A Symmetry Breaking Scenario for QCD$_3$

We consider the dynamics of 2+1 dimensional $SU(N)$ gauge theory with Chern-Simons level $k$ and $N_f$ fundamental fermions. By requiring consistency with previously suggested dualities for $N_f\leq 2k$ as well as the dynamics at $k=0$ we propose that the theory with $N_f>2k$ breaks the $U(N_f)$ global symmetry spontaneously to $U(N_f/2+k)\times U(N_f/2-k)$. In contrast to the 3+1 dimensional case, the symmetry breaking takes place in a range of quark masses and not just at one point. The target space never becomes parametrically large and the Nambu-Goldstone bosons are therefore not visible semi-classically. Such symmetry breaking is argued to take place in some intermediate range of the number of flavors, $2k<N_f<N_*(N,k)$, with the upper limit $N_*$ obeying various constraints. The Lagrangian for the Nambu-Goldstone bosons has to be supplemented by nontrivial Wess-Zumino terms that are necessary for the consistency of the picture, even at $k=0$. Furthermore, we suggest two scalar dual theories in this range of $N_f$. A similar picture is developed for $SO(N)$ and $Sp(N)$ gauge theories. It sheds new light on monopole condensation and confinement in the $SO(N)$ and $Spin(N)$ theories.


Introduction
In this paper we study QCD in 2+1 dimensions, namely non-Abelian 2+1 dimensional gauge theories coupled to fermionic matter fields. We will focus on SU (N ), SO(N ), and Sp(N ) gauge theories coupled to N f fermion flavors in the fundamental representation.
In SU (N ) each fermion flavor is in an N dimensional complex representation, in SO(N ) each flavor is in an N dimensional real representation, and in Sp(N ) each flavor is in a 2N dimensional pseudo-real representation. These are the 2+1 dimensional analogs of Quantum Chromodynamics. While there is a rather concrete picture for the dynamics of 3+1 dimensional QCD, the 2+1 dimensional counterpart is still poorly understood in part because these theories depend not just on the number of flavors and the gauge group but also on the Chern-Simons level. Our main goal is to propose a scenario for the phases of these theories as a function of the gauge group, Chern-Simons level k, and the number of flavors N f . We will follow the notation and conventions of [1][2][3][4], which we summarize now. Let us start from a single Dirac fermion of charge 1 coupled to a gauge field A, The effective action for the background field A can be computed by performing the path integral over the fermion. The effective action for A is however ambiguous, since we could add various counter-terms. More precisely, we could add to the action with k bare ∈ Z. We will fix this ambiguity by choosing the following convention for the effective action of the free fermion. If we add to the Lagrangian a positive mass term for the fermion, then our IR effective action has no Chern-Simons term for the background field. If, on the other hand, we add a negative fermion mass, then the IR effective action is − 1 4π A ∧ dA, i.e. the Chern-Simons level in the infrared is −1. This convention for the effective action breaks time reversal symmetry, but this is unavoidable in 2+1 dimensions [5]. Our specific convention amounts (up to a sign) to identifying the phase of the fermion path integral with the eta invariant (see [6] for a recent detailed discussion).
Our convention for fermions coupling to non-Abelian gauge fields is analogous.
With this convention, let us suppose that we start from N f fermions coupled to a U (1) gauge field with Chern-Simons level k bare ∈ Z. Then, for positive mass the low energy effective theory contains a Chern-Simons term of level k bare and with negative mass the infrared Chern-Simons level is k bare − N f . We could label this theory by its bare Chern-Simons term and the number of matter particles. In order to conform with the common notation used in the literature we define k = k bare − N f /2, in terms of which the infrared levels are k ± N f /2 for positive and negative mass, respectively. This choice is more symmetric. Note that while k bare was an integer, k may be a half integer, and it is a half integer if and only if the number of fermions is odd. Henceforth we label this theory by U (1) k + N f fermions. Our labelling of theories with non-Abelian gauge groups is analogous.
In all the theories that we will discuss here, the bare Chern-Simons term must always be an integer. Above, to define k, we shifted the bare Chern-Simons term by −N f /2. We can think 1 about this shift as k quantum and then k = k bare + k quantum . Before we proceed, we would like to make a general comment about the effect of the global aspects of the gauge group on the dynamics of 2+1 dimensional systems. Often our gauge symmetry G is connected but not simply connected and then the system has a magnetic global symmetry H. The charged objects under that symmetry are local monopole operators. One typical example is G = U (1), where the magnetic symmetry is H = U (1), and another example, which we will discuss in section 3, is G = SO(N ), whose magnetic symmetry is H = Z 2 . Suppose we take a subgroupH ⊂ H whereH is 1 In the general case of an interacting conformal field theory, the "quantum" Chern-Simons term was defined in [7,8]. It can be an arbitrary real number.
isomorphic to some Z n . We coupleH to a dynamical Z n gauge field. This has the effect of changing the gauge group G to a multiple cover of itG such thatG/H = G. 2 For example, starting with G = U (1) andH = Z n the new gauge group isG = U (1), but all the particle charges are multiplied by n. Similarly, for G = SO(N ) withH = Z 2 the new gauge group isG = Spin(N ).
How does the dynamics of theG theory differ from that of the original G gauge theory?
Since all we did to the G gauge theory is to gauge a discrete cyclic global symmetry, the resultingG gauge theory is anH orbifold of the original theory. It is important that in 2 + 1d such orbifolding does not change the phase diagram. The phase transitions in the G gauge theory are the same as in theG theory. The only difference between the two theories is that theG theory has fewer local monopole operators and it has more line operators from the twisted sector. If the G theory has a phase with some nonlinear sigma model (i.e. with a moduli space of vacua) on whichH acts, then theG theory has the same phase and the target space of the nonlinear sigma model is the quotient of the original target space bỹ H. We therefore see that the space of vacua could depend on global aspects of the gauge group. (This phenomenon does not occur in four dimensions.) This will be important in section 3.
As a simple example, consider the U (1) k gauge theory with N f fermions all with charge one. This theory has a magnetic H = U (1) global symmetry. Let us quotient the theory byH = Z n ⊂ U (1). This is easily done in the Lagrangian by replacing the gauge field a by na; i.e. all the fermions now have charge n and the Chern-Simons level of the new theory is kn 2 . The reasoning above shows that the phase diagram and the phase transitions between them are independent of n. In particular, the phase diagram of the U (1) 0 theory is the same as that of the R gauge theory without a Chern-Simons term [9].
In section 3 we will consider a similar example with SO(N ) and Spin(N ) gauge theories.
The dualities can be summarized as and other dualities that follow from them. (Our notation is U (k) N ≡ U (k) N,N .) The unitary dualities were conjectured to hold for 1 2 N f ≤ k, the symplectic dualities for 1 2 N f ≤ k, and the orthogonal dualities for 1 2 We will also need their time-reversed versions In all these cases the scalars have quartic interactions. We flow to the IR and allow to tune the fermion and boson masses to a critical point, if it exists. Except for some special cases, we cannot prove that such a critical point exists. Many of the statements that we will make about the phases of these three-dimensional theories are independent of whether the corresponding transitions are second order or not.
As we said, the dualities above (1.5),(1.6) were conjectured to hold for values of N f that are bounded from above by, loosely speaking, the Chern-Simons level of the fermionic theory. The central point of this note is to extend this conjecture about the behavior of these theories to larger values of N f . It is logically possible that the previous conjectures are true, but their newer extended version is not.
Of course such proposals are subject to stringent consistency checks from symmetries, anomalies, mass deformations that allow to decrease N f , and constraints from the large N f limits of 2+1 dimensional gauge theories [52].
Our conjecture is further motivated by the study of domain walls in 3+1-dimensional QCD. This will be developed in great detail in [53]. Briefly, pure Yang-Mills theory in 3+1 dimensions at θ = π has a first order transition associated with the spontaneous breaking of time-reversal. Because of anomaly considerations the domain wall at this transition carries a nontrivial TQFT -SU (N ) 1 Chern-Simons theory [54]. Suppose we now add to the 3+1-dimensional theory heavy quarks with real positive masses, such that time reversal symmetry is preserved by the Lagrangian but broken in the vacuum. For large masses the domain wall is still described by an SU (N ) 1 Chern-Simons theory, since it is hard to make a topological theory disappear. As we lower the quarks masses, the theory on the domain wall undergos a phase transition. Our scenarios below about the dynamics of the 2+1-dimensional theory are consistent by the transitions on these domain walls [53].
We define the transition point to be at m = 0. The gapped phases have topological field theories, SU (N ) k±N f /2 , which are related by level-rank duality of spin theories to U (k ± N f /2) [3]. For k = N f /2 the phase for negative m is trivial.

Summary of our proposed scenario
Let us summarize our conjecture for SU (N ) k + N f fermions, starting from the case N f ≤ 2k, which is covered by the dualities (1.5),(1.6). The theory has two phases, depending on the mass of the quarks. One phase is SU (N ) k+N f /2 pure Chern-Simons theory and the other phase is SU (N ) k−N f /2 pure Chern-Simons theory. The transition could be first or second order. The global symmetry is never broken. The theory is dual to U (k + N f /2) −N + N f bosons, which has exactly these two pure Chern-Simons phases (this can be seen by level-rank duality). For N f = 2k this description still makes sense; on one of the sides of the transition the ground state is trivial. This summarizes the first line in (1.6). We also summarize this situation in fig. 1.
For larger N f this picture breaks down because the bosonic theory has a sigma model at low energies for negative mass squared (for N f ≤ 2k the bosonic model does not have a sigma model phase in its semi-classical regimes and instead it flows to a Chern-Simons TQFT as depicted in fig. 1). By taking the mass squared of the scalars to be large and negative we can make the sigma model as weakly coupled as we like (i.e. the radius of the target space can be as large as we like). No such weakly coupled sigma model with a large target space exists in the fermionic theory and therefore the duality fails [3]. Indeed let us turn on equal masses m for all the quarks. Let us analyze it for |m| ≫ g 2 (where g is the gauge coupling). Then we can integrate out the fermions and use (1.4) to find that the low energy theory is gapped and it is described by a pure SU (N ) k IR Chern-Simons theory with level This leaves the interesting question of what the theory does for small values of m, of order Our conjecture is that there exists a window N * (N, k) > N f > 2k where N * is an unknown function of the parameters of the theory such that the U (N f ) symmetry that rotates the various flavors 3 is spontaneously broken leading to the coset .
Our proposal is that this is a purely quantum phase, not visible in the semi-classical limits of the theory and the target space of this sigma model (i.e. the complex Grassmannian (1.8)) never becomes large. It is important to mention that we need to add to the sigma model Lagrangian a certain Wess-Zumino term, which we will discuss in detail soon.
In the special case k = N f /2 − 1 this result is consistent with the study of domain walls in four dimensions [53]. This proposal for N * (N, k) > N f > 2k implies the dualities (1.5),(1.6) for lower values of N f and shifted values of k, as we explain in the bulk of the paper.
The gapped phases have topological field theories, SU (N ) k±N f /2 , which are related by level-rank duality of spin theories to U (N f /2 ± k) ∓N . The middle gapless phase is a nonlinear sigma model on a Grassmannian with a Wess-Zumino term Γ, whose coefficient is In addition, one can "derive" this picture by starting from SU (N ) 0 + N f fermions (with even N f ). It was suggested in [56][57][58][59] that in some range 0 < N f < N * (N, 0) this theory breaks its global symmetry leading to a nonlinear sigma model with target space The case k = 0 is therefore an important consistency check of our scenario. But it is not just a consistency check -we can reverse the logic and assume this scenario at k = 0 and then derive all the rest. By deforming the nonlinear sigma model with masses that correspond to fermion masses in the ultraviolet theory, one can see that this breaking must take place in a whole region of parameter space (as we vary the symmetry-preserving mass deformation) and not just at one point, and, in addition, one can derive all the other symmetry breaking patterns (1.8). In particular, as we will explain, if the symmetry breaking (1.9) does not occur, then our conjecture is necessarily wrong.
In addition, we stress again that we need to add to this discussion the Wess-Zumino term. There are various ways to see that this Wess-Zumino term is necessary, for example, by considering the quantum numbers of Skyrmions (baryons). This derivation also leads to interesting constraints on N * (N, k), which we discuss in section 2.
The sigma model (1.8) arises from a condensation of the quark bilinear. Indeed, consider the condensate with x appearing N f /2 + k times and y appearing N f /2 − k times (y = x). It breaks the Finally, it remains to discuss what happens for N f ≥ N * . We know that for sufficiently large N f the theory has to be conformal, with only Chern-Simons phases (as in fig. 1) on both sides of the transition [52]. We therefore assume that this is the behavior of the theory for all N f ≥ N * . No scalar dual is known in this regime. N * is therefore the critical number below which (and above 2k) the symmetry breaks in an intermediate, quantum phase, that is invisible semi-classically.
Our proposal for the phases of SO and Sp gauge theories is rather similar in spirit, with an intermediate symmetry breaking phase for sufficiently large (but not too large) values of N f .
In section 2 we discuss some additional details about SU (N ) k with N f fermions in the fundamental representation. In section 3 we discuss the dynamics and phases of the SO and Sp theories, where there are a few novelties.

More on SU (N ) k with N f Fermion Flavors
In this section we discuss a number of topics related to the proposed Grassmannian phase of SU (N ) k with N f fermion that have not been described in detail in the introduction.

Studying the sigma model
We will study some properties of the sigma model with target space .
One way to think about it is intrinsic to this sigma model and uses properties of this space.
This approach is standard in the literature. Alternatively, we find it convenient to view this theory as the low energy approximation of a gauged linear sigma model. In the special Specifically, we can think about (2.1) as the low energy theory of the gauge theory where we add large negative mass squared for the scalars, such that the model is weakly coupled and we can treat it classically. Of course, as we explained in the introduction, this model, when appropriately modified by Chern-Simons terms, plays a more fundamental role in the story than just an auxiliary linear sigma model. However, here we will use it to study some properties of the Grassmannian per se without referring to the more general role this model plays (upon modifying it with Chern-Simons terms and allowing it to be strongly coupled). Later we will add the Chern-Simons term and show that it leads to a Wess-Zumino term in the non-linear model.
Following the standard treatment in the CP N f −1 model, it is straightforward to write down explicitly the two-derivative nonlinear sigma model Lagrangian with target This metric can be derived from the Kähler potential Instead, we will continue to analyze the gauged linear sigma model with large, degenerate negative mass squared for the scalars, such that the model flows to (2.1).

Wess-Zumino Terms and Skyrmions
Here we describe the Wess-Zumino terms that must be added to the nonlinear sigma model Lagrangians.
We start from the simplest Grassmannians, namely the CP N f −1 manifolds, and then briefly describe the more general story. As above, we describe it in terms of a U (1) gauge theory with a gauge field b with N f scalars. In this theory we can add the Chern-Simons where the ellipses denote higher order terms 5 and w is the pull back of the Kähler form of the CP N f −1 manifold to spacetime, normalized such that its integrals over two-cycles Σ 2 For N f > 2 we can substitute (2.6) back in the Lagrangian and find where M 3 is our spacetime and M 4 is a four manifold whose boundary is M 3 . The first term can be interpreted as a Wess-Zumino term in the nonlinear model. We see that on a non-spin manifold N should be even. If N is odd, we need to have a spin c structure with a connection A. In terms of the massless pions the Wess-Zumino term induces interactions such as where the ellipses stand for higher order interaction terms (with three derivatives) that are necessary for SU (N f ) invariance.
The background fields A and B couple in (2.5) to monopole operators of the linear model. The spin of the monopole is N/2 and hence, for even N it is a boson, which couples to an ordinary background gauge field B and for odd N it is a fermion, which couples to a spin c connection A. In the nonlinear model these background fields couple to configurations with nonzero Σ 2 w. If Σ 2 is our space, then these charged objects are Skyrmions. Therefore, the Skyrmions of the model are bosons or fermions depending whether N is even or odd.
Let us consider the baryons of the fermionic SU (N ) theory. They are created by acting with the baryon operator on the vacuum. In terms of the bosonic dual, we can act on the vacuum with the (appropriately dressed) monopole operator. Since the monopole operator is dual to the baryon operator, we get dual excitations in the Hilbert space [60,10]. In our case we can further observe that the monopole operators flows to the Skyrmion operator in the sigma model phase. Therefore, the baryon particles can be identified with the Skyrmions of the non-linear sigma model. 6 As a check, the quantum numbers of these baryon operators match with those of the monopole operators of the bosonic theory and with the Skyrmion operators of the low energy theory. 7 This identification of the Skyrmions with the microscopic baryons and the way their quantum numbers are determined by N through the Wess-Zumino term are exactly as in the similar four-dimensional theory [61,62].
In addition, as in four dimensions, the Wess-Zumino term breaks a symmetry that exists accidentally in the nonlinear-sigma model. For k = 0 it is time-reversal symmetry and for k = 0 it is another discrete symmetry (exchanging the two U (N f /2) factors, which acts as transposition on the matrix of pions).
A very important exception occurs when N f = 2. Clearly, the term ww in (2.8) vanishes then. There is however still a discrete Wess-Zumino action, which can be further extended to non-spin manifolds by allowing A to be a spin c connection. The construction of this discrete term will be discussed in [63]. Note that in [64] a non-local modification of the CP 1 action was described and it was shown to affect the quantum numbers of the Skyrmions. The relation of this to the discrete invariant of [63] will be described there.

Now let us discuss briefly the general case of
, which arises in SU (N ) k gauge theory with N f fermions. The main novelty of the general case compared to CP N f −1 is that the H 4 cohomology 8 is generated by two four-forms and not just one. Therefore, there are a priori two Wess-Zumino terms. It is easiest to describe these two coefficients using the linear sigma model description, where they are the standard U (N f /2 − k) invariants, M 4 T rF ∧ T rF and M 4 T rF ∧ F . In our proposal we take only the latter one with coefficient proportional to N .
Let us finish this discussion with a comment about strings in the sigma model. Such strings exist, if the fundamental group of the target space is non-trivial. Then consider a configuration that is independent of one direction, say x 1 , and wraps the nontrivial cycle in x 2 . The energy density is localized in a small region in x 2 and therefore such a 6 See a related discussion in [59]. 7 The SU (N f ) classical gauge fields already couple to the two-derivative Lagrangian, which is the reason we do not discuss them here. By contrast, the baryon symmetry does not act on the two-derivative Lagrangian. For completeness it is however worth mentioning that the SU (N f ) gauge fields do couple to Skyrmions, since the baryon carries an SU (N f ) representation that is necessarily nontrivial, i.e. with nonzero N f -ality, if gcd(N f , N c ) = 1. 8 For a more precise treatment of this Wess-Zumino term see [65].
configuration is a strings in R 2 . This fundamental group and the corresponding strings are associated with an unbroken one-form global symmetry [66,67]. In the SU (N ) k + N f fermions theory there is no such one-form symmetry at short distances and it is therefore nice to find out that indeed π 1 (M(N f , k)) vanishes.

Flowing down in N f
We now study some mass perturbations of our model. Instead of perturbing the nonlinear model we will perturb the gauged linear model by a small mass term δm 2 ij Φ i Φ j .
One simple perturbation δm 2 ij ∼ δ ij preserves the SU (N f ) global symmetry. It just shifts the mass squared of the scalars, and in the semi-classical regime they would still condense and lead to the Grassmannian (2.1), albeit with a different size. The Grassmannian should therefore remain when a mass term proportional to the identity matrix is added to the Lagrangian. This fact has some ramifications for the mapping of parameters between the Grassmannian Lagrangian, the gauged linear sigma model, and the fermionic theory. In essence, that means that we can mix general mass perturbations, which are in the adjoint representation, with the singlet representation when we map the parameters.
Next, we analyze two mass perturbations that decrease N f to N f − 1. First, consider a mass deformation with the only nonzero entry being δm 2 11 > 0. Then the N f − 1 scalars with degenerate masses would still condense. The scalar Φ 1 is now massive around this configuration. Therefore we obtain the coset . (2.10) Second, let δm 2 11 < 0 and small. Then we condense first Φ 1 and obtain a U (N f /2 − k − 1) gauge theory coupled to N f − 1 scalars. Then, condensing them we obtain the coset .
Clearly, if we started with N f /2 − k = 1 this resulting theory is trivial.
Here we have treated the linear sigma model as a proxy for the nonlinear sigma model Returning to our general proposal, the two mass deformations show that if our proposal about the phase diagram is right for (N, N f , k), it must also be right for (N, N f − 1, k ± 1/2). This fact has two implications that we discuss now.

Constraints on N * (N, k)
We can use (2.12) in order to obtain interesting constraints on N * (N, k).
First, using the same argument as in [52], it is clear that for large enough N f with or without large k the system has only two phases separated by a second order point. For N f ≤ 2k in this range we have a proposed bosonic dual description of this phase transition.
Next, we consider finite N f and look for the Grassmannian phase with 2k < N f < N * .
Assuming we found such a point, requiring that the deformations of the Grassmannian theory by a small mass always agree (as above) in the infrared with deformation of the full theory by a large mass parameter, we get a nontrivial constraint on N * (N, k). We see that if N f < N * (N, k), then it must also be true that N f − 1 < N * (N, k ± 1 2 ), and therefore, for all N, k (with small enough k) Interestingly, the constraint (2.13) implies that the size of the window for the Grassmannian phase, namely, N * − 2k, is necessarily maximized at k = 0 (i.e. the size of the window can only decrease as k is increased). In particular, if there is no symmetry breaking One argument for it is that we can give the fermions a time-reversal invariant mass preserving this unbroken symmetry. We do that by giving N f /2 of them mass +m and N f /2 of them mass −m. This way we can gap the system in a smooth way. Such a deformation was used in [56,57,58] to argue that time reversal and the U (N f /2) ×U (N f /2) symmetry should remain unbroken. Of course, his does not mean that U (N f ) must be broken.
A more abstract point uses the fact that this theory is time-reversal invariant as well as U (N f ) invariant. However, there is a mixed 't Hooft anomaly between these two symmetries. This is just a variant of the standard mixed anomaly between time reversal and fermion number symmetry [5]. For N f ≥ N * this anomaly is represented in the IR theory by a nontrivial conformal field theory. At N f < N * the anomaly can be saturated

SO(N ) k and Sp(N ) k with N f Fermion Flavors
The situation for SO(N ) and Sp(N ) gauge theories is similar to the SU (N ) situation described above. Here we use   Again, we expect some N * (N, k) such that for N f ≥ N * there are only two nonconfining phases with a second order point between them. 10 For 2k < N f < N * we expect three phases with a gapless middle phase describing the spontaneous breaking of the global symmetry. And for N f ≤ 2k the intermediate phase is absent. For SO(N ) it is depicted in fig. 3 and fig. 4, and for Sp(N ) it is depicted in fig. 5 and fig. 6.
All the comments we made in the previous section still apply, but there are a few additional noteworthy facts. The SO(N ) fermionic theory has a Z 2 magnetic global 0form symmetry. Under the duality it is mapped to a charge conjugation symmetrythe symmetry that extends the SO( [4]. In the topological phases with SO(N ) k±N f /2 this symmetry is unbroken.
However, in the Grassmannian phase the magnetic Z 2 symmetry is spontaneously broken. This can be seen most easily using the scalar dual theories, where the Grassmannian can be seen at tree level. Without loss of generality consider the scalar theory with gauge group SO( N f 2 − k). In the Grassmannian phase the scalars have an expectation value, which, up to symmetry transformations, is of the form φ ia = xδ ia where i labels the colors and a labels the flavors, and x is nonzero. This breaks the O(N f ) global symmetry to where the first factor is a diagonal subgroup between a subgroup of the global symmetry group and the gauge group. This leads to the vacuum manifold Alternatively, we can parameterize the vacuum manifold by a gauge invariant order parameter of the form φφ. It lives iñ The true target space V is a double cover of this space because every point inṼ corresponds to two different points in V, which differ by the expectation value of a gauge invariant baryon operator of the form φφ...φ, where the color indices are contracted using 10 The value of N * (N, k) can depend on the gauge group. We use the same notation as in the SU (N ) discussion above and suppress the dependence on the gauge group.
an epsilon symbol. This order parameter is odd under the charge conjugation symmetry, thus establishing that this symmetry is spontaneously broken. Using the duality it is mapped to a monopole operator of the underlying SO(N ) gauge theory. Its expectation value shows that the magnetic symmetry of the fermionic SU (N ) theory is spontaneously broken.
We see that the magnetic symmetry of the fermionic theory is unbroken in the topological phase and it is broken in the Grassmannian phase. This is consistent with the intuitive picture that the topological phases are not confining, but the Grassmannian phase is confining (we typically think of a phase with broken magnetic symmetry as confining). A more precise way to see that uses one-form global symmetries [66,67]. None of our theories have such a symmetry. This follows from the fact that the gauge group acts faithfully on the matter fields. In particular, the space (3.2) is simply connected, which is encouraging, since that means that there are no stable strings in the macroscopic theory. However, we can change the SO(N ) theories and replace them by Spin(N ) theories (by gauging the discrete 0-form Z 2 magnetic symmetry, as described in the introduction) and then these theories have a "bonus" Z 2 one-form global symmetry. The charged objects under this one-form symmetry are Wilson lines in the spinor representation of Spin(N ). They can be used to diagnose confinement rigorously.
In order to change the gauge group to Spin(N ) we turn the global Z 2 magnetic symmetry of the SO(N ) theories to a gauge symmetry. Then the topological phases are Spin(N ) k±N/2 and the existence of nontrivial spinor Wilson lines in these phases confirms our assertion that these phases are not confining.
The Grassmannian phase is more interesting. Again, we analyze it using the two dual scalar theories. Here the Z 2 magnetic symmetry of the fermionic theory acts as charge conjugation [4] and correspondingly the gauge group is O(N f /2 ± k). 11 The effect of this extension of the gauge group on the Grassmannian is to mod it out by Z 2 turning it from V of (3.2) toṼ = V/Z 2 of (3.3). The resulting space is not simply connected π 1 (Ṽ) = Z 2 . 12 We see that the underlying Z 2 one-from global symmetry, which is associated with the center of Spin(N ), is realized as a one-form global symmetry in the macroscopic theory. 11 For a more detailed discussion see [68]. 12 The fact that the moduli space of vacua in the SO(N ), Spin(N ), and O(N ) theories are related by certain quotients and its relation to duality has already been noted in supersymmetric theories in [69,36].
Because of this nontrivial π 1 the macroscopic theory has stable Z 2 strings. These are configurations of the scalars that are independent of one spatial direction and the other spatial direction winds around the non-contractible cycle in the target space. These charged strings are interpreted as the Z 2 confining strings of the microscopic theory. We see that in the Grassmannian phase the one-form global symmetry is unbroken and as described in [67], this means that the theory is confining. A similar comment in a closely related four-dimensional context was made in [61,62].
In general, when we gauge an electric one-form symmetry we obtain in the new gauge theory a bonus magnetic zero-form symmetry. The reverse process was described in the introduction. If the electric one-form symmetry is unbroken, the theory confines. In An interesting exception occurs when one or both of the factors in the denominator of (3.2) is SO (2). In that case π 2 (V) has one or more factors of Z. For example, for N f = 4, In this case the Grassmannian theory should be supplemented with Skyrmion operators that allow Skyrmions to decay, making the global Skyrmion number only Z 2 . In the case where the bosonic dual has a SO(2) = U (1) gauge symmetry, this enhanced π 2 is associated with an enhanced U (1) T magnetic symmetry. In this case this symmetry should be explicitly broken to Z 2 by adding monopole operators of charge two to the Lagrangian in order to preserve the duality (as was done recently in other models in [55,70]). These operators have a negligible effect in the Higgs phase, except that they allow Skyrmions to decay, while preserving the remaining Z 2 charge. However, these operators can become important, if their dimension becomes relevant in the IR.
In summary, in the generic case π 2 (V) = Z 2 is nicely consistent with the expectations from the fermionic theory. In the special cases where N f /2−k = 2 or N f /2+k = 2 we have to add to the bosonic description even monopole operators. Analogously, the nonlinear sigma model Lagrangian needs to be modified by adding monopole-like operators, which would allow even Skyrmions to decay to the vacuum.
Finally, as in the SU (N ) theories, also here the Grassmannian (3.2) has to be accompanied by certain Wess-Zumino terms that are responsible for the quantum numbers of Skyrmions. We do not describe them in detail here, except to note that they follow from the Chern-Simons terms in the bosonic gauged linear models.

A Comment about SO(2)
In the discussion of fig. 4 we restricted ourselves to N > 2. In fact, our results nicely apply to N = 2 when properly interpreted. Here we would like to explain how this comes about. As we will see, not only is our general picture correct also for N = 2, this case is actually on stronger footing than the more general case and thus supports our general picture.
First, let us explain why we restricted our discussion to N > 2. In the case of SO(2) ∼ = U (1) gauge theory with N f flavors, the symmetry of the massless theory is locally SU (N f ) × U (1), where the first factor is the flavor symmetry and the second is the magnetic symmetry. This differs from the SO(N ) series with N > 2, where the magnetic symmetry is discrete (it is Z 2 ) and the flavor symmetry is SO(N f ).
For example, let us consider an SO(2) 0 ∼ = U (1) 0 gauge theory with N f = 2 fermions in the fundamental representation, i.e. two fermions of charge 1, ψ i . It was suggested in [71] and clarified in [3] that this theory enjoys self-duality; i.e. it is dual to another U (1) 0 theory with two fermions χ I . Unlike the original fermions ψ i , they are labelled by an upper case I, because they are acted upon by a different SU (2) global symmetry. The precise global symmetry, its 't Hooft anomaly, and the deformations of the theory were analyzed in detail in [18]. We will summarize below the facts that are necessary for our discussion.
When the fermions are massless (hence time reversal symmetry is not explicitly broken) and no monopole operators are added to the Lagrangian the model flows to a conformal field theory with an enhanced O(4) symmetry. This conformal field theory has several notable operators. First, an SU (2) invariant fermion bilinear in the UV (the superscript 3 will be explained shortly). In the IR it combines with a double-monopole and its conjugate In the dual description we have two fermions χ I and the flavor SU (2) that mixes them is identified in the IR with the enhanced SU (2) of the original theory. The singlet (3.4) is identified as χ I χ † I , the monopole and its conjugate are identified with the SO(4) vector. And most interestingly, the double-monopole (3.6) and the fermion bilinear (3.5) are mapped nontrivially Our analysis of the phases of the general SO(N ) theories can be naturally continued all the way to SO(2) if we properly deform the SO(2) theory such that the UV symmetries are as in the general case. This can be achieved by adding the charge-two monopole operator and its conjugate, say to the Lagrangian. (The superscripts 3, 1 where chosen without loss of generality.) This has the effect of explicitly breaking the flavor SU (2) to SO(2) and the magnetic U (1) to Z 2 , such that the UV symmetries are as in a generic SO(N ) theory in fig. 4. In the UV this operator has a very large dimension, but it becomes relevant in the IR. The duality allows us to analyze its effect in the IR. This is achieved because in the dual description this operator is a fermion bilinear. Up to an SU (2) flavor rotation of the dual theory it is [18] O 0,3 3 = χ 1 χ † 1 − χ 2 χ † 2 . (3.9) Before adding the monopole operator to the Lagrangian the UV U (2) global symmetry was enhanced in the IR to SO(4) (we suppress discrete factors). After adding the monopole operator to the Lagrangian the UV symmetry is explicitly broken to U (1) × Z 2 ⊂ U (2) and it is enhanced in the IR to U (1) × U (1) (and we again suppress some discrete factors) [18].
Adding the perturbation (3.9) in the dual theory we see that the two fermions are massive and their mass has opposite sign. As a result, when they are integrated out we remain with a U (1) gauge field and no Chern-Simons term. It is dual to a compact scalar and represents the spontaneous breaking of the magnetic U (1) in the dual theory. Using the duality map this corresponds to the spontaneous breaking of the original SO(2) flavor symmetry that remains in the presence of the monopole operator [18]. This is precisely as implied by fig. 4 upon setting N = 2 and N f = 2.
We can further consider the phase transitions in fig. 4. The horizontal axis corresponds to a deformation by the singlet mass term (3.4), which is identified as χ I χ † I . We should add the operator (3.4) in conjunction with (3.8). This means that for some value of the singlet mass χ 1 is massless and at another value χ 2 is massless. Around these two points we have U (1) 1 2 with N f = 1, which is dual to the O(2) Wilson-Fisher theory. This is exactly as predicted by the dualities in fig. 4. This conclusion is quite satisfying because of the following reason. As explained in [3], the self-duality of U (1) 0 with N f = 2 follows from the web of dualities of [2], which is supported by a lot of evidence. So this is not an additional assumption. Then, the analysis of the deformation by the double-monopole operator, which proceeds via this duality also follows from the same assumption. This means that at least for SO (2) with N f = 2 the intermediate phase scenario that we have been advocating throughout this paper follows logically from the web of dualities of [2] without further assumptions! A similar treatment applies to the U (1) 1 2 with a N f = 1 theory. This model again has a U (1) magnetic symmetry, which is not present in the general SO(N ) case. We thus interpret our phase diagram as being the result of adding a double-monopole to the Lagrangian. Again, we analyze the effect of the double-monopole operator using duality.
The model SO(2) 1/2 +ψ without monopole operators in the Lagrangian is dual to the O(2) Wilson-Fisher fixed point with a complex scalar Φ. The double-monopole perturbation is translated in this theory to Φ 2 + Φ 2 . Now, as we dial the coefficient of the invariant operator |Φ| 2 we encounter one Ising point (where one scalar is massless and the other has positive mass squared), exactly as in fig. 3 if one substitutes N = 2, k = 1 2 , N f = 1. We expect that for large enough N f this double-monopole operator is irrelevant in the IR fixed point and therefore it cannot split it as above. Again, this is consistent with our general picture. Conversely, a microscopic lattice model of these systems might not have the global magnetic U (1) symmetry. For large enough N f we recover the U (1) magnetic symmetry in the IR and the IR fixed point is not split. But for smaller values of N f , where this interaction is relevant in the IR, the fixed point is split and we find some global symmetry breaking there. This could explain why people who studied microscopic lattice models of this system, which implicitly include the double-monopole operator in the Lagrangian found global symmetry breaking for low values of N f .