Time-Reversal Breaking in QCD$_4$, Walls, and Dualities in 2+1 Dimensions

We study $SU(N)$ Quantum Chromodynamics (QCD) in 3+1 dimensions with $N_f$ degenerate fundamental quarks with mass $m$ and a $\theta$-parameter. For generic $m$ and $\theta$ the theory has a single gapped vacuum. However, as $\theta$ is varied through $\theta=\pi$ for large $m$ there is a first order transition. For $N_f=1$ the first order transition line ends at a point with a massless $\eta'$ particle (for all $N$) and for $N_f>1$ the first order transition ends at $m=0$, where, depending on the value of $N_f$, the IR theory has free Nambu-Goldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the $4d$ bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different $3d$ phases. These turn out to be the phases of the recently studied $3d$ Chern-Simons matter theories, thus relating the dynamics of QCD$_4$ and QCD$_3$, and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless $4d$ theory has an $SU(N_f)$ sigma model, the domain wall theory at low (nonzero) mass supports a $3d$ massless $CP^{N_f-1}$ nonlinear $\sigma$-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.


Introduction
As is well known, the dynamics of 4d QCD with massless quarks depends on the number of flavors N f and colors N . For N f ≥ 11 2 N it is IR free. It is meaningful only as an effective IR theory and its IR dynamics is rather simple. For N CF T ≤ N f < 11 2 N with some yet unknown N CF T (N ) the theory flows to a nontrivial fixed point. For 1 < N f < N CF T it breaks its chiral symmetry leading at low energies to a nonlinear sigma model with target space SU (N f ). For N f = 1 it is gapped with a unique vacuum. And for N f = 0 the theory has an additional parameter, θ. For generic θ it is gapped with a unique vacuum, but as we vary θ through π we cross a first order phase associated with the spontaneously broken time-reversal symmetry at that point. The vacuum is therefore doubly degenerate at θ = π.
This picture has not been rigorously derived. In fact, although the phase transition at θ = π can be derived at large N [1][2][3][4][5] and was argued to exist at finite N using anomalies [6], it could be that at small values of N the picture is different [6]. Also, it might be that for some N and N f there are additional more exotic phases. Here we will ignore these possibilities.
One purpose of this note is to make some comments about the extension of this picture to the massive theory. For simplicity, we turn on equal masses m for all the quarks. When m = 0 the theory depends on θ. However, because of the chiral anomaly, the bulk physics depends only on m N f e iθ . So, without loss of generality, throughout this note we will take m real and non-negative and we will explore the dependence of the theory on the complex number m N f e iθ .
Using a combination of arguments based on the expected behavior at m = 0, matching the large m theory with the N f = 0 theory, large N results, and anomalies, we will present a coherent picture of the phase diagram. For generic m and θ the theory has a unique gapped vacuum. It is typically the case that at θ = π the theory has a first order transition associated with the spontaneous breaking of its time-reversal symmetry there. For N f > 1 this transition persists all the way to m = 0, i.e. it is on the entire negative real axis in the complex m N f e iθ plane. It ends at m = 0, where we find the IR behavior we mentioned above (which depends on N f , i.e. a chiral Lagrangian for 1 < N f < N CF T and a free gauge theory or a conformal field theory for N f ≥ N CF T ). For N f = 1 the situation is different, essentially because the m = 0 theory does not have an enhanced symmetry compared to the m = 0 theory. For N f = 1 the first order transition runs along the negative real axis of me iθ and ends at a negative point me iθ = −m 0 . (Large N arguments imply [3,4] that the theory with m = 0 has a non-degenerate gapped vacuum. We will assume that this remains the case for lower values of N . Then, m 0 is a positive number.) The low energy theory at this point includes a single massless particle, which we will identify with the η ′ particle. The reason for that identification is that for large N the endpoint m 0 goes to zero and this particle is the Nambu-Goldstone boson of the axial U (1) symmetry.
We will study in detail domain walls and interfaces in these systems. We should distinguish between different notions.
First, whenever the theory has more than one vacuum, e.g. in the time-reversal symmetry broken situations at θ = π, there can be dynamical domain walls separating between the two vacua. These are dynamical excitations of the system. In all our examples the domain wall separates between two gapped ground states -the lowest excitation in the bulk has nonzero energy M . 1 It is often the case that there are nontrivial excitations with energy much lower than M living on the domain wall. They are described by a 3d quantum field theory. It may also be that the domain wall does not have excitations with energy much smaller than M , but it supports a 3d Topological QFT (TQFT). One of the goals of this paper is to identify these 3d QFTs. These 3d QFTs are valid only up to energies of order M . At higher energies the bulk cannot be ignored and the theory is no longer a purely 3d QFT.
Second, we will consider the system with space-dependent coupling constants. Specifically, we will let θ be position dependent and we will let it interpolate smoothly between, say θ = 0 and θ = 2πk for some k. Since θ is position dependent, this is not an excitation of the system. Yet, it is an interesting system to study. It is important that the physics of this interface follows from the UV 4d Lagrangian without additional data. 2 As we will see, the physics of the interface depends on how fast θ varies, i.e. it depends on |∇θ|.
Third, we can consider sharp interfaces. Here the parameters change abruptly. For example, we can have a sharp interface between θ 1 and θ 2 . This situation can be viewed as a limit of the previous case, as the gradient becomes larger. However, unlike the smooth interface, here the physics is not universal. There is an ambiguity in adding degrees 1 Sometime, as in the pure gauge N f = 0 theory, these two gapped phases are in different SPT phases, that is, the two gapped phases have different local-terms for background fields [6]. 2 Here we assume that the UV theory does not include additional couplings of the dynamical fields to derivatives of θ.
of freedom supported only on the interface. Anomaly considerations can constrain the dynamics, but they do not uniquely determine it. Now let us summarize the results of this paper.
The domain wall theory is comprised of the usual center of mass mode but in addition there is an SU (N ) 1 Chern-Simons theory. This is an example where the domain wall supports a nontrivial 3d TQFT [6].

2.
QCD with N f = 1. For m ≫ m 0 and θ = π the theory is well described by pure Yang-Mills theory at θ = π and therefore has two degenerate ground states related by time-reversal symmetry. The domain wall theory contains SU (N ) 1 Chern-Simons theory. As we lower the mass m this TQFT eventually disappears and we obtain a domain wall theory that contains only the center of mass mode. There is therefore a phase transition on the domain wall while the bulk is entirely gapped and smooth. 3 Here again at large m and θ = π we have SU (N ) 1 Chern-Simons theory. As we lower the mass, there is a phase transition and we obtain a CP N f −1 non-linear σ-model with some Wess-Zumino term. The CP N f −1 non-linear σ-model is clearly visible in the chiral Lagrangian. There is therefore again a phase transition on the domain wall while the bulk is gapped and smooth. 4. QCD with N CF T ≤ N f . We do not study this case in detail in this paper. But we note that a natural conjecture is that here the domain wall theory is given by an SU (N ) 1 Chern-Simons theory for all m (which should be below the Landau pole scale when The phases in all the four cases above are captured by the conjectured phases of the Chern-Simons matter theory where the fermions are in the fundamental representation of SU (N ). This model has been studied in detail in the literature on 3d Chern-Simons matter theories.
These are exactly the phases of the domain wall in QCD with one flavor, as reviewed above. Furthermore, there is a bosonic dual theory describing the transition, U (1) −N + φ, with φ having charge 1.
3. N f > 1. Here N f > |2k| and hence the above-mentioned dualities cannot be used in their simplest form [16]. However, these dualities are still useful. According to [24] for   [26][27][28], and see also [29]), etc. It would be nice to study the bulk phases, domain walls, interfaces, and corresponding dynamics and dualities in 2+1 dimensions. 4 Connections between dynamics in 3+1 dimensions and 2+1 dimensions via domain wall constructions were also studied, for instance, in [25].
In section 2 we consider the dynamics, domain walls, and interfaces of pure Yang-Mills theory. This section is mostly a review, but we make a few new observations there. In section 3 we consider one flavor QCD and again study both the bulk and domain walls phases. In section 4 we study the case of N CF T > N f > 1 degenerate quarks both at finite N and in the large N limit (following [30,31,3,4]). In section 5 we make some comments about anomaly matching on domain walls and interfaces. Some additional results about the chiral Lagrangian are collected in an appendix.

Pure Yang-Mills Theory
Consider the Lagrangian of pure Yang-Mills theory with gauge group SU (N ) where we used Euclidean signature (as we will throughout this note) and hence the factor of i in front of the θ-term. We will study this theory on a four-dimensional manifold M 4 .
The instanton number is quantized as As a result, the theory with θ is equivalent to the theory with θ + 2π. More precisely, the The symmetries of the theory include the one-form global symmetry Z N associated with the center of the gauge group [32,33] (and see references therein). It acts on the fundamental Wilson loop W F = T r F P e i A by and at θ = 0, π there is in addition also a time-reversal symmetry (equivalently, CP symmetry). The theory at θ = 0 has no anomalies and it is believed to have a trivial, unique, gapped ground state. At θ = π there is a mixed 't Hooft anomaly between the time-reversal symmetry and the Z N one-form global symmetry [6]. As a result, the theory cannot have a trivial ground state. One way to saturate this anomaly in the infrared is to break time-reversal symmetry spontaneously. In fact, this occurs in the planar limit . We will assume that this is how the anomaly is saturated also at finite N .
(For low values of N , especially for N = 2, there are also other plausible scenarios [6].) As a result of the spontaneous breaking of time-reversal symmetry at θ = π, the theory admits a domain wall. The domain wall theory cannot be trivial in the infrared because of anomaly inflow. Indeed, the bulk has a mixed 't Hooft anomaly involving time-reversal symmetry and the Z N one-form symmetry. Since time-reversal symmetry is broken in the bulk, the domain wall theory must have an 't Hooft anomaly for its one-form Z N symmetry.
In more detail, let B be the two-form Z N gauge field. Then, the 3d theory on the wall can be coupled to B, but the partition function is not gauge invariant. The non-gauge invariance can be canceled by a bulk term (More precisely, the Pontryagin square has to be used. Furthermore, this expression is valid only for even N ; for odd N see [6] and some comments below. See also [34] for an earlier discussion of similar ideas.) From the Z N anomaly (2.5) one can infer that the domain wall theory is an SU (N ) 1 Chern-Simons theory [6]. 5 Hence, the Z N symmetry is spontaneously broken on the domain wall and probe quarks are deconfined near the wall (see also [35,9]).
We can estimate the tension of the domain wall. Clearly it scales like Λ 3 . But for large N it would be important later that the tension in fact scales like This factor of N follows simply from the fact that at large N the action scales like g −2 ∼ N . 5 Note that the SU (N ) 1 theory is related by level-rank duality to U (1) −N , so one might try to claim that the theory on the wall can also be thought of as U (1) −N . However, this level-rank duality is valid only when these two theories are viewed as spin-Chern-Simons theories [16]. (In fact, for odd N the U (1) −N theory exists only as a spin-Chern-Simons theory. For even N , U (1) N is a non-spin theory but its anomaly is 2πi B ∪B rather than (2.5). For more details, see [29].) Since our microscopic theory does not need a spin manifold, this level-rank duality cannot be used.
As a result, the theory on the wall is SU (N ) 1 and should not be thought of as U (1) −N .

Digression about Anomalies in Chern-Simons Theory
In preparation for our discussion of interfaces we would like to recall a few facts concerning Chern-Simons theories. SU (N ) k Chern-Simons theory is well-defined for integer k and does not require a spin structure. The theory has a Z N one-form symmetry acting on its line operators [32,33]. We can think of this one-form symmetry as being associated with where the distinct local terms are labeled by K = 0, 1, · · · 2N − 1. (Actually, for spin manifolds with a choice of spin structure only K mod N matters.) For odd N the distinct local terms are labeled by K = 0, 2, · · · 2N − 2 [32,33].
For even N , the SU (N ) k theory leads to the anomaly (2.7) with 6 Another way to state the same facts is to note that for odd N the P SU (N ) N Chern-Simons theory is a nonspin TQFT, while for even N it is a spin-TQFT.

Interfaces
Now let us discuss interfaces in the theory. Unlike the domain walls, these are not excitations in the original system (2.1), but rather, they are obtained by letting some of the parameters be space-time dependent.
We will let θ depend on one of the coordinates x in R 4 interpolating between θ = 0 at x → −∞ to θ = 2πk with integer (positive) k as x → +∞. The two vacua at the two ends are exactly the same as long as we do not couple the theory to external background fields. They are both non-degenerate, gapped, and confining. In fact, they are related by a similarity transformation using U k with U of (2.3).
However, if we couple the theory to a Z N two-form background gauge field B for the one-form global symmetry, the vacua on the two sides are different. The theory labeled by (θ, K) is the same as the theory labeled by (θ + 2π, K − 1 + N ) [32,33,6]. Therefore, starting at (θ = 0, K = 0) and interpolating as above by changing only θ we end up with Let us now study various possibilities for varying θ between 0 and 2πk.
1. Let us first analyze the case of adiabatically varying θ(x) with |∇θ| ≪ Λ, with Λ the dynamical scale of the theory. In this case at any point in space we are approximately in the vacuum of the theory at that value of θ(x). The function of the vacuum energy is continuous as a function of θ but non-differentiable when θ ∈ π + 2πZ. This is the first order transition that we described above, where the time-reversal symmetry is broken spontaneously due to an anomaly. The energy density is therefore peaked at these points and the interface in fact breaks up to k copies of the SU (N ) 1 theory that we discussed above. The interface theory is therefore given by This clearly matches the anomaly (2.7).
2. Now let us assume that θ interpolates from 0 to 2πk rapidly, |∇θ| ≫ Λ, but with finite ∇θ. In that case the interface theory is naively This is certainly the correct answer for sufficiently small k compared to N . 7 When k becomes of the order of N , we may be able to decrease k in absolute value without changing the anomaly by subtracting N or 2N in the cases that N is odd or even, respectively. At the moment it is therefore unclear if (2.10) is correct for all k or only for k sufficiently small. We leave this interesting question for the future. In terms of Yang-Mills theory, the bi-fundamental fields Φ are simply the confining strings.
Intuitively, when the domain walls are far apart these strings are very long and their mass scales like their length (multiplied by the tension, which is of order Λ). Therefore, the bi-fundamental mass in 3d scales linearly with the inverse gradient of θ for small gradients.
As the domain walls approach each other, the strings become shorter and shorter until at some point we can intuitively imagine that they become tachyonic (which is a standard phenomenon for short open strings). Then, they condense and the theory changes to SU (N ) k . As we remarked above, this picture may break down for k of order N .

Quantum Chromodynamics with One Flavor
Here we study the theory (2.1) with an additional quark A well-known observation about this theory is that at m = 0 there is no new symmetry.
It therefore appears that m = 0 is not a special point (although it is a well-defined point 9 ). 9 One might think that since for N f = 1 the quark mass suffers from an additive renormalization, the point m = 0 is not well defined. However, at short distances this additive renormalization is softer than the bare mass m and therefore they can be easily separated. Specifically, consider the chirality violating two point function ψ αψβ at high momentum p. The bare mass m contributes However, the real me iθ line is a special subspace of the full complex plane because it preserves CP .
At large m we can integrate out the quark and we end up with pure Yang-Mills theory at θ. For generic θ it has a unique ground state and at θ = π it has two vacua. (As we said above, this could be different for small values of N .) This means that in the complex me iθ plane there is a first order line coming from infinity along the negative real axis and it must end at some point with m = m 0 . This point must be along the negative real axis (θ = π) because, as we said in the introduction, the theory for m = 0 has a trivial vacuum (this can be explicitly demonstrated at large N and we will assume that it continuous to be true at finite N even though this assumption is not crucial for us). Therefore, as we change m with θ = π, CP should be restored at m = m 0 . The transition at this point must be second order because it is the end point of a first order line. This is depicted in fig. 2.
It is natural to assume that this transition is described by a real pseudoscalar field η ′ .
Its low energy dynamics is essentially free. It can loosely be called the "4d Ising theory." m p 2 + · · ·. The additive instanton contribution includes a factor of exp − 8π 2 and therefore, up to logarithmic corrections, its contribution is ∼ 1 , which is softer than the contribution due to the bare mass m.
For θ = π and m close to m 0 its dynamics can be described by the effective low energy Euclidean theory Since η ′ is massless at m = m 0 , It is important that η ′ is a pseudoscalar because its condensation needs to break the CP symmetry spontaneously. We identify the η ′ field with the phase of fermion condensate and hence with the η ′ particle.
We conclude that the SU (N ) gauge theory with N f = 1 for any N has a value of the bare mass where the η ′ particle is massless! Our description of the emergence of the η ′ particle can be understood particularly easily in the large N limit. Indeed, in the planar limit the axial anomaly disappears and the axial symmetry is restored at m = 0. Then the Goldstone boson of its spontaneous breaking η ′ is exactly massless. 1/N corrections give the η ′ a small mass [2]. For m = 0 the low energy description of the planar theory with N f = 1 is given by [30,31,3,4] where the corrections to this Lagrangian are suppressed by powers of N . In the large N limit f 2 η ′ ∼ N Λ 2 and Λ as well as χ are fixed and do not scale with N . χ is known as the topological susceptibility. It is determined in the pure gauge SU (N ) theory and it is important that it is positive [2]. The physical mass of η ′ is therefore m 2 The potential is proportional to η ′2 for η ′ ∈ [−π, π] and beyond that the potential is defined in a periodic fashion. There is therefore a non-differentiable singularity at η ′ = π mod 2π. Now imagine adding to the Lagrangian a small mass term mψψ +c.c.. This is reflected by modifying the Lagrangian (3.6) to [30,31,3,4] where as always m is real and positive and θ is 2π periodic. We define the QCD scale Λ such that the second term in (3.7) is as we wrote it. The sign of that term is determined such that for θ = 0 the theory has a unique ground state at η ′ = 0 for all m. Of course, the Lagrangian (3.7) is valid only to leading order in m Λ and to the order we wrote in the 1 N expansion. Let us analyze the potential of (3.7) For θ = 0 it has a single minimum at η ′ = 0. For generic θ there is also a unique minimum at some value of η ′ . For θ = π the situation is more interesting. The behavior changes at where η ′ is massless. For m < m 0 there is a unique minimum at η ′ = 0. This is the analytic continuation of the situation at θ = 0. For m > m 0 there are two vacua at η ′ = ±η ′ 0 , with η ′ 0 interpolating from 0 at m > ∼ m 0 to π at m → ∞ (the chiral Lagrangian approximation breaks down beforehand).
Note that expanding around m ≈ m 0 with η ′ ≪ 1 the effective Lagrangian (3.7) goes over to (3.3). It is therefore consistent with fig. 2.
Let us discuss the case θ = π and 1 N ≪ m Λ ≪ 1 in more detail. Here the minima are at η ′ = ±η ′ 0 with η ′ 0 approaching π It is important that these two minima are not physically close to each other in field space, although they seem to be close to each other in the η ′ coordinate. Recall that at η ′ = π the potential is not differentiable. Such singularities of the effective potential typically mean that some heavy degrees of freedom are in fact important. And indeed, we expect that as we cross the singular point at η ′ = π we need to rearrange the heavy fields significantly.
As a result, despite appearances, the two ground states do not get physically close to each other as we increase the mass.
When CP is spontaneously broken, i.e. for θ = π with m > m 0 , the theory has domain walls. For sufficiently large m the domain wall is the same as in pure Yang-Mills theory at θ = π, i.e. it supports an SU (N ) 1 Chern-Simons theory. However, as m is reduced toward m 0 , this can no longer be true. Here the bulk is described by the η ′ field, which is simply a pseudoscalar field theory with two minima, and hence its domain wall theory is trivial. We see that for some finite value of the bulk mass m transition > m 0 there must be a transition on the domain wall from SU (N ) 1 to a trivial domain wall (which we can think about as SU (N ) 0 ). It is interesting that throughout this process of reducing m and going through a phase transition on the wall the bulk remains gapped and is essentially unchanged. Yet, the domain wall undergoes a phase transition.
Let us estimate the tension of the domain wall in various limits (we always take m > m 0 and θ = π so that CP is spontaneously broken and the domain wall exists).
Here the domain wall can be understood using the η ′ Lagrangian (3.7), which can be further simplified to The domain wall interpolates between . The tension of the domain wall is therefore given by solving the equation of motion for η ′ that interpolates between ±η ′ 0 and then substituting in We can rescale the coordinate x as well as the η ′ field such that it interpolates between ±1 and convert the formula for the tension to This domain wall is trivial in the sense that it has no nontrivial degrees of freedom other than the obvious center of mass.
Also in this region we can trust the chiral Lagrangian except that it is now better approximated by (3.14) In this region, as we explained around (3.10), the two minima appear to be close to each other, but they are in fact far away. It is therefore still beneficial for the domain wall to interpolate between the two vacua by having the η ′ field go around the circle, avoiding η = π. After rescaling the coordinates, the tension of this domain wall is given by These two regions 0 < m − m 0 ≪ Λ N and Λ N ≪ m − m 0 ≈ m ≪ Λ are smoothly related and give a coherent picture for m 0 < m ≪ Λ and 1 N ≪ 1. For these values of the parameters the η ′ field performs a continuously increasing field excursion on the circle as m is increased. Hence, there is no phase transition between them. Throughout this region the domain wall is trivial in the sense that it has no nontrivial degrees of freedom other than the obvious center of mass. Finally, we discuss the limit m ≫ Λ. Here we can simply integrate out the quark and we remain with a domain wall in pure Yang-Mills theory at θ = π. The tension scales like Here Λ should be the strong coupling scale of the pure YM theory that remains after integrating out the quark. The difference between it and the original strong coupling scale is subleading in N . In terms of the large N picture with η ′ we can say that at some point it becomes favourable for the η ′ field to jump over the singular point η ′ = π and not go around the whole circle. This happens when m becomes of order Λ, where the expansion in m Λ above is no longer valid. It is important to know whether the transition on the domain wall is first order or second order. From the large N discussion above it seems that for small m the η ′ coordinate interpolates through η ′ = 0 and for large m it interpolates through η ′ = π. This might indicate that the trajectory of η ′ jumps and hence the transition is first order. If this is the case, the physics of the transition involves massive modes on the wall. It could also involve massive modes in the 4d bulk of the system. If this is the case, we cannot describe the transition in terms of a 3d quantum field theory.
If, on the other hand, the transition is second order, since the bulk is gapped, it must have a purely 3d quantum field theory description.
Regardless of which of these two options materializes, we now present a 3d quantum field theory that has exactly the two phases of the domain wall. Consider the theory 10 i.e. SU (N ) 1/2 Chern-Simons theory coupled to a fundamental fermion ψ. If we give ψ a large mass, depending on the sign of the mass, we end up with either SU (N ) 1 pure Chern-Simons theory or the trivial theory. These are the expected phases of the domain wall theory. It is further conjectured (see [16,24] and references within) that these are the only two phases of the theory, with a single phase transition. We can also describe this theory in terms of a bosonic dual [12,16] U (1) −N + φ ,

Quantum Chromodynamics with N f > 1 Quarks
Consider QCD with N f quarks We will first study in detail the bulk phases of the theory and then we will consider the domain walls (and in appendix A we also briefly consider some interfaces).
For generic masses this theory has a U (1) N f /Z N global symmetry, where the quotient is by a Z N subgroup, which is part of the gauge group. This quotient will be important in section 5. For simplicity we will consider the situation with equal masses m and then the global symmetry is U (N f )/Z N . As for N f = 1, without loss of generality we can let the common mass m be real and non-negative and parameterize the theories by the complex 11 Here we can use level-rank duality because both theories need a choice of a spin structure (more generally a spin c structure) [16].
number m N f e iθ . The generalization to arbitrary masses is straightforward. 12 For generic θ the system is not CP invariant, but for θ = 0 mod π it is CP invariant. (If we rotate θ into the mass matrix, the system is CP invariant when det m is real.) As we will see, the physics at θ = 0 and at θ = π are different. Consider m ≫ Λ, where Λ is the dynamical scale of the system. Then we can integrate out the quarks. For θ = 0 there is a confining trivial vacuum. And for θ = π we obtain two ground states related by CP . In fig. 3 we summarize the phases of the theory in the m N f e iθ plane. Now consider the theory around m = 0. We restrict to N CF T > N f > 1, where the massless theory is described by a chiral Lagrangian. We will first study the finite N theory, ignoring the η ′ particle. The Lagrangian at m = 0, which is the second order transition point of fig. 3, is given by where the higher order terms are suppressed by additional derivatives and U is an SU (N f ) matrix, expressed in terms of the pions π a as U = e iπ a T a and a is an index in the adjoint of the unbroken group. Now suppose that we add a mass term m for the N f quarks.
As usual, as long as m ≪ Λ we can analyze the effect of the mass using the chiral Lagrangian, where to leading order we simply need to add a potential in terms of the fermion The distinction between f π and Λ will be important later when we study the phases of the theory at large N . The potential in the chiral Lagrangian is Let us consider the minimization of the potential for θ = 0 and θ = π. Starting with θ = 0, assuming that SU (N f ) is not spontaneously broken, the vacua are potentially at U = e 2πik/N f 1I with k ∈ Z. Minimizing (4.4) over k we find that the minimum is at U = 1I, which is CP invariant. Therefore, CP is unbroken. One can check that this vacuum is gapped and the assumption that SU (N f ) is not broken is self-consistent.
Next, we turn to θ = π. Again, we examine the SU (N f )-invariant vacua Substituting this into (4.4) we find that the potential of the configurations above is This is minimized for k = 0 and k = −1, i.e.
These two solutions are related to each other by U → e − 2πi N f U † and hence CP is spontaneously broken. (CP breaking in the chiral Lagrangian was first discussed in [36] and later by many others including [3,4] and [37,38].) Let us check whether the assumption that SU (N f ) is preserved is self consistent by studying the small fluctuations around these two minima. A general SU (N f ) matrix with entries on the diagonal is given by The energy of this configuration is For N f = 2 the potential V vanishes identically and hence it costs no energy to interpolate between the two SU (2)-preserving minima (4.7). (For a detailed discussion of this point see [39].) For N f > 2 we can expand around (4.7) and verify that they are true local minima. In the special case of N f = 2 whether SU (2) is broken or CP is broken is determined by the higher-order terms in the chiral Lagrangian. As in (4.4), adding a mass term for the fermions corresponds to adding another term to the chiral Lagrangian

Bulk Phases at Large
So we should study the potential [30,31,3,4] We parameterise the matrix U as whereŨ is an SU (N f ) matrix. This parametrization is subject to the identification N fŨ , η ′ + 2π) , (4.14) so we can limit η ′ to the range [−π, π].
We expect the global SU (N f ) symmetry to be unbroken and therefore we look for minima of the formŨ = e 2πik N f 1I for some integer k. The potential as a function of η ′ and k is As a check, for N f = 1 this coincides with (3.8).
It is instructive to examine various limits. Recall that our analysis is valid in the limits 1 N , m Λ ≪ 1. We will analyze separately the cases m Λ ≪ 1 N ≪ 1 and 1 N ≪ m Λ ≪ 1. For m Λ ≪ 1 N ≪ 1 the physics is essentially identical to the analysis for finite N , where we ignored the η ′ field. More precisely, η ′ 0 = mf 2 π χΛ 3 sin( π N f ) + · · · ∼ Nm Λ ≪ 1 and the two minima are those of (4.7).
, namely, the matrices that minimize the potential are We see that the η ′ particle is always massive and for θ = π the time-reversal symmetry is broken for every positive m.

Domain Walls and 2+1-Dimensional Dynamics
In our study here we will ignore the η ′ field and will study the domain wall (and briefly some interfaces) of QCD with N f > 1 flavors. We fix θ = π and vary m. As we said, the theory breaks its time-reversal symmetry spontaneously for every nonzero m. Hence, there is a domain wall that connects the two ground states.
For m ≫ Λ the quarks are very heavy, we can integrate them out, and we remain with pure Yang-Mills theory with θ = π. The domain wall theory in the deep infrared supports a TQFT m ≫ Λ : where we used the duality between them as spin TQFTs [16].
Now let us consider the domain wall theory for m ≪ Λ. We can study it in the chiral Lagrangian. The two vacua are given by (4.7). Clearly, each of these vacua preserves the SU (N f ) symmetry. We would like to consider a smooth configuration that interpolates between them. Since any such smooth configuration must break SU (N f ) away from the vacua, once there is one such configuration, there must be a manifold of such configurations.
We will argue in appendix A that the lowest energy domain wall theory preserves the ]. Therefore the domain wall theory is gapless, supporting degrees of freedom parameterizing the coset m ≪ Λ : There is therefore a massless nonlinear σ-model on the domain wall at small m. More precisely, for N f > 2, the chiral Lagrangian also includes a Wess-Zumino term and as a result the 2+1-dimensional domain wall theory also has an induced Wess-Zumino term.
The special case N f = 2 will be discussed below.
We see that there is a phase transition on the domain wall separating the two phases (4.17) and (4.18). As we said above, we do not know whether this transition is first or second order. And if it is a first order transition it is not clear that there should be a 3d QFT describing it. Nevertheless, we would like to present a natural 3d QFT, which has these phases. One can think about the domain wall theory as According to the conjecture of [24] this theory has two phase transitions. The domain wall theory accounts for the phases around one of these transitions, which is also described by the dual bosonic theory It is manifest that for positive mass squared in this theory we are in the phase (4.17) and for negative mass squared we have the CP N f −1 manifold; i.e. the phase (4.18). The Chern-Simons term in (4.20) gives rise to a Wess-Zumino term in the nonlinear σ-model with target space CP N f −1 [24].
We see that the domain wall theory can be naively thought of as (4.19). And then, using the conjectured dynamics in [24] it produces the topological phase (4.17) and the symmetry breaking phase (4.18). Conversely, this picture provides some more evidence to that conjectured dynamics.
Let us now discuss two slightly atypical cases. For N f = 1 the coset (4.18) is trivial and indeed the domain wall theory has a trivial phase. This is the phase where the η ′ field condenses, as explained in section 3. For N f = 2 the bulk 3 + 1-dimensional theory has no continuous Wess-Zumino term, but instead it has a Z 2 -valued θ-parameter associated with π 4 (SU (2)) = Z 2 [40,41]. It is nontrivial for odd N and makes the Skyrmions/baryons fermions for odd N . Correspondingly, the CP 1 2+1 dimensional nonlinear σ-model has no continuous Wess-Zumino term and instead it has a Z 2 valued θ-parameter [42] (and see references therein). This term is closely related to the Hopf term at θ = π [43]. As in 3 + 1 dimensions, this θ-term transforms the Skyrmions into fermions for odd N . These 2 + 1 dimensional Skyrmions originate from baryons in the bulk, which subsequently become baryons of the domain wall theory (4.19).

Anomalies in QCD and Anomaly Inflow to 2+1 Dimensions
In this section we show that when the greatest common divisor gcd(N, N f ) = 1 the theory at θ = π has a mixed 't Hooft anomaly between the continuous global symmetry and time-reversal symmetry, forcing nontrivial physics at long distances.
Our discussion here will be very similar to the analysis in [23] of the global symmetries and their 't Hooft anomalies in the analogous 3d theory. In fact, the anomaly in the 4d theory means that there must be a nontrivial 3d theory on domain walls, and that 3d theory can be one of the theories discussed in [23]. We therefore obtain the anomaly of [23] by anomaly inflow from an anomaly in 4d, which also includes time-reversal symmetry.
First, we should determine the global symmetry of our system. For θ = 0 mod π the system has time-reversal symmetry. The symmetry that commutes with the Lorentz group includes charge conjugation symmetry (which we will ignore) and a continuous internal symmetry G. Let us discuss the latter.
SU (N f )×U (1)  gauge group and hence it is a gauge symmetry. It is convenient to consider two groups that act. The group that acts on the fields in the Lagrangian is where the SU (N ) factor is the gauge group. The global symmetry group is The gauge invariant local operators are in representations of G rather than of K. They are therefore in representations of U (N f ), but that group is not represented faithfully.
We can represent K in terms of three group elements u ∈ SU (N ), v ∈ SU (N f ), ρ ∈ U (1) with the identifications which follow from the two factors in the denominator of (5.1). Similarly, we can describe G in terms of two group elements v ∈ SU (N f ), ρ ∈ U (1) subject to the identifications that follow from (5.2) The global symmetry of the system allows us to couple it background classical gauge fields. We can easily couple the system to SU (N f ) × U (1) gauge fields. But the quotients in (5.2) lead to additional options. We can couple the system to background gauge fields of G, which are not SU (N f ) × U (1) gauge fields. The quotient by Z N f is relatively simple -it is straightforward to couple the system to U (N f ) gauge fields. But the quotient by Z N is more interesting, because it acts also on the SU (N ) dynamical gauge fields. (See [44] for We see that by turning on nontrivial G classical gauge fields for the global symmetry, we can force the system to be in a nontrivial P SU (N ) bundle. It was emphasized in [32,33] that this is not a new field theory, but instead, this is an observable in the original SU (N ) gauge theory.
Note that since our matter fields are in the fundamental representation of SU (N ), our system does not have a one-form global symmetry. Yet, with appropriate twists in the flavor symmetry we can place it in P SU (N ) bundles. In [33,6] the presence of such bundles led to anomalies involving the one-form global symmetry. Here, following [23], these twisted bundles will lead to anomalies involving ordinary global symmetries. (A similar analysis can be found in [8,9].) Another way to approach the problem is to turn the SU (N f ) gauge fields to dynamical gauge fields. Then there is a one-form global symmetry, Z gcd(N,N f ) . Equivalently, we can say that the SU (N f ) gauge fields are spurions for the explicitly broken one-form symmetry in QCD. Now there could be a mixed anomaly between the one-form symmetry and timereversal symmetry. This point of view was used in [45] (see also [46,47]).
Next, we look for anomalies. Clearly, there is nothing wrong with coupling the system to G gauge fields and therefore there is no pure flavor anomaly. The situation with the timereversal symmetry at θ = π is different. Here the symmetry depends on the 2π periodicity of θ. However, nontrivial G bundles can force the gauge fields to be in nontrivial P SU (N ) bundles, which are not SU (N ) bundles. For these, (on spin manifolds) the periodicity of θ is 2πN rather than 2π and therefore the θ = π system might not be time-reversal invariant.
This is essentially the same phenomenon observed in [6], except that here we use flavor symmetries rather than one-form symmetries to create fractional P SU (N ) instantons.
In order to analyze this problem in detail we should try to add to the θ = π Lagrangian counterterms in the background fields such that it is time-reversal invariant even for the non-trivial bundles. Since all our bundles will be P SU (N ), P SU (N f ), and U (1) bundles, all the relevant θ-terms can be expressed in terms of ordinary instanton numbers, except that because of the twists they can be fractional. We have denoted the field strength of the original SU (N ) gauge field by F . Now, we add an SU (N f ) gauge field with field strength We started with θ 8π 2 Tr(F ∧ F ) at θ = π, and we add to it two local counterterms in the background fields If the background fields are simply SU (N f ) × U (1) fields, then the coefficients of the counterterms θ f and θ B are 2π-periodic and the system is time-reversal invariant when they are integer multiples of π. However, because of the Z N × Z N f quotient this is not necessarily true.
We would like to know the conditions on the integers r, s, and t in πr such that this Lagrangian is time-reversal invariant. This would be the case if twice that Lagrangian is trivial. Then, we have a consistent Chern-Simons theory (5.7) 14 Our system has fermions and therefore it involves a spin manifold with a choice of spin structure. We can generalize it by choosing a spin c structure and letting the U (1) field that couples to the fundamental quarks be a spin c connection. For simplicity, we will not do it here.
Precisely this problem was analyzed in [23]   This means that that the terms in the Lagrangian (5.5), (5.6) are Clearly, there are also other choices.
If gcd(N, N f ) = 1 we cannot satisfy the conditions (5.8) with r = 1 and therefore we cannot find a time-reversal invariant expression of the form (5.5). This means that even though the Lagrangian of the dynamical fields is time-reversal invariant, when we couple the system to background gauge fields for the symmetry G there is no way to preserve the time-reversal symmetry. This means that the system has a mixed 't Hooft anomaly between the global symmetry G and time-reversal symmetry. There should be a way to write this anomaly as coming from a five-dimensional bulk term. We do not attempt to do it here. 15 The first condition follows from the consistency of the Z N quotient. The second conditions follows from the consistency of the Z N f quotient. Together they imply that t is divisible both by N and by N f (but not necessarily by N N f ). The third condition follows from the mutual consistency of the two quotients. We need to make a Wilson line in some SU (N ) r representation times a charge t/N Wilson line of U (1) t have trivial braiding with a Wilson line in some SU (N f ) s representation times a charge t/N f Wilson line of U (1) t . This is determined by e 2πit N N f , which should be set to one [23]. Equivalently, we can choose a configuration of the fluxes in four dimensions as follows.
Take the four-manifold to be T 4 and choose a minimal 't Hooft flux [48] in the P SU (N ) bundle over the two cycle in the directions 12 (namely the integral of u 2 (P SU (N )) over the 12 cycle is 1, where u 2 is the Z N valued two-form obstruction of P SU (N ) bundles to be SU (N ) bundles).
Similarly, take a minimal 't Hooft flux in the 34 direction for P SU (N f ). All the other fluxes are taken to vanish. In this case the first and second terms in (5.6) vanish (modulo the usual integer instantons), while the third term is given by 1 Therefore one obtains that t should be a multiple of N N f , as we found above.
This anomaly constrains the bulk physics. It can be matched by either a symmetric vacuum with a nontrivial infrared theory (which could be topological) or by breaking either of the symmetries spontaneously. Indeed, as we discussed in section 4, we expect the timereversal symmetry to be spontaneously broken. Then the domain wall theory that we with θ = π. Let us use the SU (N f ) symmetry to bring U to a diagonal form U =    e iα 1 0 · · · · · 0 e iα 2 · · · · · · · · · · · · · · · · · · · · · e iα N f    , α i = 0 mod 2π .

(A.2)
Substituting this into (A.1) we obtain The ground states are clearly at α i = − 2π N f and α i = 0. These two ground states are related by time-reversal and they are both manifestly gapped (as worked out in section 4) for N f > 2.
Here we study a configuration that interpolates between the two matrices, U = 1I and The SU (N f ) symmetry must be spontaneously broken since there is no way to interpolate between these two vacua with SU (N f )-preserving matrices. Let us divide the phases α i into two groups, α 1 = ... = α k and α k+1 = ... = α N f . Suppose without loss of generality that the first group goes continuously from 0 to − 2π N f . Then the second group would end up at α k+1 = ... = α N f = − 2π N f + 2π N f −k . Therefore we end up at the required vacuum U = e − 2πi N f 1I only if N f − k = 1.
In order to study the domain wall we denote α 1 = ...α N f −1 ≡ α and α N f = −(N f − 1)α. Substituting this into (A.3) we obtain We can estimate the tension of the resulting domain wall by rescaling the coordinate orthogonal to the wall as x = x ′ / √ mΛ. Then the the tension is with α changing from 0 to − 2π N f . Therefore the tension scales like The width of the domain wall is also fixed by dimensional analysis to be of order 1 √ mΛ . In the large N limit the tension (A.6) is ∼ N m 1/2 Λ 5/2 . As we increase the mass of the quarks, the domain wall's tension rises and its width 1 √ mΛ decreases. The expressions above are valid for m ≪ Λ. For m ∼ Λ and larger the width becomes Λ −1 and the tension scales like N Λ 3 .

A.2. Interfaces in the Chiral Lagrangian
Now we discuss the problem of interfaces in the chiral Lagrangian. Here we imagine that θ varies in space along one direction such that at one end we have θ = 0 and on the other side we have θ = 2πk with integer k. (For simplicity of the discussion we limit ourselves to 0 < k ≤ N f 2 .) These two bulk vacua are identical and they are trivial and gapped. In order to minimize the bulk energy on one side we therefore have U = 1I and on the other side U = e − 2πik N f 1I. The physics of the interface depends on how fast θ changes. We start with the case that θ changes very slowly (we will soon specify slowly relative to what). Then the vacuum changes adiabatically with a transition whenever θ crosses θ = (2m + 1)π for integer m. At every such m we switch from U = e 1I. In order to do so, the system produces a domain wall with tension (A.6) and width 1 √ mΛ . Since the width of the domain wall is 1 √ mΛ , this is the preferred way for the system to minimize the energy as long as |∇θ| ≪ √ mΛ. We conclude that in this limit the theory on the interface is given by As we increase the gradient of θ, it eventually becomes preferable to make a rapid transition from U = 1I to U = e − 2πik N f 1I. Repeating the analysis of the previous subsection we find that the the eigenvalues of (A.2) should be divided into two groups such that α 1 = ... = α N f −k and α N f −k+1 = ... = α N f . The first group interpolates from 0 to − 2πk N f and the second group interpolates between 0 and − 2πk N f + 2π. We expect that the width of this configuration is again 1 √ mΛ . Here the symmetry is spontaneously broken as |∇θ| ≫ √ mΛ : .
For a related discussion see [49].
We conclude that as we change the gradient of θ through |∇θ| ∼ 1 √ mΛ the theory on the interface undergoes a phase transition from (A.7) to (A.8). For m ≪ Λ this transition can be described in the chiral Lagrangian and it appears to be first order. The model (A. 8) is closely related to the Grassmannian phase that was suggested in [24], however, we leave the details for the future.