamma-matrices are purely real. One can also check that γ µ as defined above satisfy the Clifford algebra where / D = γ µ D µ is the Dirac operator, and D µ is a covariant derivative D µ = ∂ µ +A µ , where A µ is the gauge field in the real representation of some group G, i.e. A * µ = A µ . In Euclidean space i / D is Hermitian, and so its eigenvalues are real. Let ψ n be the eigenfunctions of i / D with eigenvalues λ n . Then we can decompose where α n andᾱ n are independent Grassmann numbers, and where 38 d 4 x ψ † n ψ m = δ nm . We have that the action is given by Notice that because { / D, γ 5 } = 0 we have that for every eigenstate ψ n with eigenvalue λ n there exists an eigenstate γ 5 ψ n with the eigenvalue −λ n . So we can rewrite (assuming no λ n = 0) n (iλ n ) = so that the weight is positive definite. In addition we will see that each eigenvalue λ n is twice degenerate because it forms a Kramers doublet. Now let us move to Majorana fermions. In this caseΨ = ψ T C where C is an unitary, anti-symmetric matrix with the property Cγ µ C −1 = −(γ µ ) T .
(A.18) 38 If there are degeneracies λ n = λ m for n = m we can still chose that degenerate eigenstates are orthogonal. 39 the factor of i is there by convention, and is just an overall normalization. In this convention the weight is always positive. Now notice that the Dirac operator i / D has a degeneracy, because if ψ n has an eigenvalue λ n , then C −1 ψ * n has the same eigenvalue. Indeed, since (γ µ ) † = γ µ , we have Moreover C −1 ψ * n is orthogonal to ψ n by the anti-symmetry of C, i.e.
One can also see this as a Kramers degeneracy [42]. Indeed if K is a complex conjugation operator, we define T = C −1 K an operator which commutes with i / D. Now On the other hand if n = m then we see that the expression d 4 x(ψ i n ) T Cψ j n is anti-symmetric in i and j. We use a natural normalization d 4 x(ψ i n ) T Cψ j n = ij . (A.26) 40 We will assume that the only degeneracy in the spectrum is the Kramers degeneracy so that where the product over λ n is only over one of the Kramers doublet eigenvalue. The above is manifestly positive.

B Spectral flow
Consider a first-order differential operator where A(τ ) and I are an N × N Hermitian and identity matrices respectively. We want to look for the zero modes of the above operator. We solve the differential equation so M mn , commuting with A D must be diagonal unless A D has exact degeneracies. Let us assume that this is the case. Then the equation (B.6) implies that if we start with c n (0) not equal to zero for only some n and zero for others, it will stay that way. The diagonal matrix M is just the Barry phase of individual eigenstates. Now notice that only c n (τ ) for which λ n is positive for τ → ∞ and negative for τ → −∞ can be kept if we want normalizable ψ(τ ). Hence we conclude that the operator D has as many zero modes as the net spectral flow. If operator A(τ ) still has some degeneracies, the story is similar because we can always diagonalize M in the subspace of the degeneracies without affecting the discussion.

C β-function
The 3-loop β function for n R Weyl fermions in representation R of SU(N ) Yang-Mills theory is given by (see [55,56]) where G denotes the adjoint representation. The quadratic Casimir of representation R, C 2 (R), is t a R t a R = C 2 (R)1 R , (C.2) and C 2 (G) is the quadratic Casimir of the adjoint representation. T R is the Dynkin index, which is defined via tr t a R t b R = T R δ ab .
where dim R is the dimension of R. In particular, using the convention T R = 1 for the fundamental representation R = , we have C 2 (G) = 2N , dim G = N 2 − 1.