Matching higher symmetries across Intriligator-Seiberg duality

We study higher symmetries and anomalies of 4d $\mathfrak{so}(2n_c)$ gauge theory with $2n_f$ flavors. We find that they depend on the parity of $n_c$ and $n_f$, the global form of the gauge group, and the discrete theta angle. The contribution from the fermions plays a central role in our analysis. Furthermore, our conclusion applies to $\mathcal{N}=1$ supersymmetric cases as well, and we see that higher symmetries and anomalies match across the Intriligator-Seiberg duality between $\mathfrak{so}(2n_c)\leftrightarrow\mathfrak{so}(2n_f-2n_c+4)$.

• The case none. The 0-form symmetry and the 1-form symmetry stay separate without an anomaly.
• The case extension. Take, for example, the Spin(2n c ) gauge theory with 2n f flavors. When n c is odd, two copies of the Wilson line W in the spinor representation form a Wilson line in the vector representation. This can be screened by a dynamical fermion, which is why W 2 = 1 as far as the 1-form symmetry charge is concerned. Now let us recall that this dynamical fermion transforms nontrivially under −1 ∈ SU(2n f ). Therefore, when we further take the flavor symmetry into account, W 2 is still nontrivial.
As was discussed in [HL20], this means that the Z 2 1-form symmetry extends the SU(2n f )/Z 2 0-form symmetry in a nontrivial manner, forming a 2-group H fitting in the sequence (1.1) whose extension class is specified by βa 2 ∈ H 3 (B(SU(2n f )/Z 2 ); Z 2 ). (1.2) Here, Z 2 [1] stands for Z 2 regarded as a 1-form symmetry, SU(2n f )/Z 2 is the quotient by the subgroup {±1}, a 2 is (the representative cocycle of) the obstruction class controlling whether the SU(2n f )/Z 2 bundle lifts to SU(2n f ), and β is the Bockstein homomorphism. 4 The background field for such a 2-group is given by the background gauge field for SU(2n f )/Z 2 together with a Z 2 -valued degree-2 cochain E satisfying 5 δE = βa 2 . (1.3) A consequence of this nontrivial extension is that the theory cannot be coupled to a general SU (2n f )/Z 2 background without introducing a nontrivial E background. In short, SU (2n f )/Z 2 is not a subgroup but a quotient group of the whole symmetry 2-group, and thus gauging SU (2n f )/Z 2 alone without gauging Z 2 [1] part does not make sense.
• The case anomaly. The SO + (2n c ) gauge theory is obtained by gauging the Z 2 1-form symmetry of the Spin(2n c ) gauge theory [KS14]. The gauging of the Z 2 1-form symmetry whose 4 For a Z 2 -valued cocycle a ∈ Z 2 (X, Z 2 ), its Bockstein is defined as follows. We first construct the Z 4 -valued lift a of a by sending {0, 1} to {0, 1} ⊂ {0, 1, 2, 3}. Let us now consider δa. By construction it is 0 mod 2, and therefore 1 2 δa is a well-defined Z 2 -valued cocycle, which is defined to be βa. When βa is zero as a cohomology class, there is a Z 2 -valued cochain b such that βa = δb. This is equivalent to the fact that the Z 4 -valued cochain a := a + 2b is a Z 4 -valued cocycle. In this manner we found that [βa] = 0 means that a can be lifted to a Z 4 -valued cocycle. We now lift this Z 4 -valued cocycle to a Z 8 -valued cochain a. In this case 1 4 δa is a well-defined Z 2 -valued cocycle, which is defined to be β 2 a. This β 2 is known as a higher Bockstein operation, which we will need to use later in the paper. When β 2 a is zero as a cohomology class, we can lift a to a Z 8 -valued cocycle. We can then lift it to a Z 16 -valued cochain and define β 3 a, ad infinitum. 5 In this equation, a 2 and βa 2 need to be interpreted as cochains rather than cohomology classes. More generally, cochains and cohomology classes will not be carefully distinguished explicitly in this paper. Hopefully this bad practice would not cause too much confusions.  Table 1: How the Z 2 1-form symmetry and the SU(2n f )/Z 2 0-form symmetry are combined in massless so(2n c ) QCD. 'none' implies that they remain a direct product without mixed anomaly; 'anomaly' means that they remain a direct product but with mixed anomaly; and 'extension' is when they combine into a nontrivial 2-group. The orange lines show how the duality of Intriligator and Seiberg acts on this set of theories.
background field is E is done by introducing another Z 2 -valued degree-2 closed cochain B, adding the interaction 2πi · 1 2 BE, (1.4) and summing over all possible E. In this particular case, the background field E is the second Stiefel-Whitney class w 2 of the SO(2n c ) gauge bundle, and summing over them gives the SO + gauge theory.
As we will see, the contribution to the anomalies from the fermions significantly complicates the analysis. Neglecting this contribution, we see that the coupling (1.4) is not closed due to (1.3), and has the variation 2πi · 1 2 Bβa 2 . (1.5) This means that the Z 2 1-form symmetry of the SO(2n c ) gauge theory and the SU(2n f )/Z 2 0-form flavor symmetry remains a direct product, but with a mixed anomaly given by (1.5).
We will carefully analyze how the Z 2 1-form symmetry and the SU(2n f )/Z 2 0-form symmetry are combined in the rest of the paper. The novelty in our paper over the analysis in [HL20] is that we take fermionic contributions into account. The derivation will be detailed in the following, and here we simply summarize the result in Table 1.
So far we assumed that the fermions are massless. It is also useful to see what happens when the fermions are massive. When we give equal masses to all N f = 2n f fermions, the flavor symmetry is reduced from SU(2n f )/Z 2 to SO(2n f )/Z 2 . The crucial simplification is that βa 2 appearing in the anomaly or the extension becomes cohomologically trivial when n f is odd, because a 2 now lifts to a Z 4 class controlling whether an SO(2n f )/Z 2 bundle lifts to a Spin(2n f ) bundle. Stated differently, the contributions from the fermions vanish since the fermions can be made massive, so that then the analysis of [HL20] applies. The results are shown in Table 2, which is significantly simpler than the behavior in Table 1.
(n c , n f ) Spin SO + SO − (even, even) none none none (odd, even) extension anomaly extension (even, odd) none none none (odd, odd) none none none Table 2: How the Z 2 1-form symmetry and the SO(2n f )/Z 2 0-form symmetry are combined in massive so(2n c ) QCD. Our conventions follow that of Table 1.
Application to the Intriligator-Seiberg duality: Our result thus far is equally applicable in the case of N =1 supersymmetric QCD, since they are connected to the non-supersymmetric QCD by a continuous deformation preserving all the symmetries we care about. Now, let us recall that Intriligator and Seiberg found in [IS95] a duality exchanging so(N c ) and so(N f − N c + 4), which in our notation sends n c to n c = n f − n c + 2, keeping n f fixed. Following a crucial set of observations in [Str97] that spinors in the original theory are mapped to magnetic monopoles in the dual theory, the Intriligator-Seiberg duality of N =1 so theories was refined in [AST13], to account for the global form of the gauge group and the discrete theta angle. It was concluded there that Spin is exchanged with SO − while SO + maps to itself. This mapping was given a further confirmation by using supersymmetric localization on S 3 /Z n × S 1 in [RW13]. Our analysis allows us to check this duality by comparing how the 0-form symmetry and the 1form symmetry are combined in the dual pairs. We superimposed the action of the duality on our main Table 1 for the massless case and Table 2 for the massive case. It is satisfying to see that the duality action correctly preserves the behaviors 'none', 'anomaly' and 'extension'. In the last couple of years, the study of higher symmetries and their anomalies of supersymmetric theories has seen some activity, 6 but mostly from the point of view of string theory or M-theory. The authors hope that this paper paves a way toward a more field-theoretical analysis of these matters.
Organization of the paper: The rest of the paper is organized as follows. In Sec. 2, we determine exactly when the electric / magnetic / dyonic Z 2 1-form symmetries and the SU(2n f )/Z 2 flavor 0-form symmetry form a nontrivial 2-group, by examining the charges of line operators in each theory. In Sec. 3, we exploit the SL(2, Z 2 ) actions on theories with Z 2 1-form symmetries, including our so(2n f ) QCDs. This will allow us to determine the 't Hooft anomalies they possess. Combining the results with those obtained in Sec. 2, one can completely determine the structures of symmetries and anomalies of so(2n f ) QCDs, and can further confirm that they are indeed compatible with the Intriligator-Seiberg duality. Although the result itself is satisfactory, the analysis leading to it is somewhat ad-hoc, so in Sec. 4, we partially complement it with a more direct computation of fermion anomalies.
In Appendix A, we discuss how we can understand the 2-group structure in general by studying line operators and line-changing point operators, and find a relation to the crossed module extensions classifying H 3 . Finally, we have the two appendices providing technical details of the mathematical facts used in the main part; in Appendix B, we compute relevant bordism groups capturing the anomalies of spin QFTs associated with various symmetries; and in Appendix C, we describe some subtleties concerning the Pontrjagin square.
Before proceeding, we list the obstruction classes which will be frequently encountered in this paper. In general, given a group G, a subgroup Z n in the center of G, and a G/Z n bundle on a manifold X, there is a obstruction class in H 2 (X; Z n ) controlling whether this bundle lifts to a G bundle. For G = Spin(N c ) and G/Z 2 = SO(N c ) this is the familiar second Stiefel-Whitney class w 2 . The classes we use are listed in Table 3. Table 3: The names we use for the obstruction classes ∈ H 2 (X, Z n ) controlling whether a G/Z n bundle on X lifts to a G bundle.

2-group structure
Let us first study whether the Z 2 1-form symmetry and the SU(2n f )/Z 2 0-form flavor symmetry form a nontrivial 2-group or not. This can be found rather physically by studying the line operators.

Spin
We start by discussing the Spin(2n c ) gauge theory with 2n f fermions in the vector representation.
The results presented in this subsection was originally found in [HL20,Sec. 4.4]. First, recall that the center of Spin(2n c ) is Z 2 × Z 2 or Z 4 depending on whether n c is even or odd. This corresponds to the fact that the tensor square of a spinor representation contains the identity representation when n c is even while it contains the vector representation when n c is odd. Now, consider the Wilson line W in the spinor representation. When n c is even, W 2 contains the identity representation, and therefore we simply have a Z 2 1-form symmetry independent of the flavor symmetry, and there is nothing more to see here.
When n c is odd, W 2 contains the vector representation. This can be screened by the dynamical fermion, which however carries the fundamental representation of SU(2n f ) flavor symmetry, and in particular transforms nontrivially under −1 ∈ SU(2n f ). In other words, the flavor Wilson line in the fundamental representation of SU(2n f ) can now be considered as the square of the gauge Wilson line in the spinor representation of Spin(2n c ). This means that we have the following extension of groups As the groups of charges of SU(2n f ) 0-form symmetry and Z 2 1-form symmetry are combined nontrivially, the symmetry groups themselves are also combined nontrivially. Let us see this point by considering their background fields. (We will discuss another general method to relate this extension to 2-groups in Appendix A.) The fermion fields are simultaneously in the vector representation of the gauge so(2n c ) and the fundamental representation of the flavor su(2n f ), and therefore are in a representation of . Given a G bundle on a manifold X, there is an SO(2n c )/Z 2 bundle and an SU(2n f )/Z 2 bundle associated with it. Let us denote by v 2 , a 2 ∈ H 2 (X; Z 2 ) the obstruction classes controlling whether they lift to an SO(2n c ) bundle and an SU(2n f ) bundle respectively. Then we have v 2 = a 2 for a G bundle. The flavor Wilson line in the fundamental representation is charged under −1 ∈ SU(2n f ) in the center, and a 2 can be considered as the background field for this Z 2 1-form symmetry. Now, without the flavor background, the background E ∈ H 2 (X; Z 2 ) for the electric Z 2 1form symmetry of the Spin(2n c ) theory sets the Stiefel-Whitney class w 2 ∈ H 2 (X; Z 2 ) of the SO(2n c ) gauge bundle to be E = w 2 , which controls whether it lifts to a Spin(2n c ) bundle. When the flavor background a 2 is nontrivial, the obstruction class v 2 controlling the lift from an SO(2n c )/Z 2 bundle to an SO(2n c ) bundle is nontrivial. In this situation, when n c is odd, w 2 can no longer be defined as a closed cochain; rather it satisfies δw 2 = βv 2 , where β is the Bockstein operation, since 7 together they specify the obstruction class x 2 ∈ H 2 (X; Z 4 ) controlling the lift from an SO(2n c )/Z 2 = Spin(2n c )/Z 4 bundle to a Spin(2n c ) bundle. As E = w 2 and v 2 = a 2 , we conclude that the background field satisfies δE = βa 2 . (2.2) In general, a 2-group H combining a 1-form symmetry A and a 0-form symmetry G, which fits in the exact sequence with the extension class α ∈ H 3 (BG; A), is defined as a symmetry whose background field is given by a pair of a degree-2 cochain E ∈ C 2 (X, A) and a background G field g : X → BG satisfying δE = g * (α). Here A[1] means the Abelian group A regarded as a 1-form symmetry, 7 Indeed, let v 2 the Z 4 -lift of the cochain v 2 , where the value {0, 1} are lifted to {0, 1} ⊂ {0, 1, 2, 3}. βv 2 is by definition 1 2 δv 2 , as we explained in footnote 4. The Z 2 -reduction of the cochain x 2 is v 2 , and x 2 − v 2 is divisible by 2, so we can identify w 2 = 1 2 (x 2 − v 2 ). As 1 2 δx 2 is zero as a Z 2 -valued cochain, we find δw 2 = βv 2 , as desired. and we drop the pull-back symbol g * when its presence is clear from the context. In our case, we see that the Z 2 1-form symmetry and the SU(2n f )/Z 2 0-form flavor symmetry form the 2-group H fitting in the sequence with the extension class being βa 2 ∈ H 3 (B(SU(2n f )/Z 2 ); Z 2 ). 8 Note that having the extension of groups of charges of line operators as in (2.1) is equivalent to having a nontrivial 2-group extension (2.4) whose background field satisfies (2.2). The situation can be summarized as the following commuting diagram: Here, the sequences of the form 0 → G → G → G → 0 in the columns and the rows are to be interpreted as having fibration sequences BG → BG → BG among the respective classifying spaces. 9 We note that the map w 2 : SU(2n f )/Z 2 → Z 2 [1] extracts the information of the obstruction class a 2 ∈ H 2 (B(SU(2n f )/Z 2 ); Z 2 ). We also note that the 2-group H is uniquely determined , it is trivial. Therefore, to determine the 2-group extension, we can simply study the group of charges A H of line operators, which we will carry out for SO ± gauge theories next.

SO ±
We would like to study how the magnetic / dyonic Z 2 1-form symmetry of the SO(2n c ) ± gauge theory is combined with the SU(2n f ) flavor symmetry. We first discuss the case SO + in detail; the minor changes needed to take SO − into account would be described later.
In accord with the discussions in the previous subsection, we consider what happens when we take two copies of the 't Hooft line operator H and fuse them. At the very naive level, H 2 can be screened by dynamical monopoles, but dynamical monopoles can receive flavor / gauge center charges from the fermion zero modes.
Making deformations: To study these issues, it is useful to deform the theory and make it simpler by performing the following steps: • Reduce the flavor symmetry from SU(2n f ) to USp(2n f ). The fundamental representation still transforms nontrivially under −1 ∈ USp(2n f ), which is enough for our purposes.
• Add an adjoint scalar Φ [ab] and the interaction ψ ai α ψ bj β J ij Φ ab αβ + c.c.. Here a, b and i, j are vector indices of SO(2n c ) and USp(2n f ), α, β are the spinor indices, and J [ij] is the constant invariant matrix for the USp(2n f ).
• Give a generic vacuum expectation value (vev) to Φ ab and break SO(2n c ) to SO(2) nc .
The 't Hooft lines in the resulting SO(2) nc theory can be labeled by their magnetic charges (m 1 , . . . , m nc ) ∈ Z nc . The dynamical monopoles have the charges in the 'adjoint class', which are in the root lattice Λ of SO(2n c ). Then, the group of the magnetic charges of 't Hooft lines up to screening by the dynamical monopoles is which agrees with the 1-form symmetry before the deformation. We now would like to study how this Z 2 is combined with the flavor / gauge center Z 2 charge.
Analysis of the so(4) case: The vev of the adjoint scalar in this basis can be written as (a 1 , a 2 ), which we assume to be a 1 > a 2 > 0. Here, the fermion is in the vector representation of so(4). Under the monopole in su(2) 1 , it is a doublet coupled to an adjoint vev of size a 1 with bare mass a 2 , and similarly for the monopole in su(2) 2 . Now, the explicit analysis in [Cal78, Sec. IV] concerning the number of zero modes in the 't Hooft-Polyakov monopole says that a doublet fermion coupled to an adjoint vev of size a with bare mass µ has a zero mode if |a| > |µ| and has no zero modes if |a| < |µ|. With our assumption a 1 > a 2 > 0, this means that the monopole in su(2) 1 has a zero mode, while the monopole in su(2) 2 does not. In our original basis, this means that the monopole with (0, . . . , 1, −1; q nc−1 ) does not produce any zero modes and q nc−1 = 0, while the monopole with (0, . . . , 1, +1; q nc ) has two zero modes per flavor. The 1-form symmetry group is obtained by dividing Z 2 × Z 2 by the subgroup generated by (1, −1; 0) and (1, +1; q nc ). This is Z 2 × Z 2 or Z 4 depending on whether q nc is 0 or 1.
Let us determine q nc , the center charge of the monopole in su(2) 1 . We saw that there are two zero modes per flavor; this means that there are fermionic zero modes transforming in where R 2n f is the fundamental representation of usp(2n f ), while V 2 is the doublet of su(2) 2 , 10 and we need to impose the reality condition using the pseudo-reality of both factors, so that there are 4n f Majorana fermion in total.
To determine the flavor / gauge center charge q nc of the monopole, it suffices to consider the case n f = 1; the general case is given simply by multiplying it by n f . When n f = 1, there are 4 Majorana fermions. Quantizing them, we find the monopoles in (2.10) It has the 'vector' charge under usp(2) su(2) flavor symmetry or is a doublet under su(2) 2 , which corresponds to the 'vector' charge under so(4) gauge symmetry. In either case, they have the flavor / gauge center charge 1 ∈ {0, 1} = Z 2 . Therefore we conclude the flavor / gauge center charge q nc is simply given by n f mod 2.
Summary: Combining the intermediate steps we described above, we conclude the following: for the SO(2n c ) + gauge theory, the group Z 2 of magnetic charges of 't Hooft lines is extended by the flavor / gauge center symmetry Z 2 to become Z 4 when n f is odd, while they remain separate when n f is even.
The analysis of the SO(2n c ) − gauge theory is largely the same; the only difference is that the discrete theta angle gives an additional gauge center charge 11 to the simple dynamical monopole with the magnetic charge (0, 0, . . . , 1, +1), so that q nc = n f + n c mod 2. Therefore, we conclude 10 It is actually broken to u(1), but keeping su(2) 2 representation is useful in organizing the answer. 11 To see this, note that the original interaction 2πi 1 4 P(w 2 ) induces the interaction 2πi 1 4 ( nc i=1 c (i) 1 ) 2 in the SO(2) nc theory. This gives the electric charge (1, 1, . . . , 1) to the monopole with the magnetic charge (0, 0, . . . , 1, +1). Under −1 ∈ SO(2n c ) such a state transforms by (−1) nc . the following: for the SO(2n c ) − gauge theory, the group Z 2 of magnetic charges of 't Hooft lines is extended by the flavor / gauge center symmetry Z 2 to become Z 4 when n f + n c is odd, while they remain separate when n f + n c is even.
The result of the analysis is summarized in Table 4. There, 'product' means that the Z 2 1-form symmetry and the SU(2n f )/Z 2 flavor symmetry are kept separate and form a direct product, while 'extension' means that they form a nontrivial 2-group. We remark that the nontrivial 2-group is always given by the extension (2.4) whose background fields satisfy (2.2).
(n c , n f ) Spin SO + SO − (even, even) product product product (odd, even) extension product extension (even, odd) product extension extension (odd, odd) extension extension product Table 4: How the Z 2 1-form symmetry and the SU(N f )/Z 2 0-form flavor symmetry are combined in so(2n c ) QCD. The label 'product' means that they form a direct product, while the label 'extension' means that they form a nontrivial 2-group.
3 SL(2, Z 2 ) action and the anomalies In the last section we determined the 2-group structure of the so(2n c ) gauge theories with 2n f flavors, by studying the group of the charges of line operators. Here we determine the anomalies of these symmetries, utilizing the SL(2, Z 2 ) action on the set of QFTs with Z 2 1-form symmetry.
3.1 SL(2, Z 2 ) action and so gauge theories Let us say that we are given a four-dimensional spin QFT Q with Z 2 1-form symmetry. We denote its partition function on a manifold X by Z Q [E], where we suppress the dependence on X in the notation, and E ∈ H 2 (X; Z 2 ) is the background field for the Z 2 1-form symmetry. We then define SQ and T Q to be QFTs with partition functions given by the formula where P : H 2 (−; Z 2 ) → H 4 (−; Z 4 ) is a cohomology operation called the Pontrjagin square. We can show that S 2 = T 2 = 1 and (ST ) 3 = 1, meaning that they generate SL(2, Z 2 ). This operation was introduced in [GKSW14] as an analogue of the SL(2, Z) action on 3d QFTs with U(1) symmetry of [Wit03] and then further studied in [BLT20]. Importantly, Spin(2n c ) and SO(2n c ) ± gauge theories with 2n f flavors with the same (n c , n f ) form a single orbit under this SL(2, Z 2 ) action. More precisely, we need to make a distinction between Spin(2n c ) and T (Spin(2n c )), and similarly between SO(2n c ) ± and T (SO(2n c ) ± ) respectively, where the theories with T prepended are different from the original ones only by its discrete theta coupling to the background. Then we have the following chain of actions: 3.2 SL(2, Z 2 ) actions with extra background Let us now study what happens if we perform this SL(2, Z 2 ) action when the Z 2 1-form symmetry in question is part of a larger symmetry group. So far we have been considering the effect of SU(2n f )/Z 2 0-form flavor symmetry, but the discussions in the last section show that, at a formal level, only the background field a 2 ∈ H 2 (X; Z 2 ) matters, which controls the lift from an SU(2n f )/Z 2 bundle to an SU(2n f ) bundle. Let us regard a 2 as the background field for a flavor Z 2 1-form symmetry.
Then, it is combined with the original Z 2 1-form symmetry into either Z 2 × Z 2 or Z 4 , and we perform the SL(2, Z 2 ) action by picking a Z 2 subgroup. The symmetry and the anomaly of the resulting theory can be determined by a formal argument independent of the dynamics of the theory, once those of the original theory and the action of the anomaly-free subgroup to be gauged are given, as discussed in [Tac17].
Let us work at the level of anomalies described by cohomology, since we do not need to deal with more general anomalies described by bordism. We consider a d-dimensional QFT with a symmetry group G with an anomaly specified by a cochain α ∈ C d+1 (BG; U(1)). We pick a subgroup H ⊂ G such that α trivializes in it, so that one can find its trivialization µ ∈ C d (BH; U(1)) satisfying α| H = δµ. We then gauge H, using µ as the action.
What determines the symmetry and the anomaly of the gauged theory is the data (µ, α). Clearly, given ν ∈ C d (BG; U(1)), the pair (µ, α) and the pair (µ − ν| H , α − δν) should give the same result, since we merely added the counterterm ν to the action. This allows us to always choose the pair of the form (0, α ) equivalent to a given (µ, α), by taking ν to be an arbitrary lift of µ from H to G. This is convenient in discussing the SL(2, Z 2 ) action, since our S operation is defined in the convention that µ = 0.
At this stage, the residual identifications (0, α ) ∼ (0, α ) are of the form α = α + δν, where ν ∈ C d (BG; U(1)) is required to satisfy ν| H = 0. Their equivalence classes form the relative cohomology group H d+1 (BG, BH; U(1)). 12 The four choices: Now, what are the possible choices of (µ, α) ∼ (0, α ) we need to discuss? Let us first consider Z 2 × Z 2 1-form symmetry. As detailed in the Appendix B, the only possible anomaly for 4d spin QFTs with this symmetry is where B, E ∈ H 2 (Y ; Z 2 ) are the background fields on the bulk 5d spin manifold Y , and we use Q/Z-valued cochains to describe the anomaly. Its restriction to Z 2 1-form symmetry subgroup is trivial i.e. α| H=Z 2 = 0, and thus the possible choice of µ is simply the discrete theta angle where P is the Pontrjagin square. This µ can be lifted from the Z 2 subgroup to the entire Z 2 × Z 2 group as a closed cochain, and therefore does not affect the gauging process. Therefore, we only have to consider pairs (0, 0) and (0, α). Next, we consider Z 4 1-form symmetry. In the Appendix B, we show that there is no anomaly for Z 4 1-form symmetry. Therefore we can pick α = 0. Then the only possible choice of µ for the Z 2 1-form subgroup is again the discrete theta angle (3.4). One difference here is that the discrete theta angle (3.4) cannot be lifted as a closed cochain to the entire Z 4 1-form subgroup. As discussed in the Appendix C, with δE = βa 2 where a 2 ∈ H 2 (X; Z 4 /Z 2 ), one finds where β 2 is the higher Bockstein operation associated with the short exact sequence and a 2 is the lift of a 2 to a Z 4 -valued cochain; see also footnote 4. We conclude that the pairs we need to consider for the Z 4 1-form symmetry are (0, 0) and (µ, 0) ∼ (0, α ). Summarizing, we need to consider the following four choices, namely: • For Z 2 × Z 2 , the pairs (0, 0) and (0, α), which we call 'none' and 'anomaly' • For Z 4 , the pairs (0, 0) and (µ, 0) ∼ (0, α ), which we call 'extended' and 'extended T '.
SL(2, Z 2 ) action on the four choices: Let us now determine how the SL(2, Z 2 ) action affects these data. The case 'none' is very easy. The additional Z 2 factor plays no role, and we find the chain of actions given by In the rest of this subsection, we will establish the chain of actions We already explained above that T (i.e. adding the discrete theta angle (3.4)) leaves 'anomaly' unchanged, while it exchanges 'extended' and 'extended T '. To establish the chain above, we then need to show that S exchanges 'extended' and 'anomaly' while leaves 'extended T ' unchanged.
That S exchanges 'extended' and 'anomaly' was in fact already reviewed in the Introduction, around (1.4) and (1.5), where we started from 'extended', gauged the Z 2 subgroup of Z 4 , and found the 'anomaly', as first demonstrated in [Tac17].
That S leaves 'extended T ' unchanged was established in [HL20]. We will provide a slightly different explanation than the one given there. Recalling that 'extended T ' can be obtained by performing the T transformation on 'extended', its S transformation then involves the coupling exp 2πi where E is the variable to be gauged and B is the newly introduced background field. As Z 2 to be gauged is the Z 2 subgroup of Z 4 1-form symmetry, E is not necessarily closed, but rather satisfies the relation δE = βa 2 , (3.10) where a 2 is the background field for the quotient Z 4 /Z 2 1-form symmetry. Then the second term in (3.9) is not closed, and to even talk about the first term in (3.9), one first needs to extend the definition of the Pontrjagin square P to non-closed cochains, as we discuss in Appednix C.
To make the coupling (3.9) well-defined, we consider adding a counterterm 1 4 P(B) depending solely on the newly introduced field B to (3.9), i.e. we perform a further T transformation. The total coupling is now exp 2πi This theory is perfectly well-defined and has no anomaly, if the newly-introduced background field B also satisfies δB = βa 2 , (3.12) since δ(B + E) = 0. This means that, starting from 'extended' and performing T , S, and T , we come back to 'extended'. Therefore, simply performing S for the theory of the type 'extended T ', one finds 'extended T '. This establishes the chain of actions shown in (3.8).

Anomalies from SL(2, Z 2 ) action
Let us now combine our result in Table 4, which summarizes our knowledge whether the 0-form symmetry and the 1-form symmetry form a nontrivial 2-group, and the SL(2, Z 2 ) actions (3.7) and (3.8) on the four choices we determined above. We first need to double each column of Table 4, since we need to distinguish Spin from T (Spin) and SO ± from T (SO ± ). The entry 'product' in Table 4 corresponds to either 'none' or 'anomaly', and the entry 'extension' there corresponds to either 'extended' or 'extended T '. We now demand that the SL(2, Z 2 ) action (3.2) on so QCD to be compatible with the SL(2, Z 2 ) action on the labels, (3.7) and (3.8). The only consistent assignment is given in Table 5. As the way we determine the symmetry structures were somewhat indirect, we confirm the structure of the Spin case in the next section in a different means.
(n c , n f ) (even, even) none none none none none none (odd, even) extended T extended anomaly anomaly extended extended T (even, odd) anomaly anomaly extended extended T extended T extended (odd, odd) extended extended T extended T extended anomaly anomaly Table 5: The symmetry structure of so(2n c ) QCD with 2n f flavors, as deduced from the 2-group structures found in Sec. 2 and from the SL(2, Z 2 ) action discussed in this section. The symmetry structure of the Spin case, colored in purple, will be checked independently in Sec. 4. The action of Intriligator-Seiberg duality is also superimposed using orange arrows.
We can also use this Table 5 to give a further check of the Intriligator-Seiberg duality, which is known to act as follows, as shown in [GKSW14,Sec. 6]: (3.13) We displayed this action in Table 5 using orange arrows; we see that the symmetry structures are indeed preserved across the duality.

Fermion contribution to anomalies
So far, we first determined the 2-group structure in Sec. 2 by studying the charges of line operators, and then determined the anomalies in Sec. 3 by matching it to the action of SL(2, Z 2 ). Going over the entries on the column Spin of Table 5, we find that the anomaly is trivial when (n c , n f ) is (even, even) or (odd, even), while it is α given in (3.3) or α given in (3.5) when (n c , n f ) is (even, odd) or (odd, odd), respectively. Since the 1-form symmetry background in the Spin theory is simply the Stiefel-Whitney class w 2 of the SO(2n c ) gauge bundle, these anomalies should simply come from the anomalies of fermions charged under Here we use USp(2n f ) instead of SU(2n f ), because under the latter we also have perturbative anomalies, which would complicate the analysis.
For even n c , the anomaly should be given by where w 2 , a 2 ∈ H 2 (X; Z 2 ) controls the lifts from an SO(2n c ) bundle to a Spin(2n c ) bundle and from a USp(2n f )/Z 2 bundle to a USp(2n f ) bundle, respectively. For odd n c , the anomaly cochain should be given by where x 2 ∈ H 2 (X; Z 4 ) is the class controlling the lift from an SO(2n c )/Z 2 = Spin(2n c )/Z 4 bundle to a Spin(2n c ) bundle. We note that, as explained in the previous section, α is exact as a cocycle on B( ) but defines a nontrivial element in the relative cohomology ), BSO(2n c ); U(1)). As such, this cochain still affects the gauging process. The aim of this last section is to give a check of these anomalies from a different point of view. We will proceed as follows. Starting from the theory where the fermions are charged under SO(2n c ) × USp(2n f ), we add scalar fields which are adjoint under USp(2n f ) in the system, and break it down to a subgroup. We then determine the effective interaction induced by the fermion zero modes. The next step is to see what happens when the symmetry group is changed from SO(2n c ) × USp(2n f ) to its Z 2 quotient; we will see that the effective interaction will have the required anomalies.
Before proceeding, we have two remarks. First, this method was first used in [Wit95,Sec. 4] to understand 'a curious minus sign' appearing in the topologically-twisted Seiberg-Witten theory, which was more recently recognized as determining an anomaly in [CD18,Sec. 2.4.3]. It was also used in [WWW18, Sec. 3.1 and 5.1.2] to relate the 'new' SU(2) anomaly with the effective interaction in the U(1) theory. Second, in this section we can only say that the effective interaction we find is compatible with the anomalies as found in Sec. 3, and will not be able to determine the anomalies completely. This is mostly due to the fact that the computation of the spin bordism group Ω spin ) which governs the anomaly is quite hard, because even the integral cohomology of the classifying space in question is hard to compute, at least to the authors. Only in a couple of cases we can say more, as we comment along the way.

Effective interaction
We break USp(2n f ) down to U(n f ) using a scalar field, such that the fundamental representation of USp(2n f ) splits into the fundamental plus the anti-fundamental representation of U(n f ). The monopole charge is given by the first Chern class c 1 of the low-energy U(n f ) flavor symmetry.
Take a standard 't Hooft-Polyakov monopole associated with U(1) ⊂ USp(2) and embed it into U(n f ) ⊂ USp(2n f ). The fermion zero modes form a vector representation of SO(2n c ), whose quantization leads to the spinor representation. As first discussed in [Tho14] and also used in [WWW18,Sec. 3.1], this means that there is an effective interaction 1 2 w 2 (SO(2n c ))c 1 (U(n f )). (4.3) One way to understand it is as follows. We started from a system which has SO(2n c ) symmetry, but the spinor representation is only a projective representation of this symmetry. There is an anomaly at the core of the monopole, which needs to flow in from the bulk. Indeed, taking the spacetime to be X = R ≥0 × R t × S 2 around the monopole, and reducing the bulk term (4.3) on S 2 with S 2 c 1 = 1, we have the effective interaction 1 2 Y w 2 on the half-space Y = R ≥0 × R t , with the monopole living on the boundary. Therefore, the degree of freedom on the boundary is in the projective representation characterized by w 2 ∈ H 2 (BSO(2n c ); Z 2 ).

Anomalies
We now change the symmetry group from SO(2n c ) × USp(2n f ) to SO(2nc)×USp(2n f ) Z 2 by taking the Z 2 quotient. Note that π 1 (U(n f )/Z 2 ) = Z × Z 2 or Z depending on whether n f is even or odd. We denote by a 2 the obstruction class to lift a U(n f )/Z 2 bundle to a U(n f ) bundle. This implies the following: • When n f is even, c 1 (U(n f )) = c 1 (U(n f )/Z 2 ) and a 2 (U(n f )/Z 2 ) = a 2 (USp(2n f )/Z 2 ).
(4.9) This is the anomaly we wanted to see. When n c = 2 and n f = 1, we can confirm that this is indeed the entire anomaly, since we can compute Hom(Ω spin 5 (B SO(4)×USp(2) Z 2 ), U(1)) and show that this is the only nontrivial element there. For details, see Appendix B.3.
(n c , n f ) = (odd, odd) : Now we make the replacement on both sides and therefore the effective interaction is 1 8 x 2 (SO(2n c )/Z 2 )c 1 (U(n f )/Z 2 ) (4.10) and δ 1 8 which is the pull-back of Recall that the symmetry we are now considering is , and therefore there is a single degree-2 obstruction cochain which equals both v 2 and a 2 , and therefore the anomaly cochain is This is what we wanted to show.
A 2-group structure, line-changing operators, and crossed module extensions In this paper we have encountered the 2-group extensions such as (2.4) in massless so QCD. Here we put our observation there into a more general framework. A similar remark was made very recently in [Bha21, Sec. 2].

A.1 Physics setup
Let us generally consider a theory with a 0-form symmetry G and a discrete 1-form symmetry A. The Pontrjagin dual of the 1-form symmetry group A can be identified with the following group: 13 where the quotient via ∼ means that we identify two line operators L 1 and L 2 if there exists a line-changing operator between them. 14 Two line operators can be connected by a line-changing operator, but the operator is not necessarily consistently acted on by the 0-form symmetry group G, which is defined to act faithfully on the local operators. In this situation, we can also define the group where the quotient by ∼ is similar to the previous one by ∼, but here only the line-changing operator consistently acted on by the 0-form symmetry group G is considered. This groupÂ fits in the following short exact sequence which dually forms the short exact sequence The lines inĈ are equivalent to trivial lines under the equivalence relation ∼. Therefore, a line labeled byĉ ∈Ĉ can end on a point operator which is in a nontrivial projective representation of G, andĉ controls the projective phase. Equivalently, such a point operator is in a representation of G which is an extension of G by C: 13 There can be nontrivial p-form symmetries that act trivially on all of the p-dimensional objects in the theory. One of the examples is the 0-form symmetries of a 3d Chern-Simons TQFT. Another example for Z 2 1-form symmetry is found in [HT19]. Such symmetries (in general topological operators) are called the condensations [GJF19]. Here we ignore these symmetries.
14 To be precise, we identify L 1 and L 2 if there exists a line operator L 3 such that there exists a point operator connecting L 1 , L * 2 , L 3 , L * 3 with * being the orientation reversal. The freedom to include L 3 is necessary to make A a group in general, for example in a 3d TQFT, but can be ignored in our non-topological gauge theory example.
Combining, we have an exact sequence of groups where G is the group faithfully acting on the whole set of line-changing operators. Now, the extension (A.5) is characterized by an element w 2 ∈ H 2 (G, C). We can then use the Bockstein operator β associated to (A.4) to obtain an element βw 2 ∈ H 3 (G, A), which is the data characterizing the 2-group extension.

A.2 Mathematical remark
Since the dawn of time, humans wondered how to find an interpretation for H 3 (G, A) and higher cohomology groups analogous to the fact that H 2 (G, A) classifies extensions This was achieved e.g. in [Hol79]. 15 The statement goes as follows. Given G and A, one considers all extensions of the form where N is not necessarily Abelian, and we furthermore require that N is a crossed module over G, i.e. there is an action of g ∈ G on n ∈ N which we denote as g n, such that a(n) n = nn n −1 , a( g n) = ga(n)g −1 . (A.9) Let us denote such an extension by (N, G). For two such extensions we denote by (N, G) ⇒ (N , G ) if we can make the following diagram commute: where the first and the fourth down arrows are isomorphisms and the second and the third are homomorphisms. Then, we say (N, G) ≈ (N , G ) when there is a chain where the last arrow can be oriented in either direction. The fundamental result proved in [Hol79] is that the extensions of the form (A.8) satisfying (A.9) under the equivalence relation ≈ form the group H 3 (G, A). It was further shown in [Hol79,Prop. 2.7] that we can always choose N to be Abelian. In this case, the conditions (A.9) reduce to the fact that A and A are G-modules and the sequence (A.8) is compatible with the G action. Therefore, our setup in Sec. A.1 actually covers all possibilities of extension classes α ∈ H 3 (G, A). In particular, there always is a choice of a coefficient sequence 0 → A → A → C → 0 (A.4) such that α = βw for an element w ∈ H 2 (G, C) with β the Bockstein operation.

B Bordism group computations
The bordism groups Ω spin • (X) for X = B p+1 G are known [FH16,YY21] to capture the anomalies of p-form symmetry G. More precisely, the anomalies of d-dimensional spin QFT are characterized by (d + 1)-dimensional spin invertible QFTs, whose deformation classes form a group Inv d+1 spin (X) which sits in the middle of the following short exact sequence Note that the information on global (non-perturbative) anomalies is encoded in the part while that on local (perturbative) anomalies is encoded in the part both of which correspond to bordism invariants.
In this appendix, we compute these bordism groups Ω spin • (X) for various classifying spaces, using the Atiyah-Hirzebruch spectral sequence associated with the trivial fibration In short, the spectral sequences have the E 2 -terms given by ordinary homology groups H p X; Ω spin q , and they converge to the desired bordism groups. For a more detailed introduction especially aimed at physicists, see e.g. [GEM18] and references therein.
The (reduced) bordism group Ω spin d (X) to be computed characterizes the anomalies of Z 2 × Z 2 1-form symmetry in spin QFTs. Since B 2 (Z 2 × Z 2 ) = B 2 Z 2 × B 2 Z 2 , the necessary information on (co)homology is derived from those of the Eilenberg-MacLane space B 2 Z 2 = K(Z 2 , 2). Here, the Z 2 -(co)homology is known [Ser53] to be H * (K(Z 2 , 2); Z 2 ) = Z 2 [u 2 , Sq 1 u 2 , Sq 2 Sq 1 u 2 , · · · ], (B.4) where Sq i are the Steenrod operations, among which Sq 1 coincides with the Bockstein homomorphism β associated with the short exact sequence while the Z-homology of K(Z 2 , 2) can be read off from [Cle02]. Then, with the help of the Künneth formula which says that, for a principal ideal domain (PID) R, there are short exact sequences which are split, the E 2 -page of the Atiyah-Hirzebruch spectral sequence is filled as The horizontal and vertical axes correspond to p and q respectively; this will be the convention throughout the appendix.
The necessary information on (co)homology can be obtained by using the Leray-Serre spectral sequence, whose E 2 -terms are H p (B; H q (F ; Z)) and converges to H • (E; Z) for the fibration For the case of interest, the relevant fibration is where the cohomology of the fiber is known to be while that of the base is derived from together with the use of the Künneth formula. As a result, the E 2 -page is filled as It turns out that the differential d 5 : E 0,4 → E 5,0 must be nontrivial to account for the allowed instanton numbers. 16 As a result, we end up with the following integral cohomology structure d 0 1 2 3 4 5 6 · · · H d (B SO(4)×SU(2) 16 To explain this point in more detail, note first that the E 2 -page implies that H 4 (B SO(4)×SU(2)

Z2
contains three su(2) factors, let c 2 , c 2 be the instanton numbers of two su(2) factors of the SO(4)/Z 2 part, so that p 1 = 4c 2 and p 1 = 4c 2 are the generators of E 4,0 . Similarly, let c 2 be the instanton number of the SU(2) part, i.e. the generator of E 0,4 . Now, in B SO(4)×SU(2)

Z2
; Z) = Z ⊕3 is obtained by extending H 4 (B(SO(4)/Z 2 ); Z) = Z ⊕2 by the Z generated by 2 c 2 . This means that the differential d 5 in question needs to be a mod-2 reduction i.e. nontrivial.

C Coboundary of Pontrjagin square for non-closed cochains
The aim of this section is to determine the coboundary of the Pontrjagin square of non-closed cochains. Recall that the Pontrjagin square for an element x ∈ C • (−; Z 2 m ) is defined to be where x ∈ C • (−; Z) is an integral lift of x, and ∪ 1 is the higher cup product of Steenrod. The variation of interest is then given by If x is a Z 2 m -cocycle, then x is a cocycle mod 2 m i.e. δ x = 0 (mod 2 m ), and the RHS of (C.2) is 0 mod 1, which then means that P(x) is a Z 2 m+1 -cocycle. However, when x is not a cocycle but merely a cochain, P(x) is also not a cocycle. For our purpose, we limit ourselves to the case δx = βy (C.3) for a cocycle y ∈ Z 2 (−; Z 2 ). Recalling that the Bockstein operation β is associated with the short exact sequence 0 −→ Z 2 2 −→ Z 4 −→ Z 2 −→ 0, (C.4) x and y combine to define a cocycle z ∈ Z 2 (−; Z 4 ), such that z = y (mod 2) and z = 2 x when y = 0. This motivates us to consider the term 1 4 · 1 4 P( z), (C.5) which reduces to 1 4 P( x) when y = 0, as its general replacement. Using (C.2), we find δ 1 4 · 1 4 P( z) = 1 2 · 1 8 · 2 · δ z ∪ z − δ z ∪ 1 δ z where β 2 is the higher Bockstein operation associated with the short exact sequence defined for cocycles y ∈ Z 2 (−; Z 2 ) which are Z 4 -liftable to z ∈ Z 2 (−; Z 4 ).