Comments on Global Symmetries, Anomalies, and Duality in (2+1)d

We analyze in detail the global symmetries of various (2+1)d quantum field theories and couple them to classical background gauge fields. A proper identification of the global symmetries allows us to consider all non-trivial bundles of those background fields, thus finding more subtle observables. The global symmetries exhibit interesting 't Hooft anomalies. These allow us to constrain the IR behavior of the theories and provide powerful constraints on conjectured dualities.

The main boson/fermion dualities that we will study are [35,39] conjectured to hold for N f ≤ N (our notation is U (N ) k ≡ U (N ) k,k ), and [45] SO(N ) k with N f real scalars ←→ SO(k) −N+ (1. 3) The first duality is the celebrated particle/vortex duality of [1,2]. The second duality maps an interacting bosonic theory to a free fermion [38]. The theory in the third duality has a 1 See [48] for some recent tests. 2 We will follow the notation and conventions of [49,38,39,45] and will not repeat them here.  quantum SO(3) global symmetry [45] (see fig. 1). In all these cases the monopole operator of U (1) k in the theory on the left side of the duality, whose spin is k 2 , is an important operator in the theory on the right side. It is the scalar in the first case, it is the free fermion in the second case, and it is the new current of the enhanced SO(3) symmetry in the third case. 3 3 One might wonder whether the theory of U (1) 3 with a scalar, which has a monopole operator of spin 3 2 and a global U (1) symmetry, could have N = 2 supersymmetry in the IR. It has a dual description as SO(3) − 3 2 ∼ = SU (2) −3 /Z 2 with a fermion in the adjoint [45], which seems to have N = 1 supersymmetry. However, this supersymmetric theory is expected to be gapped [50] with a low energy SO(3) −1 ∼ = SU (2) −2 /Z 2 trivial TQFT. As we vary the fermion mass, we can find a transition to another gapped phase with a TQFT SO(3) −2 ∼ = SU (2) −4 /Z 2 ↔ SU (3) −1 ↔ U (1) 3 .
The duality statement could mean that the theory at this transition point is dual to the U (1) 3 theory with a scalar. However, since we needed to change the fermion mass from the supersymmetric point, we broke supersymmetry explicitly and there is no reason to believe that the IR All these dualities are IR dualities. We start at short distances with a renormalizable Lagrangian and impose some global symmetry on its terms. Then, we scan the relevant deformations that are consistent with the global symmetry. These are typically mass terms, but there are also others. For generic values of these parameters the low-energy theory is gapped. As these parameters are varied there could be phase transitions between different phases and the phase transition points occur at fine-tuned values of the scanned parameters. We will assume that, as we vary these parameters, the phase transitions can be second order. Then the long-distance physics is described by a fixed point of the renormalization group, which is a continuum conformal field theory. The statement of the IR duality is about this fixed point and its neighborhood. If, on the other hand, the IR theory is always gapped with possible first-order transitions between phases, the statement of the duality is significantly weaker and it applies only to the gapped phases.

Global symmetries
Our starting point is to identify the correct global symmetry of a quantum field theory.
For the moment we ignore discrete symmetries like time reversal T and higher-form global symmetries [51,52]. We will discuss them later.
We should distinguish between the global symmetry of the UV theory G UV and the global symmetry of the IR theory G IR . Although there might be elements in G UV that do not act on the IR degrees of freedom, we should still pay attention to them in the IR.
The IR effective action might contain topological local counterterms for background gauge fields coupled to those elements.
Conversely, there could be new elements in G IR that are not present in the UV. These lead to an accidental or quantum symmetry in the IR. These symmetries are approximate and are violated by higher-dimension operators in the IR theory. Examples of such quantum symmetries are common in (1 + 1)d field theories and have played an important role in supersymmetric dualities, in particular in (2 + 1)d mirror symmetry [7].
A noteworthy simple example [45] is summarized in fig. 1, where four different UV theories, some with G UV = O(2) and some with G UV = SO(3), flow to the very same IR fixed point with SO(3) global symmetry (we will discuss this example in Section 3). theory is actually gapless, it could be dual to the U (1) 3 theory with a scalar, in which case it will also have enhanced N = 2 supersymmetry. These considerations are extremely important in the context of duality. Two dual theories T A and T B that flow to the same IR fixed point must have the same global symmetries. In some cases the UV symmetries are the same G UV A = G UV B . But it is also common that the UV symmetries are different G UV A = G UV B , and yet they are enhanced to the same IR symmetry G IR . Again, the example in fig. 1 demonstrates it and gives interesting consistency checks on the various dualities. We will see several examples of that in Section 3.
When we discuss the global symmetry G (either G UV or G IR ) we should make sure that it acts faithfully on the operators. Specifically, we will see many examples where all the local gauge-invariant operators in the theory transform in certain representations of the naive global symmetry group G naive , but the true global symmetry G-which acts faithfully-is a quotient G = G naive /C by an appropriate C.
A key tool in the analysis of a quantum field theory is its coupling to background gauge fields for the global symmetry. If we misidentify the global symmetry and couple the system to background G naive gauge fields, we miss important observables. In particular, if all the local operators transform trivially under C ⊂ G naive we can couple the system to G = G naive /C bundles, which are not G bundles.
For example, consider the SU (2) 1 theory with a scalar in fig. 1. The naive global symmetry is G naive = SU (2). However, in this case all gauge-invariant operators in the theory are in integer isospin representations of this group and therefore the true global symmetry is G = G naive /Z 2 = SO(3). This means that the system can be coupled to additional background fields-SO(3) gauge fields, which are not SU (2) gauge fields. The response to such more subtle backgrounds leads to interesting observables, which give us more diagnostics of the theory.
More explicitly, we can couple the matter fields to gauge fields. This is consistent because the matter fields do not sense the Z 2 quotient. Below we will see many generalizations of this example. We will encounter dynamical fields b for a gauge group G dyn and background fields B for the true global symmetry of the model G = G naive /C for some C. As in the example (1.4), the dynamical and classical fields can be combined to a gauge field B with group If the classical fields B are in G naive bundles, the dynamical fields b are in G dyn bundles.
But when B are in nontrivial G = G naive /C bundles, also the dynamical fields b are in G dyn /C bundles rather than in G dyn bundles. The consistency of the theory under gauge transformations in (1.5) and possible anomalies in these transformations will be extremely important below.
We should point out that the authors of [53,54] have examined such anomalies for discrete groups from a different perspective.
We will be particularly interested in the theories in (1.1) and (1.2), so let us discuss their UV symmetry G UV . In the fermionic case that is the actual UV symmetry of the theory, while in the bosonic case that is the symmetry that we impose on the quartic potential. The naive UV global symmetry G naive is 4 U (N f ) ⋊ Z C 2 (where the second factor is charge conjugation) in the theories with SU gauge group, SU (N f ) × U (1) M ⋊ Z C 2 (where the second factor is the magnetic symmetry) for U gauge group, O(N f ) × Z M 2 × Z C 2 for SO gauge group, 5 and U Sp(2N f ) for U Sp gauge group. However, we will find that the faithfully-acting symmetry G UV is (we will not discuss the SO case here): where r is an integer in the theory with scalars and an integer plus N f 2 in the theory with fermions. Here by U (N f )/Z N we mean the quotient by e 2πi/N 1I. In the special 4 In this discussion we mostly neglect time-reversal symmetry T. 5 In this case Z C 2 is an element of O(N ) not connected to the identity. When N, N f are both odd, Z C 2 is already contained in O(N f ) (up to a gauge transformation), and should not be listed as an independent symmetry. cases of U (N ) 0 with N f scalars and U (k) N f 2 with N f fermions, the global symmetry is One should be careful at small values of the ranks. For instance, SU (2) r with N f fermions has U Sp(2N f )/Z 2 symmetry as manifest in the U Sp(2) r description, while the symmetry of SU (2) r with N f scalars depends on what we impose on the quartic potential. This will be analyzed in Section 3.

Anomalies
It is often the case that the global symmetry G has 't Hooft anomalies. This means that the correlation functions at separated points are G invariant, but the contact terms in correlation functions cannot be taken to be G invariant. Related to that is the fact that the system with nonzero background gauge fields for G is not invariant under G gauge transformations. Often, this lack of G gauge invariance of background fields can be avoided by coupling the system to a higher-dimensional bulk theory with appropriate bulk terms.
Let us discuss it more explicitly. Since we denote the classical gauge fields by uppercase letters, A, B, etc., we will denote the coefficients of their Chern-Simons counterterms [55,56] by K. 7 They should be distinguished from the Chern-Simons coefficients of dynamical fields a, b, etc., which we denote by lower case k. It is important that k and K should be properly normalized as (2 + 1)d terms. As we will see below, it is often the case that the proper normalization of these coefficients involves a nontrivial relation between K and k.
It might happen that imposing the entire symmetry G there is no consistent value of K. In that case we say that G has an 't Hooft anomaly and we have two options. First, we consider only a subgroup or a multiple cover of G and turn on background fields only for that group. Alternatively, we allow gauge fields for the entire global symmetry group G, but extend them to a (3 + 1)d bulk. In this case the partition function has a dependence See also footnote 10. 7 We will use uppercase N in the gauge group of dynamical fields and Chern-Simons levels of dynamical fields depending on N and N f . We hope that this will not cause confusion.
on how the background fields are extended to the bulk. It is important, however, that the dynamical gauge fields are not extended to the bulk.
We will not present a general analysis of such anomalies. Instead, we will first mention two well known examples. Then make some general comments, and later in the body of the paper we will discuss more sophisticated examples.
A well known typical example in which we can preserve only a subgroup G ⊂ G is the time-reversal anomaly of (2 + 1)d free fermions. Here G includes a global U (1) symmetry and time reversal, but they have a mixed anomaly. One common option is to preserve G = U (1), but not time reversal. Alternatively, in the topological insulator we extend the background U (1) gauge field to the bulk and we turn on a (3 + 1)d θ-parameter equal to π [57,58], such that the entire global symmetry G is preserved.
In this case the bulk term with θ = π is time-reversal invariant on a closed fourmanifold, but not when the manifold has a boundary: a time-reversal transformation shifts the Lagrangian by a U (1) 1 Chern-Simons term. This is an anomaly in time reversal.
The fermion theory on the boundary has exactly the opposite anomaly, such that they cancel each other and the combined (3 + 1)d theory is anomaly free.
Another well known example, where we can preserve a multiple cover G naive of the global symmetry G, is the following. Consider a quantum mechanical particle moving on S 2 with a Wess-Zumino term with coefficient k. (This is the problem of a charged particle on S 2 with magnetic flux k.) The global symmetry of the problem is G = SO(3), but as we will soon review, for odd k this symmetry is anomalous.
One way to represent the theory uses two complex degrees of freedom z i with a potential forcing z 1 2 + z 2 2 = 1. This system has an O(4) global symmetry. Next we introduce a dynamical U (1) gauge field b coupled to the phase rotation of z i . The resulting theory is the CP 1 model whose target space is a sphere. We can add to the theory the analog of a Chern-Simons term, which is simply a coupling kb. In terms of the effective CP 1 model this is a Wess-Zumino term with coefficient k [59]. The spectrum of the theory is well known: it is ⊕ j H j , where H j is the isospin j representation of SU (2) and the sum over j runs over j = k 2 , k 2 + 1, ... Naively, the global symmetry is G naive = SU (2) which rotates z i . However, the global symmetry that acts faithfully is G = SU (2)/Z 2 ∼ = SO (3). To see that, note that the coordinates z i are coupled to a U (1) gauge field b, can be further coupled to an SU (2) classical field B, but then b and B combine into a gauge field B. For even k the Hilbert space includes integer j representations and represents SO (3) faithfully. In this case there is no anomaly. But for odd k all the states in the Hilbert space have half-integer j and the global symmetry acts projectively-it represents the double cover G naive .
What should we do about this anomaly? One option is to say that the global SO(3) symmetry acts projectively, or equivalently, the global symmetry is SU (2). A more interesting option is to introduce a (1 + 1)d bulk M 2 (with boundary the original timeline), and add a bulk term that depends on the SO(3) gauge field B.
Explicitly, the original degrees of freedom z i couple to a U (2) gauge field B. Therefore, the CS term kb should be written as k 2 Tr B. Although this is properly normalized as a CS term for a U (1) gauge field b, for odd k it is not properly normalized for a U (2) gauge field The perspective on this phenomenon that we will use below is the following. The boundary theory-in this example a particle in the background of an odd-charge magnetic monopole-is anomalous and its action is not well-defined in (0 + 1)d in the presence of G depends on additional data. It is anomalous. This anomaly is exactly canceled by the anomaly in the boundary theory, such that the combined system is well defined.
Below we will see higher-dimensional generalizations of these examples. Using the notation discussed around (1.5), the dynamical fields b will typically have Chern-Simons couplings k while the background fields B will have Chern-Simons couplings K. In addition, for U (1) factors in the two groups there can be mixed Chern-Simons couplings. The way to properly define these couplings is by writing them as (3 + 1)d bulk terms of the form θ Tr F B ∧ F B or θ Tr F B ∧ Tr F B with various θ's, where F B are the field strengths of the gauge fields B. In addition, we will also encounter discrete θ-parameters, like those in [60]. In this form we have a well defined expression for gauge fields of (G dyn × G naive )/C.
As in the quantum mechanical example of a particle on S 2 , it is crucial that these bulk terms must be independent of the bulk values of b at fixed B. This guarantees that b is a dynamical field living on the boundary. If the bulk terms are also independent of the bulk values of B, we say that the global symmetry G is anomaly free. Instead, if there is a dependence on the bulk values of B, the global symmetry suffers from 't Hooft anomalies.  Some of these characteristic classes are related to various discrete θ-parameters. We have already seen such a discrete θ-parameter in (1.10). Below we will encounter the discrete θ-parameter of [60], which is associated with the Pontryagin square operation P(w 2 ) [61,62]. As in [51,52], these can be represented by a two-form field B with a (3 + 1)d coupling B ∧ B. This coupling is gauge invariant on a manifold without boundary. But when a boundary is present, this term has an anomaly. The anomaly is canceled by having an appropriate boundary theory, which has the opposite anomaly. For a (0 + 1)d boundary we have already seen that around (1.10), while below we will see examples with a (2 + 1)d boundary.
It is well known that in (3 + 1) dimensions, 't Hooft anomaly matching conditions lead to powerful consistency constraints on the IR behavior of a theory and on its possible dual descriptions. Consider first the simpler case of G UV A = G UV B . Then the 't Hooft anomaly, which is the obstruction on the theory to be purely (2 + 1)-dimensional, must be the same on the two sides of the duality. In other words, if we need to couple the theory to a (3 +1)d bulk and add some bulk terms with coefficients θ, these bulk terms should be the same in the two dual theories. Such θ-parameters can be ordinary or discrete ones. More precisely, θ should be the same, but the boundary counterterms K A and K B in the two theories can be different, provided they are properly quantized. This condition is the same as the celebrated 't Hooft anomaly matching.
In the more interesting case that G UV A = G UV B , we can use the constraint in the UV by coupling background fields to the common subgroup G UV A G UV B . Their θ must be the same on the two sides of the duality. The IR theory can then be coupled to G IR A = G IR B gauge fields and this analysis also allows us to determine the value of θ for these fields.
Again, we will see examples of that below.

Outline
In Section 2 we check 't Hooft anomaly matching in the dualities (1.1)-(1.2). This is both an example of our methods and a nontrivial new test of those dualities.
In Section 3 we focus on some interesting special cases of the dualities with gauge group U (1) ∼ = SO(2) and SU (2) ∼ = U Sp (2), either in the fermionic or the bosonic side.
Such theories participate in more than one duality in (1.1)-(1.2). This leads to new tests of the dualities and to deeper insights into their dynamics. We also use those special cases to analyze theories with a surprising quantum SO(3) global symmetry in the IR, as in fig. 1.
In Section 4 we follow [46,39] and consider in detail a fermion/fermion duality that leads to an enhanced O(4) global symmetry. We extend previous discussions of this system by paying close attention to the global structure of the global symmetry and to the counterterms. This allows us to find the precise anomaly in O(4) and time-reversal, and to restore those symmetries by adding appropriate bulk terms.
In Section 5 we analyze the phase diagram of systems with global SO(5) symmetry and clarify some possible confusions about various fixed points with that global symmetry.
Appendix A derives the induced Wess-Zumino term in the model of Section 5, while Appendix B describes carefully the duality of [47] paying attention to the proper quantization of CS couplings, to the spin/charge relation, to the global structure of the symmetry group, and to the bulk terms. In Appendix C we discuss more examples of 't Hooft anomalies.

't Hooft Anomalies and Matching
We start by determining the 't Hooft anomalies in the following theories: The dualities are valid only for N f ≤ N , but we will determine the symmetries and anomalies for generic integer values of N , k, N f . Here and in the following, to be concise, we indicate complex scalars as Φ, real scalars as φ, complex fermions as Ψ and real fermions as ψ.

All four theories have a naive global symmetry
where the last factor is charge conjugation. In the theories with SU gauge group, the first two factors combine into a manifest U (N f ) acting on the scalars or fermions. In the theories with U gauge group, SU (N f ) acts on the scalars or fermions, while the Abelian factor is the magnetic U (1) M , whose charge is the monopole number. However the faithfully-acting symmetry G is a quotient thereof, which as we will soon see is U (N f )/Z N ⋊ Z C 2 in the first line of (2.1) and U (N f )/Z k ⋊ Z C 2 in the second line, as summarized in (1.6). 9 For N f ≤ N this is a check of the dualities.
There might be an obstruction-an 't Hooft anomaly-to turning on background gauge fields for G. We will show that the obstruction is the same on the two sides of the dualities, thus providing a nontrivial check of them.

Global symmetry
The first step is to identify the global symmetry that acts faithfully on the four theories in (2.1). To do that, we analyze the local gauge-invariant operators.
Let us start with SU (N ) k with N f scalars. There is a U (N f ) symmetry that acts on the scalars in the fundamental representation, but only U (N f )/Z N acts faithfully on gauge invariants. In the absence of a magnetic symmetry, monopole operators do not change this result (since GNO flux configurations [63] are continuously connected to the vacuum).
There is also a charge-conjugation symmetry Z C 2 that exchanges the fundamental with the antifundamental representation, therefore the symmetry is  [64]. 10 Charge conjugation acts 9 In the special cases of By the same argument, U (N ) k with N f scalars has a faithfully-acting symmetry

Background fields
Now we turn on a background for the SU (N f ) × U (1) symmetry of the four theories in (2.1), which can always be done, and analyze under what conditions the background gauge fields can be extended to Consider SU (N ) k with N f scalars. Turning on background gauge fields with generic CS counterterms we obtain the theory The Z N quotient acts anti-diagonally on SU (N ) and the Abelian factor by a phase rotation e 2πi/N , while Z N f acts anti-diagonally on SU (N f ) and the Abelian factor by e 2πi/N f . The quantization conditions on CS counterterms are 3) The first condition comes from the SU (N f ) factor. The second and third conditions come from the separate quotients by Z N and Z N f , respectively. The last condition ensures that the generators of Z N and Z N f have trivial braiding and one can take the simultaneous quotient.
The equations in (2.3) have solutions in L, J, if and only if k = 0 mod gcd(N, N f ). If this is not the case, there is an 't Hooft anomaly and the theory with background is not consistent in (2 + 1)d. One can make sense of the theory on the boundary of a (3 + 1)d bulk, but then there is an unavoidable dependence on how the classical background fields are extended to the bulk. We will express the anomaly below.
We stress that the magnetic U (1) is coupled to U (k) by a mixed CS term. The quotient by Z N f acts on SU (N f ) and the two Abelian factors. We have chosen to parametrize the CS counterterms in a way that matches the dual description (2.2) when the duality is valid.
Then the topological symmetry U (1) K f has CS counterterm 11 To see that, we mass deform the scalar theory by ±|Φ| 2 and the fermionic theory by ∓ΨΨ. The two resulting topological theories are identified, exploiting level-rank duality on the dynamical fields. 12 The map of CS counterterms for the U (1) global symmetry was already discussed in [39].
consider the Abelian factors: where we have indicated asâ1I k the Abelian factor in U (k). The equations of motion are as follows (neglecting the matter contribution): We are after a Z N f one-form symmetry-then the matter contribution is canceled by a rotation in the center of SU (N f ). An integer linear combination of the equations in (2.6) gives N f k dâ + J N dB = 0, which describes a Z N f one-form symmetry, if and only if J ∈ N N f Z. This reproduces the fourth condition in (2.3). The generator of the one-form symmetry is the line To perform the Z N f quotient in the fermionic theory we combine The Z N f quotient is well-defined if its generator has integer or half-integer total spin, It can be interpreted as the Z N quotient of U (1) J −N k . 12 We cannot use level-rank duality on the background fields, which are not integrated over in the path-integral.
which reproduces the third condition in (2.3). Thus the 't Hooft anomaly is the same on the two sides of the duality (2.1).
The discussion in the other two cases is similar.
fermions first. Turning on background gauge fields we have Taking into account the bare CS levels, the quantization conditions are Next consider U (N ) k with N f scalars. With background gauge fields we have The CS counterterms are chosen to match with those in (2.9) when the duality is valid, The SU (N f ) and U (1) factors give the quantization conditions L ∈ Z and K s ∈ Z, respectively. An integer linear combination of the equations of motion for the This has to be combined with the generator in SU (N f ) L , and the condition that the total spin be in 1 2 Z reproduces the third condition in (2.10). Thus, all conditions in (2.10) are reproduced and the anomaly matches across the duality.
We should emphasize again that if we are only interested in the naive global symmetry , which does not act faithfully, there is no problem turning on background gauge fields. The issue is only in considering gauge fields of the quotient group. In that case we can attach the system to a bulk, extend the fields to the bulk and replace the Chern-Simons terms by F ∧ F type terms there. Then the point is that the resulting theory depends on the extension. From this perspective, the 't Hooft anomaly matching is the statement that we can use the same bulk with the same background fields there and attach to it either of the two dual theories on the boundary.
Consider the theory SU (N ) k with N f scalars in (2.2). To express the dependence on the bulk fields, we proceed as follows. A U (N f )/Z N bundle can be represented by two correlated bundles, P SU (N f ) and for some class w 2 . Therefore the dependence on the bulk fields is completely fixed by the classical U (N f )/Z N background. Such a dependence is described by The integral is on a closed spin four-manifold M 4 , and P is the Pontryagin square operation [61,62] such that P(w (for more details see [60] and references therein). We say that e iS anom captures the phase dependence of the partition function on the bulk extension of the U (N f )/Z N bundle, in the sense that given two different extensions one can glue them into a closed manifold M 4 and then e iS anom is the relative phase of the two partition functions.
If we choose J ∈ DZ, then we can substitute the square of (2.12) into (2.13) to which is well-defined modulo 2π. From this expression it is clear that if we can solve the constraints in (2.3), then e iS anom = 1 and there is no anomaly. On the other hand, it is always possible to make a suitable choice of L, J such that S anom reduces to (2.15) 13 If J ∈ DZ then (2.13) contains more information than w (N ) 2 and w We can regard this as a minimal expression for the anomaly.
As we have shown, the anomaly in U (k) −N+ N f 2 with N f fermions is the same as in (2.13). However one has to remember that the U (1) in (2.13) is an N -fold multiple cover of U (1) M . The special case N = 0 is discussed in Appendix C. The other two cases are similar, with an obvious substitution of parameters, and are presented in Appendix C.
Although we checked the anomaly matching separately for the two dualities, in fact they are related by performing S, T operations on the U (1) symmetry [65,39]. Since the operations add equal terms on both sides, the change in the bulk dependence on both sides must be equal, and thus the anomaly must still match. The anomaly also matches for other dualities obtained from them by S, T operations, such as the last two dualities in (1.1).
In general, the anomaly is characterized by bulk terms that are meaningful on closed manifolds, but anomalous when there is a boundary. 14 This is true for the anomaly (2.13) where P(w 2 ) is meaningful only on a closed manifold, and it is also true for the two examples discussed in Section 1.2.
Although we do not need it for the dualities, it is nice to demonstrate our general analysis of the anomaly by specializing it to a U (1) gauge theory of scalars with k = 0.
Ignoring charge conjugation, the global symmetry is More precisely, b satisfies N f b = Tr B. Therefore, the coupling to the magnetic U (1) M background field B M is the ill-defined expression 1 2πN f (Tr B)dB M that needs to be moved to the bulk. This highlights that the global symmetry suffers from 't Hooft anomalies, which are characterized by the bulk term This discussion is analogous to a similar example in [66]. See Appendix C for more details.

Symplectic gauge group
We conclude this section by briefly analyzing the 't Hooft anomalies in the two theories Again, the dualities are valid only for N f ≤ N , but we will study these theories for generic integer values of N, k, N f . Since there is no magnetic symmetry, the faithfullyacting symmetry G is the one acting on gauge invariants constructed out of the scalars or fermions, which is U Sp(2N f )/Z 2 in both cases.
Coupling the two theories to a generic background, we obtain Recall that the scalars and fermions are in a pseudo-real representation, therefore they are subject to a symplectic reality condition. The CS counterterms are chosen in such a way that they match when the theories are dual. The quantization conditions are together with L ∈ Z in both theories. This provides 't Hooft anomaly matching for the duality [45].
When N k is odd and N f is even, (2.19) cannot be solved and we have an 't Hooft anomaly. The anomaly is captured by the bulk term

Quantum Global Symmetries from Special Dualities
Infrared dualities provide alternative descriptions of the same IR physics. It might happen that one description, say T A , makes a symmetry transformation manifest all along its RG flow, while the same symmetry is not present in the other description, say T B . Then, duality predicts that T B develops the symmetry quantum mechanically in the IR, because of strong coupling. In this section we survey various dualities at our disposal [35,38,39,45] and examine in what cases they predict a quantum enhancement of the global symmetry in the IR.
The theories we consider have N f scalars or fermions in the fundamental representation. For gauge group U they have a naive global symmetry For scalar theories there are two subtleties to take into account. First, when using these theories in dualities N f is restricted (N f ≤ N in SU/U and U Sp dualities, while Second, in the scalar theories we turn on quartic couplings in the UV and we must analyze their global symmetries. However, this theory does not participate in the SU/U dualities. The SO dualities use this theory for k > 2, but they require only SO(2) ⊂ SU (2) invariance (in addition to the U (1) M global symmetry). There are two quartic couplings that respect that symmetry, symmetry, the previous one and Φ ai Φ † aj Φ bj Φ † bi . These two couplings are assumed to be present in the theories with SU/U duals.
In the following, we analyze in detail these low-rank cases.
The SU/U duality requires N f = 1. Therefore consider N f = 1 and k ≥ 2. There is only one quartic term in the U (1) k ∼ = SO(2) k scalar theory and the following fixed points are all the same: In the generic case the fixed point has U (1) ⋊ Z C 2 ∼ = O(2) symmetry, which is a quantum symmetry in the fermionic SO(k) − 3 2 theory. In the special case k = 2 the symmetry becomes SO (3), which is visible in the SU (2) − 1 2 fermionic theory while it is a quantum symmetry in all other descriptions. This case is precisely the one in fig. 1, indeed the scalar theory is the third example in (1.3).

U
Then the following fixed points coincide: In the generic case there is a U (N f )/Z N ⋊ Z C 2 symmetry, which is a quantum symmetry in the SO(N ) 2 bosonic description. In the special case N = 2 and N f = 1, the fixed point coincides with (3.1) with k = 2 (this case is the one in fig. 1 and in the third line of (1.3)).
The symmetry becomes SO (3), which is visible in the SU (2) 1 bosonic theory while it is a quantum symmetry in the other descriptions.

SU (2) k with 1 Φ
We exploit SU (2) k ∼ = U Sp(2) k . In the case N f = 1 both the SU/U and U Sp dualities are valid. The scalar theory has only one quartic gauge invariant, thus the two dualities share the same fixed point: The two theories in the first row have manifest SO (3)

SU (2) k with 2 Φ
We could write the theory as U Sp(2) k with 2 Φ, which has N = 1 and N f = 2, however the U Sp duality requires N f ≤ N and so it is not valid. The SU/U duality, instead, is valid. In such a duality the scalar theory has two quartic terms, singlets under U (2)/Z 2 ⋊ Z C This example, discussed at length in Section 5, does not develop quantum symmetries.

SU (2)
−N+ N f 2 with N f Ψ Both SU/U and U Sp dualities require N f ≤ N . The two dualities have common fermionic theory and thus the fixed points are the same: The fixed point has U Sp(2N f )/Z 2 symmetry, which is a quantum symmetry in the bosonic U (N ) 2 theory. When N = N f = 1, the fixed point coincides with (3.1) with k = 2 (as in fig. 1 and the third line of (1.3)).

Examples with Quantum SO(3) Symmetry and 't Hooft anomaly matching
Consider the examples with enhanced SO(3) symmetry, specifically the family of The first two columns are special cases of the discussion above (and had already been considered in [45]). The two theories in the last column can be coupled to a U (1) background for the maximal torus of SO(3) with Lagrangians (Tr a)d(Tr a) where the parameter K s is identified with the level of the SU (2) K s /Z 2 ∼ = SO(3) K s /2 background in the upper middle description in (3.6). The needed CS counterterms have been computed in [39]. In all six cases, the CS counterterms are well-defined for K s + 1 ∈ 2Z, providing a check of the dualities.

Example with Quantum O(4) Symmetry: QED with Two Fermions
In this section we consider three-dimensional QED, i.e. U (1) 0 , with two fermions of unit charge. As first observed in [46], this model enjoys self-duality. The analysis of [39] paid more attention to global aspects of the gauge and global symmetries and to the Chern-Simons counterterms. Here we continue that analysis and discuss in detail the global symmetry and its anomalies. In particular, we will show that the IR behavior of this model has a global O(4) symmetry and time-reversal invariance T, but these symmetries have 't Hooft anomalies. As in previous sections, various subgroups or multiple covers of this symmetry are anomaly free and can be preserved in a purely (2 + 1)d model. We also add bulk terms to restore the full global symmetry.

QED 3 with two fermions
We consider a pair of dual UV theories flowing to the same IR fixed point. As in [39], we start with a purely (2 + 1)d setting and study 15 15 CS grav is a gravitational Chern-Simons term defined as M=∂X CS grav = 1 192π X Tr R ∧ R. In this section we also use that the partition function of U (N ) 1 is reproduced by the classical Lagrangian −2N CS grav . See [49,38] for details.
We would like to identify the global symmetry of the model. The UV theory in the left side of (4.1) has a global SU (2) X × O(2) Y symmetry. The explicit background field Similarly, the UV theory in the right side has a SU (2) Y × O(2) X symmetry which includes a C X transformation. We will soon see that they do not act faithfully.
Before we identify the global symmetry of the IR theory, we should find the precise global symmetry of the UV theories (4.1). First we study how local operators transform under SU (2) X × U (1) Y in the left side of the duality (4.1). A gauge-invariant polynomial made out of Ψ i , Ψ i and derivatives has even U (1) X charge corresponding to SU (2) X isospin j X ∈ Z, and it is neutral under U (1) Y . A monopole of a has U (1) Y charge Q Y = 1 and U (1) a charge 1. In order to make it gauge invariant, we must multiply it by a fermion, thus making the operator have j X = 1 2 . 16 More generally, it is easy to see that all gauge invariant operators have 2j X + Q Y ∈ 2Z.
As in the previous sections, this means that the dynamical U (1) a and the classical SU (2) X × O(2) Y should be taken to be U (1) a × SU (2) X × O(2) Y /Z 2 and the global symmetry that acts faithfully is SU (2) X × O(2) Y /Z 2 .
A similar argument can be used in the right hand side of (4.1) showing that the global symmetry there is SU (2) Y × O(2) X /Z 2 . The duality (4.1) means that the IR theory should have the union of these two symmetries SO(4) ∼ = SU (2) X × SU (2) Y /Z 2 . 17 Also, the duality means that the theory is invariant under a transformation Z C 2 that exchanges X ↔ Y , thus the global symmetry is really O(4) ∼ = SO(4) ⋊ Z C 2 . The Lagrangians in (4.1) use only U (1) X × U (1) Y gauge fields and in terms of these the global symmetry is 16 It is easy to see that the basic monopole operators can have spin zero. More generally, our theory satisfies the spin/charge relation with a the only spin c connection. Therefore, all gauge-invariant local operators must have integer spin. In addition, in the absence of background fields (i.e. as long as we consider correlators at separate points) the theory is clearly time-reversal invariant: 18 where T(a) = a, T(X) = X, T(Y ) = −Y . Of course, we can combine this transformation with C Y and/or with an element of SU (2) X . With a background, the theory is timereversal invariant up to the anomalous shift 2 4π (XdX + Y dY ). This anomaly should not be surprising. The U (1) X symmetry is embedded into SU (2) X and in terms of that, the functional integral over Ψ leads to an η-invariant (that can be described imprecisely as ) which has a time-reversal anomaly. Note that in the other side of the duality this transformation must act as T(ã) = −ã.
Next, we would like to examine whether the Z 2 quotient of U (1) X ×U (1) Y is anomalous or not. Since we should take the quotient U (1) a ×U (1) X ×U (1) Y /Z 2 (and similarly with U (1)ã), this means that the fluxes of a,ã, X, Y are no longer properly quantized, but a±X, a ± Y are properly quantized spin c connections and X ± Y are properly quantized U (1) gauge fields. A simple way to implement it is to change variables a → a − X,ã →ã − Y in (4.1) such that a,ã become ordinary spin c connections: Except for the first term in each side, namely − 2 4π Y dY and − 2 4π XdX, all the terms are properly normalized Chern-Simons terms under the quotient gauge group.
The existence of these terms means that the two dual UV theories (4.1) have an 't Hooft anomaly preventing us from taking the Z 2 quotient.
We can change this conclusion by adding appropriate counterterms, e.g. 2 4π XdX, to the two sides of the duality (4.1) or equivalently (4.3). Denoting the Lagrangians in these equations by L 0 (X, Y ) ←→ L 0 (Y, X), we set XdX . This removes the first term in the right side of (4.3) and makes also the left side consistent with the quotient. Then, we can place the UV theory in U (1) X ×U (1) Y /Z 2 backgrounds.
In the left side of the duality this term represents adding SU (2) X 1 while in the right side this interpretation is meaningful only in the IR theory. After this shift, the IR theory can be placed in nontrivial SO(4) backgrounds. However, now the IR duality symmetry, which exchanges X ↔ Y , is anomalous: i.e. under the Z C 2 transformation the IR theory is shifted by SU (2) X −1 × SU (2) Y 1 . To summarize, the global symmetry that acts faithfully is O(4), but we cannot couple the system to background O(4) gauge fields. Starting with (4.1) we can couple it to P in ± (4) background fields, 19 or starting with (4.4) we can couple it to SO(4) background fields.

Mass deformations
We can check the duality (4.1) by deforming both sides with fermion bilinear operators in either the singlet or vector representation of the SU (2) flavor symmetry factors.
The deformation by the SO(4)-singlet mass term mΨ i Ψ i was discussed in [39]. The theory flows to the Lagrangians for m < 0 . The SU (2) X triplet mass term 20 m( In fact, the duality (4.1) can be derived by combining two fermion/fermion dualities involving a single fermion (e.g. see Section 6.3 of [39]), and from there one finds that the SU (2) X triplet mass term maps to the SU (2) Y triplet mass term 19 Since in the IR there are no operators transforming in spinor representations of Spin(4), we can extend O(4) to both P in ± (4). 20 We thank D. Gaiotto for a useful discussion about this deformation. Deforming the CFT (4.1) by this mass term leads to the low energy Lagrangians We see that the theory is not gapped: the photon a is massless and its dual is the Goldstone boson of a spontaneously broken global symmetry. From (4.7) we see that the unbroken symmetry is a diagonal mixture of U (1) X × U (1) Y . Under both deformations (4.6) and (4.7) we find consistency of the duality.
We could entertain the possibility that the symmetry of the CFT be SO(5) ⊃ O(4).
That would imply that at the fixed point the O(4) invariant operator Ψ i Ψ i Ψ j Ψ j sit in the same representation 14 of SO(5) as Ψ i (σ 3 ) i j Ψ j , and share the same dimension. As we just discussed, we can assume that the operator Ψ i (σ 3 ) i j Ψ j , which is relevant in the UV, is relevant in the IR as well: this leads to a coherent picture. This would imply that also the 4-Fermi interaction is relevant in the IR, and since it is irrelevant in the UV, it would be a dangerously-irrelevant operator. Then, in order to reach the putative CFT with SO(5) symmetry, one would need to tune the irrelevant operator Ψ i Ψ i Ψ j Ψ j in the UV. The theory we have been discussing in this section-QED with two fermions-does not have such a tuning, and therefore it would not reach the SO(5) fixed point even if the latter existed.

Coupling to a (3 + 1)d bulk
We have seen that the IR behavior of the UV theories (4.1) has an O(4) global symmetry and time-reversal T. But these symmetries suffer from an 't Hooft anomaly. We cannot couple them to background gauge fields for these symmetries. We saw that depending on the choice of counterterms we can have either P in ± (4) or SO(4) background fields, but we cannot have O(4) background fields and in either case we do not have time-reversal symmetry.
However, we can couple our (2+1)d system to a (3+1)d bulk and try to add background gauge fields in the bulk such that the full global symmetry is realized.
Let us start with the O(4) ∼ = SU (2) X × SU (2) Y /Z 2 ⋊ Z C 2 symmetry. The bulk couplings of these gauge fields are characterized by two θ-parameters, θ X and θ Y . Because of the Z 2 quotient, they are subject to the periodicity (4.8) and the semidirect product restricts to (θ X , θ Y ) ∼ (θ Y , θ X ).
Consider a bulk term S B with (θ X = −2π, θ Y = 0). For a closed four-manifold with X and Y being P in ± (4) gauge fields, this bulk term is trivial. When X and Y are O(4) gauge fields the partition function e iS B is ±1 and depends only on w 2 of the gauge fields.
(More precisely, the sign is determined by the Pontryagin square P(w 2 )/2.) This means that even for O(4) gauge fields the partition function is independent of most of the details of X and Y in the bulk.
The Z C 2 transformation, which exchanges X and Y , shifts the bulk term S B by the term (θ X = 2π, θ Y = −2π). On a closed four-manifold this shift has no effect on the answers. But in the presence of a boundary it shifts the boundary Lagrangian by the Chern-Simons terms of SU (2) 1 × SU (2) −1 /Z 2 . In other words, in the presence of a boundary the bulk term S B has an anomaly under Z C 2 . Starting with the boundary theory (4.1) we add the boundary term in (4.4) and the bulk term S B . Naively, this did not change anything. The bulk term might be thought of as an SU (2) X −1 boundary Chern-Simons term and therefore it seems like it removes the term added in (4.4). However, because of the quotients this conclusion is too fast. Instead, the bulk term is meaningful for SO(4) fields and has an anomaly under Z C 2 . The boundary term we added in (4.4) made the boundary theory meaningful for SO(4) fields and created an anomaly under Z C 2 . Together, we have a theory with a bulk and a boundary with the full O(4) symmetry. Now that we have achieved an O(4) symmetry we can try to add additional terms to restore time-reversal symmetry. We would like to add a bulk O(4) term that even with a boundary does not have a Z C 2 anomaly, but such that it compensates the anomaly in time reversal (4.2). Clearly, we need to add a bulk term S ′ B with (θ X = π, θ Y = π). Without a boundary this term is T and Z C 2 invariant. With a boundary it does not have an anomaly under Z C 2 but it has a T anomaly which exactly cancels that of the boundary theory (4.2). Note that the time-reversal anomaly (4.2) was not modified by adding the boundary term in (4.4) and the bulk term S B with (θ X = −2π, θ Y = 0). These two terms almost completely cancel each other. To summarize, the theory with the added boundary term in (4.4) and a bulk term S B + S ′ B with (θ X = −π, θ Y = π) has the full symmetry of the problem.
We should make a final important comment. As we said above, the bulk term S B with (θ X = −2π, θ Y = 0) leads to dependence only on some topological information of the bulk fields. Instead, the bulk term S ′ B with (θ X = π, θ Y = π) depends on more details of the bulk fields.

Example with Global SO(5) Symmetry
In this section we would like to study in some detail the theory U Sp(2) k ∼ = SU (2) k with two scalars , (5.1) and the relation with its SU/U dual U (k) −1 with two fermions. Since N f = 2, there are various quartic terms we can include in the potential, and depending on the choice we reach different IR fixed points. We will use mass deformations to check the duality, and exploit the 't Hooft anomaly matching for general SU/U dualities discussed in Section 2.

A family of CFTs with SO(5) global symmetry
Let us first consider U Sp(2) k with two Φ. As a U Sp theory, it has maximal global symmetry SO(5) ∼ = U Sp(4)/Z 2 . We can classify relevant deformations accordingly. We describe the scalars through complex fields ϕ ai with a = 1, 2, i = 1, . . . , 4 subject to the reality condition ϕ ai ǫ ab Ω ij = ϕ * bj (where Ω is the U Sp(4) symplectic invariant tensor). The quadratic gauge invariants are collected into the antisymmetric matrix M ij = ϕ ai ϕ bj ǫ ab , which decomposes under SO(5) as Here the subscript is the SO(5) representation and we suppress the indices. Given the However, since the gauge group has rank one, it turns out that O 2 1 = 4O 2 5 and so there is only one quartic singlet.
In U Sp(2) k with two Φ we insist on SO(5) global symmetry: we turn on O 1 and O 2 1 with a fine-tuning on O 1 and we assume that it flows to a nontrivial fixed point T (k) 0 . Such a fixed point has SO(5) global symmetry. We can couple the theory to SO(5) background gauge fields as with some CS counterterm with level L. The conditions to have a well-defined (2 + 1)d action are L ∈ Z and k − 2L ∈ 2Z. As we discussed in Section 2.3, for k even the equations can be solved, but for k odd they cannot and there is an 't Hooft anomaly. The anomaly is captured by the bulk term For a closed manifold M 4 this is trivial for k even, in the sense that e iS anom = 1, but it is ±1 for k odd. For M 4 with a boundary this term is anomalous for odd k and corrects the anomaly in the boundary theory.
We can consider relevant deformations of T describes an S 7 , which is an SU (2) Hopf fibration over S 4 , therefore gauging SU (2) leaves the S 4 NLSM.
As was shown in [59] (we review it in Appendix A), the S 4 NLSM has a Wess-Zumino interaction term kS WZ that originates from the level k Chern-Simons term in the UV. The Wess-Zumino term can be written as where the integral is over a four-manifold with boundary, σ are the NLSM fields, and ω 4 is the volume form of S 4 normalized to total volume 1. We can couple the theory to SO (5) background gauge fields (gauging in general dimension was discussed in [67]). From our derivation in Appendix A it is clear that for odd k, the WZ action depends on how the SO(5) background fields are extended to the bulk, and the dependence is captured by the very same term (5.5). 21 This is 't Hooft anomaly matching along the RG flow.
Let us stress that the far IR limit of the S 4 NLSM is given by 4 free real scalar fields.
At higher energies there are irrelevant interactions that turn it into the S 4 NLSM with WZ term. As we show in Appendix A, at the same scales there are also other irrelevant 21 We thank Todadri Senthil for pointing out to us the relevance of P(w 2 ) in this context and for mentioning [68].
(higher-derivative) interactions that break time reversal T (for k > 0). Therefore such an S 4 NLSM has only SO(5) global symmetry, as the UV theory.

Two families of CFTs with SO(3) × O(2) global symmetry
Let us now consider SU (2) k with two Φ (i.e. the same gauge group and matter content as before), but imposing only SU (2) × U (1) symmetry on the quartic terms, as it is the case in the SU/U dualities. Then there is another quartic deformation we can add in the UV: where η is a U (2)-invariant tensor. This operator is contained in O 14 from the decompo- . It turns out that the preserved symmetry acting faithfully is SO(3) × O(2). 22 If we want to avoid the appearance of directions in field space where the potential is unbounded from below, the absolute value of the coefficient λ 14 of O (14) should not be too large compared to that of O 2 1 (which is positive). Since at λ 14 = 0 there is a phase transition with enhanced SO(5) symmetry, we expect two different RG flows for λ 14 ≷ 0 that can lead to two families T We can learn about the properties of the fixed points T  by O 1 , as in Section 5.1, and then by O (14) . The TQFT SU (2) k is not affected by O (14) , because SO(5) only acts on massive particles and thus O (14) is decoupled. In the S 4 NLSM we use coordinates ρ 1,...,5 with 5 I=1 ρ 2 I = 1. Then we have O (14) = −3(ρ 2 1 + ρ 2 2 ) + 2(ρ 2 3 + ρ 2 4 + ρ 2 5 ). If we deform the potential by O (14) with positive coefficient, we flow to an S 1 NLSM, while a negative coefficient leads to an S 2 NLSM. We conclude that, for µ 1 < 0, the µ 1 O 1 deformation of T (k) + gives an S 1 NLSM, while gives an S 2 NLSM. Notice that when the NLSM maps are restricted to an equatorial S 1 or S 2 , the WZ term (5.6) vanishes.
We can provide two different descriptions of T (k) + through the SU/U duality where the theory on the left has O 2 1 and O (14) quartic couplings both with positive coefficient. 23 This gives evidence that the fixed points T (dual to SU (2) k ) for negative fermion mass, and U (k) 0 (whose low energy limit is the fail to give a dual pair (as advocated in [45] by a different argument). In this example T (1) ± appear in the same RG diagram, but are indeed distinct. 23 As in [39], U (1) ⊂ U (k) is a spin c connection and we must add a transparent line to the theories in order for the duality to be valid. This transparent line does not affect the critical behavior.

A family of RG flows with O(4) global symmetry
We can consider a different deformation of the SO(5) invariant theories T (k) 0 , obtained by using a quartic operator in O 14 that preserves an O(4) . We will call this operator O (14) .
It can be written in terms of a Spin(4) invariant tensor η ′ as 24 by O (14) breaks SO(5) → O(4). Thus, we study the theory U Sp(2) k with 2 Φ and quartic deformations O 2 1 and O (14) . More easily, this is SU (2) k with two scalars and a potential V = α |Φ 1 | 4 + |Φ 2 | 4 + 2β|Φ 1 | 2 |Φ 2 | 2 with α = β. In order to have a theory with a potential bounded from below, O 2 1 should have positive coefficient while the coefficient λ 14 of O (14) should not be too large in absolute value. As before, we expect two different RG flows for λ 14 ≷ 0, separated by T  fig. 2. With positive masssquared we flow to the TQFT SU (2) k , which is not affected by O (14) because the latter is decoupled. With negative mass-squared we flow to deformations of the S 4 NLSM. In the NLSM coordinates, O (14) = −(ρ 2 1 + ρ 2 2 + ρ 2 3 + ρ 2 4 ) + 4ρ 2 5 . Therefore, λ 14 > 0 leads to an S 3 NLSM, while λ 14 < 0 leads to two gapped vacua with spontaneous breaking of Z C 2 . The WZ term kS WZ in the S 4 NLSM descends to a θ-term πkQ in the S 3 NLSM, where Q ∈ Z is the wrapping number in π 3 (S 3 ) = Z (in other words θ = kπ). 25 In the presence of the deformation O (14) , with either sign of its coupling λ 14 , a tuning on O 1 may or may not lead to a fixed point. At the moment we do not have candidate dual descriptions for those fixed points, and we leave the question open.
There are more general deformations of T 25 Restricting the NLSM maps σ to an equatorial S 3 in S 4 , the WZ term gives 0 on a map that does not wrap S 3 , and π on a map that wraps S 3 once. By linearity, S WZ = πQ(σ). (14) , O (14) , For instance, if we want to preserve only SO(4) ⊂ SO (5), in terms of the two SU (2) doublets Φ 1 and Φ 2 we can turn on the following relevant deformations: there are two mass terms |Φ 1 | 2 and |Φ 2 | 2 , and three quartic terms |Φ 1 | 4 , |Φ 2 | 4 , |Φ 1 | 2 |Φ 2 | 2 . This is a different basis than the one in Table 1. With many operators at our disposal, the precise breaking pattern depends on the ratios between the various terms.

Relation with a Gross-Neveu-Yukawa-like theory
We can compare the U Sp(2) k theory with two scalars with a different model, discussed in [70], which also exhibits SO(5) global symmetry and a phase described by the S 4 NLSM with WZ term.
Consider a Gross-Neveu-Yukawa-like theory (GNY) with 5 real scalars, 4k complex fermions and schematic Lagrangian [70] L = (∂φ) 2 + Ψ/ ∂Ψ − φ 4 + φ a ΨΓ a Ψ . is odd under T and is thus set to zero by imposing that symmetry). We could also think of the fixed point as the IR limit of the O(5) Wilson-Fisher fixed point with 4k complex decoupled fermions perturbed by the relevant operator φ a ΨΓ a Ψ.
As discussed in [70], if we deform (5.12) by a negative scalar mass-squared, the scalars condense breaking spontaneously SO(5) → SO(4) and leading to an S 4 NLSM. In addition, because of the Yukawa interaction the fermions become massive. Integrating them out produces a WZ interaction kS WZ [71]. Deformation by a positive mass-squared leads to 4k complex massless free fermions. 26 The GNY fixed point (5.12) and the fixed point T The one obtained from (5.12) has the time-reversal symmetry T, preserved by S WZ , while the one from SU (2) k with two scalars has higher-derivative corrections that break T, because time-reversal symmetry is not present in the UV. In addition, the phase obtained by positive mass-squared is different in these two models.
Appendix A. Derivation of the Wess-Zumino term in the 3D S 4 NLSM Here we show that when SU (2) k with two scalars flows to the S 4 NLSM by mass deformation, the Chern-Simons term induces a Wess-Zumino interaction at level k at low energies [59].
Insisting on SO(5) global symmetry and turning on a negative mass-squared, the minima of the potential lie along a,i |Φ ai | 2 = λ (a = 1, 2, i = 1, 2) which is S 7 (here λ is a mass scale). The SU (2) action corresponds to the Hopf fibration SU (2) → S 7 → S 4 , thus what we gauge is the SU (2) fiber. Recall that SU (2) bundles over S 4 are completely classified by π 3 SU (2) = Z which is the second Chern class, and S 7 has minimal class: where G = dC − iC 2 is the curvature of the SU (2) bundle and C(Φ) is a function of Φ.
The 3D gauge theory has a CS term where S 3 is the topology of spacetime and the second definition is the proper one. At low energies the scalars are constrained to |Φ| 2 = λ and we integrate out the gauge field.
Starting with the schematic Lagrangian the equation of motion for a is This equation contains a µ as well as its first derivative, and it is non-linear. If we drop the last term, the equation is simply J µ ≡ −iΦ ↔ D µ Φ † = 0 setting to zero the SU (2) gauge current. This means that a µ is identified with the connection C(Φ) of the SU (2) bundle over S 4 . To take into account the last term, we notice that the first four terms in (A.4) are of order λ (because |Φ| 2 ∼ λ) while the last term is of order λ 0 and it contains a derivative.
We can then solve the equation as a series expansion in λ −1 , and since λ −1 is dimensionful, the series is actually a derivative expansion. Thus in the IR limit we have where the dots are higher-derivative corrections.
Having identified the field strength F in (A.2) over the extended spacetime manifold Finally, consider coupling the UV theory to SO(5) background fields, namely consider the theory SU (2) k × U Sp(4) L /Z 2 with a bifundamental scalar. As discussed in Section 2.3, for odd k the action has a sign dependence on the extension of the SO(5) background fields to M 4 . By (A.6), this implies that also the WZ term coupled covariantly to SO (5) background fields [67] has the same anomalous dependence.

Appendix B. Comments on Self-Dual QED with Two Fermions
Building on the interesting fermion/fermion duality of [4][5][6], the authors of [46] proposed the self-duality of a U (1) theory with two fermions. This was later generalized in [47] to the self-duality of a U (1) gauge theory with two fermions, one with charge 1 and one with charge k odd. As emphasized in [49,38,39], the coefficients in the Lagrangians in [4][5][6] are improperly quantized. This was fixed in [38] by adding more fields and more terms to the Lagrangian. Then, a proper derivation of the self-duality of the theory with k = 1 was given in [39]. That perspective was also consistent with the spin/charge relation and described the proper coupling of background gauge fields. Here we will present a similar derivation of the self-duality of the theory with generic odd k. This will lead us to a more detailed analysis of the global symmetries and 't Hooft anomalies of the problem.
We start with the fermion/fermion duality of [38]: Next, we follow the steps in [39]. We take a product of the theory in (B.1) and of its timereversed version in which we substitute A → kA − 2X (X is a background U (1) gauge field): AdA .

(B.2)
Note that for odd k this is consistent with the spin/charge relation. We add the following AdA + 2CS grav , to the two sides of the duality. Here N = (k 2 + 1)/2 and Y is a background U (1) field. The specific counterterms and the value of N were picked such that we can integrate out most of the fields on the right hand side. Then we can promote A to a dynamical field (more precisely, a spin c connection) a. On the left hand side we find We will call this Lagrangian L 0 (X, Y ). On the right hand side there are several gauge fields, but we can integrate most of them out. We redefine a = a ′ +2u 1 and u 2 = u ′ 2 +ku 1 + k+1 2 a ′ , then the Lagrangian is linear in u 1 and it can be integrated out to set a 1 = ka 2 − 2Y .
Finally we can integrate out a ′ to find where we relabeled a 2 =ã. Note that all terms are properly quantized withã being a spin c connection. We see that (B.3) and (B.4) are related by relabeling the dynamical fields and by exchanging X ↔ Y . This establishes the self-duality of the model, namely L 0 (X, Y ) ←→ L 0 (Y, X). 27 27 In order to compare with [47], for every fermion coupled with / D A we should add the terms − 1 8π AdA − CS grav . This turns (B.3) and (B.4) into where we removed the gravitational Chern-Simons term. Up to the last counterterm (which we cannot remove because of the spin/charge relation) this agrees with the equations in [47].
As a check, for k = 1 we can substitute a → a + X in (B.3),ã →ã + Y in (B.4) and subtract the counterterm 1 2π XdY from both sides, to find the same duality (4.1) as in [39]. Let us examine the global symmetry of the problem. First, there is a U (1) X × U (1) Y .
Second, there is a charge-conjugation symmetry acting as C(a) = −a, C(X) = −X, C(Y ) = −Y (and C(ã) = −ã in the dual). We will denote the combined group for these two symmetries as S O(2) X × O(2) Y . Third, because of the duality there is the Z C As in the other cases, in particular the one in Section 4, we have different options.
1. We can leave the (2 + 1)d Lagrangian L 0 (X, Y ) as it is, but then we can only couple it to S O(2) X × O(2) Y ⋊ Z C 2 background fields with no fractional fluxes. 2. We add to the two sides (B.3)-(B.4) of the duality the Chern-Simons counterterms − 1 4π (XdX − Y dY ). These counterterms violate the spin/charge relation. Now we can have S O(2) X × O(2) Y /Z 2 backgrounds, but Z C 2 is violated. 3. We can attach the theory to a (3 + 1)d bulk, add suitable bulk terms and obtain a well-defined theory on general backgrounds, but whose partition function depends on the extension of the background fields to the bulk.
In order to preserve time-reversal as well, we add another boundary term S ′ B with (θ X = 2π, θ Y = 2π) and also − k 2 −1 192π Tr R ∧ R. In the presence of a boundary, the variation of S ′ B under T precisely cancels the one of L 0 (while the variations of the added boundary term to get L 1 and of S B cancel among themselves).

Appendix C. More 't Hooft anomalies
We list here the 't Hooft anomalies for other cases discussed in the main text. Consider the theories U (N ) k with N f scalars and SU (k) −N+ N f 2 with N f fermions. The global symmetry is U (N f )/Z k and charge conjugation that we will neglect. Following the same steps as in Section 2.2, one finds that for generic choices of the CS counterterms and with the same conventions as in (2.9) and (2.11), the anomaly is With the choice J ∈ DZ, using the square of the previous relation the anomaly simplifies to S anom = 2π 3) The case k = 0 is special and the formulae above do not directly apply.
So, consider the theory U (N ) 0 with N f scalars. In this case the global symmetry is P SU (N f ) × U (1) M , as well as charge conjugation and time reversal that we neglect.
The scalars are coupled to a U (N f ) gauge field B (where U (1) ⊂ U (N f ) is dynamical) and a dynamical gauge field b, with N f Tr db = N Tr dB. The coupling to the magnetic U (1) background field B M is described by the ill-defined expression N 2πN f (Tr B)dB M which needs to be moved to the bulk. This highlights that the global symmetry suffers from an 't Hooft anomaly. Including a CS counterterm at level L for SU (N f ) (which could be set to zero), the anomaly is characterized by the bulk term where we have identified 1 2π Tr dB = w (N f ) 2 mod N f . This expression can be regarded as a singular limit of (C.1).
Similarly, the theory U (k) N f 2 with N f fermions has global symmetry U (N f )/Z N f , besides charge conjugation that we neglect. The expression (2.13) for the anomaly does not directly apply (since N = 0). Following similar steps as before, we find that the anomaly is characterized by the bulk term The other time-reversal invariant theory is U (k) 0 with N f fermions, which requires N f to be even. The UV symmetry is U (N f )/Z N f /2 together with charge conjugation and time reversal. Applying (2.14) with N = N f /2, the anomaly is where F satisfies (