Toric 2-group anomalies via cobordism

2-group symmetries arise in physics when a 0-form symmetry $G^{[0]}$ and a 1-form symmetry $H^{[1]}$ intertwine, forming a generalised group-like structure. Specialising to the case where both $G^{[0]}$ and $H^{[1]}$ are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such `toric 2-group symmetries' using the cobordism classification. As a warm up example, we use cobordism to study various 't Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of $B|\mathbb{G}|$ where $\mathbb{G}$ is any 2-group whose 0-form and 1-form symmetry parts are both $\mathrm{U}(1)$, and $|\mathbb{G}|$ is the geometric realisation of the nerve of the 2-group $\mathbb{G}$. By leveraging a variety of algebraic methods, we show that $\Omega^{\mathrm{Spin}}_5(B|\mathbb{G}|) \cong \mathbb{Z}/m$ where $m$ is the modulus of the Postnikov class for $\mathbb{G}$, and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) $\mathrm{U}(1)$ global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure.


Introduction
In quantum field theory, anomalies are loosely defined to be quantum obstructions to symmetries. More precisely, anomalies can themselves be identified with (special classes of) quantum field theories in one dimension higher than the original theory, via the idea of 'anomaly inflow'. 1 This modern viewpoint led to an algebraic classification of anomalies via cobordism, which was made rigorous by Freed and Hopkins [3] following many important works (including [4][5][6][7][8]). The cobordism classification includes all known anomalies afflicting chiral symmetries of massless fermions in any number of dimensions, as well as other more subtle anomalies that involve discrete spacetime symmetries (see e.g. [9][10][11]). The cobordism group that classifies anomalies, for d spacetime dimensions and symmetry type S, 2 is the Anderson dual of bordism [3], where M T S denotes the Madsen-Tillman spectrum associated to symmetry type S, IZ denotes the Anderson dual of the sphere spectrum (with Σ d`2 denoting the d`2fold suspension), and rX, Y s denotes the homotopy classes of maps between a pair of spectra X and Y . In fact, this cobordism classification goes beyond what we would normally think of as 'anomalies' to cover all reflection positive invertible field theories (with or without fermions) with symmetry S in d`1-dimensions.
To unpack the meaning and significance of the cobordism group H d`2 IZ pM T Sq, it is helpful to recall that it fits inside a defining short exact sequence [3] Ext 1 Z pπ d`1 pM T Sq, Zq ãÑ H d`2 IZ pM T Sq ↠ Hom pπ d`2 pM T Sq, Zq , (1.2) 1 Mathematically, any anomalous theory in d dimensions can be described by a relative quantum field theory [1] between an extended field theory in one dimension higher (the 'anomaly theory'), and the trivial extended field theory (see e.g. [2]). 2 A given symmetry type S, as defined precisely in [3], includes both the spacetime symmetry and an internal symmetry group, as we as maps between them, extended to all dimensions.
where π k pXq is the k th stable homotopy group of a spectrum X. For the special case of chiral fermion anomalies in d dimensions, the invertible field theory in question is precisely the exponentiated η-invariant [8] of Atiyah, Patodi and Singer [12][13][14] in dimension d`1. In that case, integrating the gauge-invariant anomaly polynomial Φ d`2 provides an element of the right factor Hom pπ d`2 pM T Sq, Zq. Theories with Φ d`2 " 0, which are in the kernel of the right map and thus in the image of the left map, correspond to residual 'global anomalies' which are thus captured by the left-factor. For theories with symmetry type S " SpinˆG, where G is the internal symmetry group, we have that Ext 1 Z pπ d`1 pM T Sq, Zq -Hom´Tor Ω Spin d`1 pBGq, R{Z¯classifies global anomalies, where Ω Spin ‚ denotes spin bordism. A straightforward corollary is that if the group Tor Ω Spin d`1 pBGq vanishes there can be no global anomalies, which has been applied to various particle physics applications in recent years [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].
Concurrent with this development of the cobordism classification of anomalies, in the past decade the notion of 'symmetry' has been generalised beyond the action of groups on local operators, in various exciting directions (for recent reviews, see Ref. [33,34]). This includes the notion of p-form symmetries [35], which act not on local operators (the case p " 0) but more generally on extended operators of dimension p. These symmetries couple to background fields that are pp`1q-form gauge fields. Such higher-form symmetries are found in a plethora of quantum field theories. The most well-known examples include a pair of Up1q 1-form symmetries in 4d Maxwell theory, and a Z{N 1-form symmetry in 4d SUpN q Yang-Mills theory; the latter exhibits a mixed anomaly with charge-parity symmetry when the SUpN q θ-angle equals π, which led to new insights regarding the vacua of QCD [36]. Other important classes of generalised symmetry include non-invertible and categorical symmetries, as well as subsystem symmetries. These shall play no role in the present paper.
Soon after the notion of p-form symmetries was formalised in [35], it was realised that higher-form symmetries of different degrees can 'mix', leading to the notion of pgroup symmetry. The most down-to-earth example of this symmetry structure, which in general is described using higher category theory (see e.g. [37]), is the notion of a 2-group symmetry whereby a 1-form symmetry H r1s and a 0-form symmetry G r0s combine non-trivially. This occurs, for example [38,39], when a bunch of Wilson lines (which are charged under a discrete 1-form centre symmetry) can be screened by a dynamical fermion (which is charged under a 0-form flavour symmetry). 2-group symmetries were first observed, in their gauged form, in string theory in the context of the Green-Schwarz mechanism, in which a Up1q 1-form symmetry combines nontrivially with a diffeomorphism [40][41][42][43]. Their appearance as global symmetries in field theory was appreciated first by Sharpe [44], before many new examples were identified in Refs. [45,46]. Further instances of 2-group symmetry have since been discovered and analysed in 4d [38,39,[47][48][49], 5d [46,48,[50][51][52], and 6d [53][54][55] quantum field theories. A 2-group symmetry structure has recently been studied in the Standard Model [56], in the limit of zero Yukawa couplings.
In this paper we study how the cobordism classification of anomalies can be applied to 2-group symmetries. As a first step, we here specialise to 2-groups for which both G r0s and H r1s are compact, connected, abelian groups ergo tori -Up1q n ; we refer to such a structure, at times, as a 'toric 2-group'. In this case, there are 1-form and 2form Noether currents j p1q G and j p2q H associated with the 0-form and 1-form symmetries respectively, and the non-trivial 2-group 'mixing' between these symmetries is reflected in a fusion rule for these currents, schematically for a known (singular) function Kpxq. Toric 2-group symmetries appear, for example, in chiral Up1q gauge theories in 4d when there are mixed 'operator-valued' anomalies between global chiral symmetry currents and the gauge current [45], as well as in 3d [57] and in hydrodynamics [58]. Toric 2-group symmetries thus naturally appear in fermionic systems, for which anomalies can be more fruitfully analysed using cobordism (compared to purely bosonic anomalies). For such a toric 2-group G, we study the bordism theory of manifolds equipped with 'tangential G-structure'. This can be defined analogously to a tangential S-structure for ordinary symmetry type S, given now the particular 2-group G plus an appropriate form of spin structure. When the 2-group symmetry does not further mix with the spacetime symmetry, this reduces to studying the spin bordism groups of the topological space that classifies G-2-bundles.
Thankfully much is known about the classifying space of G-2-bundles that will be of use to us, given the physics applications we have in mind. Quite generally, the classifying space of a 2-group G can be computed as B|G|, where |G| is the geometric realization of the nerve of the 2-group G viewed as a category [59,60], which is itself a topological 1-group -albeit often an infinite-dimensional one. This means that Freed and Hopkins' classification of invertible field theories with fixed symmetry type can be applied directly, taking the tangential structure to be Spinˆ|G|.
In the special case of a pure 1-form symmetry valued in an abelian group H r1s " A, this formula for the classifying space of the corresponding 2-group reduces to BpBAq, recovering the well-known result for classifying abelian gerbes [61]. This special case, which is of significant physical interest, has already been used in the physics literature to study 1-form anomalies, for many examples with variously Spin, SO, and O structures in Ref. [18]. An extension of this to a pure 2-form symmetry and its anomalies in 6d gauge theories was studied via cobordism in Ref. [24].
More generally for any 2-group G, the classifying space B|G| always sits in a 'defining fibration' B`BH r1s˘Ñ B|G| Ñ BG r0s , (1.4) which allows one to leverage the usual spectral sequence methods to compute e.g. the cohomology and/or the (co)bordism groups of B|G|. For 'non-trivial' 2-groups, i.e. those for which the fibration (1.4) is non-trivial, there is to our knowledge only a smattering of calculations in the literature; in particular, the case BpBZ{2 r1s q Ñ B|G| Ñ BO r0s was studied by Wan and Wang [18] (Section 4), and BpBUp1q r1s q Ñ B|G| Ñ BSUp2q r0s was studied by Lee and Tachikawa [24] (Appendix B.6). In this paper we extend these works with a dedicated cobordism analysis of theories with 2-group symmetry and their anomalies. The content of this paper is as follows. After reviewing 2-groups and their appearances in quantum field theory in §2, we then describe the topological spaces B|G| that classify 2-group background fields, and show how these spaces can be computed, in §3. By considering appropriate tangential structures for 2-group symmetries, this then leads to a definition of the cobordism groups relevant for describing 2-group symmetries and their anomalies ( §3.2). We then apply this cobordism theory to study a small selection of examples in detail, focussing on toric 2-group symmetries as an especially tractable case. The main examples that we study in this paper are as follows. In §4, which is a warm up example, we study anomalies involving the pair of 1-form symmetries in 4d Maxwell theory, defined on non-spin manifolds. As well as the Z-valued 'local' mixed anomaly between the two 1-form symmetries [35,[62][63][64], there is a trio of Z{2-valued global anomalies (involving gravity) that only appear on nonspin manifolds; the associated phases have been studied in [65]. In §5 we turn to our main example of toric 2-group symmetries, with non-trivial Postnikov class, occurring in 4d abelian chiral theories. We see, using cobordism, how turning on the Postnikov class transmutes a Z-valued local anomaly into a torsion-valued global anomalyà la the Green-Schwarz mechanism -we precisely recover the physics results derived by Córdova, Dumitrescu and Intriligator in Ref [45]. Finally, in §6 we drop down to two dimensions, where anomalous 0-form symmetries can be recast as 2-group symmetries by using the trivially conserved 1-form symmetry whose 2-form current is simply the volume form [44]. Our cobordism analysis highlights the prominent role played by spin structures; when the Up1q r0s anomaly coefficient is odd, the Green-Schwarz mechanism cannot in fact absorb the anomaly completely but leaves a Z{2-valued anomaly, which we identify as an anomaly in the spin structure; we also show how this Z{2-anomaly is trivialised upon passing to Spin c bordism. 3 2 Review: 2-groups and their classifying spaces In this Section we recall the notion of a 2-group G ( §2.1), and briefly reprise their appearance as generalised symmetry structures in quantum field theory ( §2.2).

What is a 2-group?
We begin by sketching some equivalent definitions of a 2-group, for which our main reference is Ref. [60] by Baez and Stevenson. First and most concise is probably the definition of a 2-group G as a higher category. Loosely, a 2-group G is in this language a 2-category that contains a single object, with all 1-morphisms and 2-morphisms being invertible.
(Slightly) less abstractly, a 2-group can also be described using 'ordinary' category theory, without recourse to higher categories, and this can be done in two ways; either as a groupoid in the category of groups, or as a group in the category of groupoids. 4 Expanding on the former viewpoint, a 2-group is itself a category whose objects form a groupG r0s , and whose morphisms also form a group, where the latter can be written as a semi-direct product betweenG r0s and another groupH r1s , viz.
ObjpGq "G r0s , MorpGq "H r1s¸Gr0s , (2.1) such that all the various maps involved in the definition of a groupoid are continuous group homomorphisms (see §3 of [60]). We will usually be interested in situations where bothG r0s andH r1s are Lie groups. This definition involving a pair of groups pG r0s ,H r1s q is closely related to yet another, somewhat more practical, definition of a 2-group as a topological crossed module. The data specifying a topological crossed module consists of a quadruplet G " pG r0s ,H r1s , t, αq, whereG r0s andH r1s can be identified with the groups that appear in (2.1), and t : H r1s ÑG r0s , α :G r0s Ñ AutpH r1s q are continuous homomorphisms such that the pair 3 These examples emphasize that bordism computations are useful tools for keeping track of such subtle mod 2 effects, by automatically encoding various characteristic classes' normalisation that can differ in spin vs. non-spin theories -for example, the class c 2 1 is even on a spin manifold but needn't be on a non-spin one. 4 It will be implicitly assumed that all groups and groupoids discussed carry a topology.
of conditions t pαpgqphqq " gtphqg´1, αptphqqph 1 q " hh 1 h´1 (2.3) hold @g PG r0s , h, h 1 PH r1s , which can be thought of as equivariance conditions on the maps. To define a 2-group from this data, one takes the group of objects ObjpGq "G r0s and the group of morphisms MorpGq "H r1s¸Gr0s , where the semi-direct product is defined using the map α as ph, gq¨ph 1 , g 1 q " phαpgqph 1 q, gg 1 q, and the various source, target, and composition maps from MorpGq, as well as identity maps from ObjpGq, are defined using simple formulae (for which we refer the interested reader to §3 of [60].) We can tie together this circle of definitions by linking this final crossed module definition of G to the first definition of G as a 2-category; in that picture,G r0s is the group of 1-morphisms, andH r1s is the group of 2-morphisms from the identity 1morphism to all other 1-morphisms [66]. WhenG r0s andH r1s are both Lie, and when the maps t and α are both smooth, G is a Lie 2-group.
The final description of a 2-group G that we just gave, as a topological crossed module, is still of somewhat limited use in physics because the groupsG r0s andH r1s are not preserved under equivalence of 2-groups as 2-categories. Moreover, a physical system with a 2-group symmetry (possibly with associated background fields) is only sensitive to the equivalence class of the 2-group at long distance, not the 2-group itself, as described by Kapustin and Thorngren [66]. It is therefore more convenient to work with equivalence classes of 2-groups from the beginning, which we describe next.
For completeness, we also define the notion of 2-group homomorphisms, in terms of ordinary group homomorphisms between different elements of the associated crossed modules [60]. Represent two 2-groups G and G 1 by crossed modules pG r0s ,H r1s , t, αq and pG 1r0s ,H 1r1s , t 1 , α 1 q. A 2-group homomorphism f : G Ñ G 1 , which is a functor such that f : ObjpGq Ñ ObjpG 1 q and f : MorpGq Ñ MorpG 1 q are continuous homomorphisms of topological groups, can be represented by a pair of maps h :H r1s ÑH 1r1s and g :G r0s ÑG 1r0s such that the diagram is commutative and that h, g are compatible with α, α 1 : for all a PG r0s , b PH r1s .

2-group symmetries in quantum field theory
Physically, an equivalence class G " pG r0s , H r1s , α, βq of 2-groups describes a symmetry structure appearing in quantum field theories which have both a 0-form symmetry group G r0s and a 1-form symmetry group H r1s . When the two symmetries are not independent, the Postnikov class β of the corresponding 2-group symmetry is non-trivial, and the 0-form and 1-form parts cannot be analysed in isolation. While we will eventually specialise to the case of toric 2-groups in the bulk of this paper, we first recap how 2-groups appear in field theory more broadly. It is convenient to distinguish two broad categories of 2-groups which in general have different physical origin: This class of 2-groups arises in field theory when, for example, the gauge transformation law for the 1-form symmetry background gauge field is modified so that there is no operator-valued 't Hooft anomaly involving the background gauge field for the 0-form symmetry, in an analogue of the Green-Schwarz mechanism for global symmetries [45]. The 0-form symmetry can here be abelian or non-abelian, connected or disconnected, and compact or non-compact. In this paper we focus our attention on the special case where both the 0-form and 1-form symmetry are connected, compact, and abelian groups, which we refer to as a 'toric 2-group'.

2.
A discrete 2-group G is one with a discrete 1-form symmetry group H r1s . This class of 2-group symmetries arises in gauge theories when the gauge Wilson lines, which are charged objects under the 1-form symmetry H r1s , are not completely screened in the presence of the background gauge field for the 0-form symmetry group G r0s . The local operator O that screens the gauge Wilson lines when the 0-form background gauge field is turned off is also charged under the 0-form symmetry. Hence, when the background gauge field of G r0s is turned on, O transmutes a gauge Wilson line into a flavour Wilson line [39,46,47].

Cobordism with 2-group structure
In this Section we describe the background gauge fields on associated principal G-2bundles which play an important role in field theories with 2-group symmetries. We focus on describing the topological spaces B|G| that classify these background fields, and show how to calculate such classifying spaces in elementary examples, before introducing the cobordism groups that are central to this paper in §3.2.

Background fields and their classifying spaces
A theory with a 2-group symmetry G can be coupled to a background gauge field (which a physicist might wish to decompose into components of an ordinary 1-form gauge field and a 2-form gauge field), 5 which is a connection on a principal G-2-bundle in the sense defined in [59]. Mathematically, Bartels moreover shows that such 2-bundles over a manifold X are classified by theČech cohomology groupȞ 1 pX, Gq with coefficients valued in the 2-group G. Baez and Stevenson prove [60] that there is a bijectioň where rA, Bs denotes the set of homotopy classes of maps from A to B, and |G| is the geometric realization of the nerve N G of the 2-group G when viewed as a groupoid as in (2.1). The nerve N G of the category G with ObjpGq " G and MorpGq " H¸G is a set of simplices that we can construct out of these objects and morphisms -we will see some explicit examples shortly. Quantum field theories with 2-group symmetry G are thus defined on spacetime manifolds X equipped with maps to B|G|, just as a theory with an ordinary symmetry G is defined on spacetimes equipped with maps to BG.

Elementary examples of B|G|
Since the classifying space B|G| will play a central role in what follows, we pause to better acquaint the reader with B|G| and how it can be computed by looking at some simple examples. The calculations in this subsection are purely pedagogical -readers who are familiar with (or not interested in) such constructions might wish to skip ahead to §3.1.2.

Pure 0-form symmetries
In the case that there is no 1-form symmetry, and the 2-group symmetry simply defines an ordinary 0-form symmetry G, we expect to recover the usual classifying space of G. In this case, the 2-group corresponds to the quadruplet G " pG, 0, 0, 0q, to use the topological crossed module notation, and indeed B|pG, 0, 0, 0q| " BG. To see this from the nerve construction, we first view G as a category, which is very simple in this case: ObjpGq " G with only identity morphisms at each element. So there are no non-degenerate n-simplices when n ą 0, while the set of 0-simplices is G. Hence, the geometric realisation of the nerve |G| is simply the group G itself, and its classifying space is BG, as claimed.

Pure 1-form symmetries
The case of a 'pure 1-form symmetry', in which there is no 0-form symmetry at all, corresponds to 2-groups of the form G " p0, H, 0, 0q ": Hr1s, where H is the (abelian) 1-form symmetry group. The corresponding classifying space is [59] B|Hr1s| " BpBHq, which we often denote simply B 2 H, which coincides with the well-known classifying space of an abelian gerbe [61]. For example, when H " Up1q, B|Hr1s| is an Eilenberg-Maclane space KpZ, 3q.
An example: pure Z{2 1-form symmetry. For an explicit example that illustrates how to actually calculate the geometric realization of the nerve of such a G " Hr1s, we consider the simplest case where H " Z{2. Viewing this as a category as in (2.1), we have that ObjpZ{2r1sq consists of only one element because G is the trivial group, denoted simply by ‚, while the group MorpZ{2r1sq is just isomorphic to H " Z{2. Diagrammatically, this structure can be represented as The nerve of Z{2r1s is then a simplicial set N Z{2r1s built out of these objects and morphisms, whose low dimensional components are given as follows. etc.
This is enough information for us to work out the CW complex cell structure for its geometric realisation, which recall is the topological space that we denote |Z{2r1s|. , tells us that the 2-cell is constructed by identifying the 1-cell S 1 as the boundary of a 2-ball, with antipodal points on the boundary identified. This gives the topology of the 2-dimensional real projective space RP 2 . Building inductively in this way, one can show that |Z{2r1s| is topologically RP 8 . On the other hand, we know that this is the same as the classifying space of Z{2, i.e. |Z{2r1s| -BZ{2. Therefore, the classifying space of the Z{2 1-form symmetry bundle is B|Z{2r1s| -B 2 Z{2. This argument can be generalised to any 2-group of the form Hr1s " p0, H, 0, 0q. For any group H, it is proven that the geometrisation of the nerve |Hr1s| is the topological space BH [68], and the classifying space of Hr1s-2-bundles is thus B 2 H, coinciding with the known classification of H-gerbes when H is abelian.

2-groups with a trivial map
Moving up in complexity, let us now consider a 2-group with both the constituent 1-groupsG andH being non-trivial, but with at least one of the maps in the crossed module definition being trivial. To wit, consider a particular 2-group G " pG,H, t, αq, written using the notation of (2.2), where the map α :G Ñ AutpHq is trivial i.e. αpgq is the identity automorphism for any g PG.
An example:G " Z{2 andH " Z{2. Probably the simplest example of a 2-group of this kind is one whereG andH are both Z{2. Since AutpHq -AutpZ{2q is trivial, α is automatically trivial. IfG andH do not interact at all, meaning that the map t is also trivial (the module is 'uncrossed'), then the 2-group factorises as a product of a 0-form and a 1-form symmetry: |G| -Z{2ˆBZ{2, whose classifying space is just BZ{2ˆB 2 Z{2.
Since t :H ÑG must be a homomorphism, the only other option for t is the identity map t :´1 Þ Ñ´1. Setting G " pZ{2, Z{2, id, 0q, then as a category ObjpGq -Z{2 and MorpGq " GˆH -Z{2ˆZ{2, whose action can be fully captured in the following diagram:´1 Since the morphisms shown in the diagram are all the morphisms in this category, we can identify the morphisms p1, 1q and p1,´1q as the identity morphisms at the objects 1 and´1, respectively.
The components of the nerve N G in low dimensions are given by From these data, it is easy to see that the 0-cell is just a pair of points, the 1-cell , and the 2-cell is a 2-sphere S 2 , constructed by joining ‚´1 ‚ 1 and ‚ 1 ‚´1 as two "hemispheres", with the 1-cell at the equator.
Continuing ad infinitum we obtain |G| -S 8 . But the infinite sphere is contractible, which means it is homotopy equivalent to a point. Hence, the classifying space for G must be trivial. This result should not be a surprise because the crossed module under consideration, namely G " pZ{2, Z{2, t, 0q with t an isomorphism, is equivalent to the trivial 2-group because both G r0s " coker t and H r1s " ker t are trivial!

General B|G| as a fibration
Our discussion so far has been mostly pedagogical; the idea was to acquaint the reader with some basic computations of the classsifying spaces of 2-groups. Thankfully, more powerful tools are available to us for the computations that we will be interested in. First and foremost, for any 2-group G there is a fibration 6 Note that, because the 1-form symmetry H r1s is always an abelian group, BH r1s is itself a (topological) group, and so BpBH r1s q can be defined using the ordinary definition of the classifying space of a group. We will see in the next Subsection that this fibration is sufficient for our purpose of anomaly classification.

Cobordism description of 2-group anomalies
In order to classify anomalies associated to a 2-group G via cobordism, most of the structure contained in G as a 2-group can be discarded. As discussed in the previous Subsection, G-2-bundles are classified by the topological space B|G|, where |G| is the geometric realization of the nerve (henceforth just 'the nerve') of G, regardless of other information contained in G. Thus, we can use the nerve |G|, itself a topological group, to define a tangential structure on spacetime just as we would for an ordinary 0-form symmetry. For instance, if we are interested in a fermionic theory with internal 2-group symmetry G (with no mixing with the spacetime symmetry), we can take the symmetry 6 The extensions encoded in (3.3) are in fact classified by H 3 SM pG r0s , H r1s q [69], the Segal-Mitchison group cohomology in degree 3 (a degree that usually classifies 2-step extensions). type to be 7 S " Spinˆ|G| . (3.4) Directly applying Freed-Hopkins' classification [3] of invertible, reflection-positive field theories to this symmetry type, suggests that the cobordism group correctly classifies anomalies in d-dimensional fermionic theories with the 2-group symmetry G. Our detection of anomaly theories is therefore distilled into computing the spin bordism groups of B|G|, for which we can use the same methods as for an ordinary symmetry.
In particular, our general strategy will be to build up the spin bordism groups of B|G| by using the 'defining fibration' described above, B 2 H r1s Ñ B|G| Ñ BG r0s , as follows. We first use the Serre spectral sequence (SSS) [70] to compute the cohomology of B|G| from the fibration. Then, one can use this newly computed cohomology to compute the spin bordism groups of B|G|, for example via the Adams spectral sequence (ASS) [71]. Alternatively, we can convert the result to homology using the universal coefficient theorem, and use it as an input of the Atiyah-Hirzebruch spectral sequence (AHSS) [72] E 2 p,q " H p pB|G|; Ω Spin q pptqq . for another fibration pt Ñ B|G| Ñ B|G|. Kapustin and Thorngren already used the SSS associated to (3.3) in [66] to compute the cohomology of B|G|. Moreover, this approach has been used to calculate bordism groups relevant to 2-group symmetries with non-trivial Postnikov class; Wan and Wang did so using the ASS in Ref. [18], while Lee and Tachikawa used the AHSS in Ref [24]. We usually adopt the AHSSbased approach in this paper.

Maxwell revisited
As a (rather lengthy) warm-up example, let us discuss Maxwell theory. We consider a 4d Up1q gauge theory without matter, and variants thereof in which we couple the action to various TQFTs without changing the underlying symmetry structure. The action for vanilla Maxwell theory is where a denotes the dynamical Up1q gauge field and ‹ denotes the Hodge dual, assuming a metric g on M 4 . This theory has two Up1q 1-form global symmetries [35], referred to as 'electric' and 'magnetic' 1-form symmetries. Since each is associated with a continuous group, one can write down the corresponding conserved 2-form currents, which are The former is conserved (d ‹ j e " 0) by Maxwell's equations d ‹ da " 0, and the latter by d 2 " 0 (and using ‹‹ " 1). 8 It was already observed in [35] that there is a mixed 't Hooft anomaly between the two 1-form symmetries, which is a local anomaly that can be represented by an anomaly polynomial dB e^d B m , where B e and B m are background 2-form gauge fields for the electric and magnetic 1-form symmetries respectively. This kind of anomaly and its consequences concerning various quantum numbers are discussed in Ref. [63,64] (see also [62]). In Ref. [65], a variety of 5d SPT phases that have Maxwell-like theories on the 4d boundary, protected by electric and magnetic 1-form symmetries, were derived. Here we show how these results are simply captured by the cobordism classification.

Cobordism classification of 1-form anomalies
Maxwell theory has two global Up1q 1-form symmetries, and no 0-form global symmetries. The classifying space of this global symmetry structure G is that of an abelian H-gerbe with H " Up1q eˆU p1q m . From (3.2), Since there are no fermions, we do not need a spin structure, and so can define pure Maxwell theory on any smooth, orientable 4d spacetime manifold. 9 The 5d invertible field theories with this symmetry type are therefore classified by the generalized cohomology group where this sequence splits, but not canonically. In terms of anomaly theories, the factor on the right of this SES captures local anomalies, and the factor on the left captures global anomalies; the abelian group that detects all anomalies is isomorphic to the direct product of the two.

Local anomalies
In Appendix B.1 we compute that Ω SO 6 ppB 2 Up1qq 2 q " Z, and so corresponding to the integral of the degree-6 anomaly polynomial dB e Y dB m on a generator of Ω SO 6 ppB 2 Up1qq 2 q. This detects the local mixed 't Hooft anomaly between the two 1-form symmetries discussed in [35].

Global anomalies
In Appendix B.1 we also compute that Ω SO 5 ppB 2 Up1qq 2 q " pZ{2q 3 . Thus, the group classifying global anomalies is In terms of characteristic classes, these three factors of Z{2 correspond to where w 2,3 denote the second and third Stiefel-Whitney classes of the tangent bundle, and τ e,m is a unique generator of H 3 pB 2 Up1q e,m ; Z{2q (c.f. Appendix A.1) which can be thought of as a mod 2 reduction of an integral cohomology class represented by dBe,m 2π .

Phases of non-spin Maxwell theory
The w 2 w 3 invariant corresponds to a global gravitational anomaly seen on non-spin manifolds, related to that in [11]. The w 2 τ e,m bordism invariants correspond loosely to 't Hooft anomalies for each of the 1-form symmetries, that prevents them from being gauged on certain (non-spin) gravitational backgrounds. We discuss these in more detail next.
To see the global anomaly that is captured by the w 2 Y τ e , 10 we must go beyond the vanilla Maxwell theory described by (4.1), and couple 4d Maxwell to a topological term. The modified action is where r¨s Z denotes an integral cohomology class and ρ 2 is mod 2 reduction, and where we shall write c 1 for both the first Chern class of the Up1q gauge bundle and its mod 2 reduction. The topological term couples a background magnetic 2-form gauge field to the Z{2 part of the magnetic 1-form symmetry, and then equates it to the second Stiefel-Whitney class of the tangent bundle. The electric 1-form symmetry shifting a Þ Ñ a`λ, where λ is a closed 1-form, remains intact. This theory describes fermionic-monopole electrodynamics; the topological term ş w 2 Y c 1 forces all monopoles to become fermionic [73,74]. To see this, following [11], consider adding a magnetic monopole via an 't Hooft line operator of charge 1 along ℓ Ă M 4 that we excise from M 4 with the boundary condition that ş The theory is now defined on the complement M 1 4 of ℓ in the presence of the 't Hooft operator, which is a manifold with boundary BM 1 4 -S 2 ℓˆℓ . For the integral ş M 1 4 w 2 pT M 1 4 q Y c 1 to make sense on a manifold with boundary, we need a trivialisation of the integrand on BM 1 4 . 11 Now, since c 1 is non-trivial on S 2 ℓ due to the non-zero monopole charge, we have to trivialise the w 2 pT M 1 4 q factor. A trivialisation of the second Stiefel-Whitney class is nothing but a spin structure. Since there is a unique spin structure on S 2 , a spin structure on S 2 ℓˆℓ is the same as a spin structure along ℓ. Therefore, to define the additional phase, we must choose a spin structure along the monopole worldline: the monopole is a fermion. 10 For simplicity, we consider the case where the 1-form symmetry is electric. The corresponding story for the magnetic 1-form symmetry can be obtained by applying the electric-magnetic duality. 11 This is because for an n-manifold with boundary M 1 , the fundamental homology class rM 1 s is in H n pM 1 , BM 1 q. The integration of a cohomology class on M 1 is in fact a pairing between a cohomology class and the fundamental class rM 1 s, so when we write ş M 1 c for a cohomology class c P H n pM 1 q, we need to first find a class in H n pM 1 , BM 1 q that "corresponds" to c to make sense of the pairing. This is always possible because the long exact sequence in cohomology implies that any class c P H n pM 1 q can always be lifted to a class C P H n pM 1 , BM 1 q. The choice of this lift is the choice of trivialisation of c on BM 1 .
In order to see the 't Hooft anomaly afflicting Up1q e , we now couple the theory to a background electric 2-form gauge field B e , and promote the 1-form Up1q e global symmetry transformation above to a 'local' transformation by relaxing the condition that the 1-form parameter λ has to be closed. In fact, λ does not strictly need to be a 1-form on M 4 -more precisely, we now consider shifting the gauge field a by any connection λ. For a connection λ with non-zero curvature dλ, the action (4.7) shifts under a Þ Ñ a`λ, by which encodes the 't Hooft anomaly associated with certain 'large 2-form gauge transformations', and on certain gravitational backgrounds. For example, take M 4 to be CP 2 . As usual, we can parametrize CP 2 with three complex coordinates z 1 , z 2 , and z 3 , such that ř 3 i"1 zi z i " 1 and with the equivalence z i " e iα z i for α P R{Z. Define the 2-form ω :" i 2 BB logpz 2 1`z 2 2 q, which is just the volume form on the S 2 -CP 1 Ă CP 2 submanifold defined by z 3 " 0. Its cohomology class rωs Z can be taken as a generator for H 2 pCP 2 ; Zq, and likewise a :" ρ 2 rωs Z can be taken as a generator for H 2 pCP 2 ; Z{2q. We also have that w 2 pT CP 2 q " a. If we shift a by a connection λ with " dλ 2π ‰ Z " rnωs Z " nrωs Z , the shift in the action is where x¨,¨y here denotes the pairing between mod 2 homology and cohomology, thus realising the Z{2-valued global anomaly for odd values of n. As usual, there is a dual description of this anomaly in terms of the 5d SPT phase that captures it via inflow to the 4d boundary. If M 4 were nullbordant, the phase would here be S SPT " πi ş X w 2 Y τ e where recall τ e " ρ 2 " dBe 2π ‰ Z , for a 5-manifold X such that BX " M 4 and to which the background fields are extended. To see this, first note that one could cancel the anomalous shift (4.8) by adding a 'counter-term' where x w 2 is a closed 2-form constructed from w 2 , 12 recalling that the gauge transformation for B e is B e Þ Ñ B e`d λ. However, this '4d action' is not properly quantised. (It is 'half-quantised', precisely because there is a Z{2 anomaly.) One must instead write it as the 5d action S ct. "´πi ş X w 2 Y τ e . Thus the fermionicmonopole electrodynamics has exactly the right anomaly to be a boundary state of the 12 To construct x w 2 , first note that there is an integral lift W 2 P H 2 pM 4 ; Zq of w 2 P H 2 pM 4 ; Z{2q because w 3 " 0 for any orientable 4-manifold. (One sees this from the long exact sequence in cohomology associated to the coefficient sequence 0 Ñ Zˆ2 Ý Ý Ñ Z ρ2 Ý Ñ Z{2 Ñ 0, for which the Bockstein connecting homomorphism β : H 2 p¨; Z{2q Ñ H 3 p¨; Zq : w 2 Þ Ñ βpw 2 q where ρ 2 βpw 2 q " w 3 ). Next, given the integral class W 2 , construct any complex line bundle over M 4 with c 1 " W 2 , and take x w 2 to be the curvature 2-form of that bundle.
Pure electrodynamics mixed U(1) [1] e × U(1) Figure 1. A map of the four possible 't Hooft anomalies that can afflict a 4d theory with a pair of Up1q 1-form global symmetries, defined on orientable manifolds without a spin structure. Each 't Hooft anomaly corresponds to a factor in the cobordism group H 6 IZ -ZˆpZ{2q 3 , and is exhibited by Maxwell theory on its own (top, corresponding to the local Z-valued anomaly) or coupled to a particular TQFT (left, bottom, and right, corresponding to the trio of global Z{2-valued anomalies). The red arrows illustrate the relations between these four theories. For the global anomalies, these versions of 4d electromagnetism can also be realised as the boundary theories for 5d SPT phases.
SPT phase given by S SPT " πi ş X w 2 Y τ e (see also Ref. [65]). 13 Bordism generator for the w 2 Y τ e anomaly. Of course, it is important to note that CP 2 is not nullbordant in Ω SO 4 , given the signature σ " 1 is a 4d bordism invariant. Given we saw the anomaly explicitly on CP 2 , one cannot in fact realise it via the counterterm S ct. "´πi ş X w 2 Y τ e because there is no 5-manifold X bounded by M 4 . But as is well-known [75], the phase of the partition function Z on such a non-nullbordant manifold suffers from an ambiguity, since it can always be shifted by a choice of generalized theta angle, i.e. a coupling to a non-trivial TQFT corresponding to an element in HompΩ SO 4 p¨q, R{Zq. Thus, one can fix arg ZrM 4 , B e s " θ 0 to a reference phase, and then arg Z is uniquely defined on any other 4-manifold pM 1 4 This phase can be calculated by constructing a 5d mapping torusX by taking a cylinder that interpolates between pM 4 , B e q and pM 1 4 , B 1 e q, and gluing its ends to make a closed 5-manifold. ThisX will be a representative of the generator of the Z{2 factor in Ω SO 5 ppB 2 Up1qq 2 q that we have claimed is dual to w 2 Y τ e . In more detail, take M 4 " CP 2 and let B 0 e denote any reference choice of background 2-form gauge field for the electric 1-form symmetry on M 4 . Next take the product manifold CP 2ˆr 0, 2πs with a product metric. Over the interval I " r0, 2πs one implements (for example linearly) the 'large' 2-form gauge transformation B e Þ Ñ B 1 e :" B e`n ω, n P Z. 14 Since B e and B 1 e are gauge-equivalent, the manifold CP 2ˆr 0, 2πs can be glued at 0 and 2π to make a closed 5-manifold with Up1q-2-bundle: nω, (4.10) where t P I.
To verify that this mapping torus can be taken as a generator of the bordism group Ω SO 5 ppB 2 Up1qq 2 q, it is enough to evaluate the bordism invariant 'anomaly theory' w 2 Yτ e on rXs and find a non-trivial value for the phase. Physically, evaluating xw 2 Y τ e , rXsy computes the phase accrued by the partition function on pΣ, B e q upon undergoing the 2-form gauge transformation B e Ñ B 1 e . Doing the computation, we have that τ e " " dt 2π ‰ Y rnωs " y Y nx P H 3 pX; Z{2q, where x is the non-trivial element of H 2 pX; Z{2q obtained by pulling back a along the projectionX Ñ CP 2 , and y is the non-trivial element of H 2 pX; Z{2q obtained by pulling back the generator of H 1 pS 1 ; Z{2q along X Ñ S 1 . For n odd, we have that τ e " y Y x, the unique non-trivial element of H 3 pX; Z{2q. Thus w 2 Y τ e " x 2 y P H 5 pX; Z{2q for n odd, and so the anomaly theory evaluates to 1 mod 2 on this mapping torus, while it is trivial for n even.
The w 2 Y τ m phase An identical account can be given for the magnetic 1-form symmetry, if one replaces the topological coupling πi ş denotes the first Chern class of the electromagnetic dual of the gauge field. In that case, the 4d theory suffers from a 't Hooft anomaly afflicting the magnetic 1-form symmetry, which obstructs Up1q m r1s from being gauged on non-spin manifolds such as CP 2 . The 5d anomaly theory is ş M 5 w 2 Y τ m . Physically, this same anomaly can be understood from a different perspective. Suppose one couples vanilla Maxwell theory (4.1) to a charge-1 fermion, defined on all orientable 4-manifolds using a Spin c structure. The electric 1-form symmetry is explicitly broken, but one can still probe anomalies involving the magnetic 1-form symmetry and the gravitational background. The magnetic 1-form symmetry remains intact, even though we effectively have a 'charge-1 2 ' monopole due to the constraint c 1 " w 2 mod 2 2 that follows the definition of Spin c , because this monopole is not dynamical but rather acts as a constraint on the field configurations that are summed in the path integral. The symmetry type is now Spin cˆG m , with B|G m | " B 2 Up1q m , for which the cobordism group H 6 IZ pM T pSpin cˆG m qq -Z{2ˆZ 2 . 15 Of course, all the anomalies involving the electric 1-form symmetry are absent, as is the w 2 w 3 anomaly because the Spin c requirement trivialises w 3 , but the w 2 Y τ m global anomaly remains.
The final Z{2-valued global anomaly that is detected by (4.6) corresponds to the 5d SPT phase To exhibit this anomaly, one starts from vanilla Maxwell theory and couples w 2 pT M 4 q as a background gauge field to both the electric and magnetic Up1q 1-form symmetries. This SPT phase is purely gravitational and has been extensively analysed in the literature. Its boundary states include the theory of all-fermion electrodynamics [11,76] and a fermionic theory with an emergent SUp2q gauge symmetry [11].

Scalar QED and 1-form anomaly interplay
Now let us couple Maxwell theory, defined as before with an SO structure, to a charge-2 boson. This coupling to matter breaks the electric 1-form symmetry down to a discrete remnant, Up1q e r1s Ñ Z{2 e r1s , (4.11) while the full Up1q m r1s magnetic 1-form symmetry is preserved. This version of scalar QED therefore furnishes us with two global 1-form symmetries, one discrete and one continuous, and no 0-form symmetries. The classifying space of the global symmetry G 1 is therefore 12) and the 5d invertible field theories with this symmetry type are classified by the sequence (4.4) but with |G| replaced by |G 1 |. In Appendix B.2 we compute the relevant bordism groups to stitch together this generalized cohomology group.
Local anomalies: We compute from Appendix B.2 that and so there are no local anomalies; clearly, the mixed local 't Hooft anomaly 'dB e Y dB m ' can no longer be realised now that one of the 1-form symmetries is broken to a discrete group, because the associated 2-form gauge field now has zero curvature.
Global anomalies: On the other hand, we find that Ω SO 5 pB 2 Z{2ˆB 2 Up1qq " pZ{2q 4 . The group classifying global anomalies is now and we notice that there is an extra factor of Z{2 corresponding to an extra global anomaly. At the level of characteristic classes, we can represent these four global anomalies in terms of products of Stiefel-Whitney classes and cohomology classes of H ‚ pB 2 Up1qˆB 2 Z{2; Z{2q that are in degree-5. There are five of them in total: where u 2 is the unique generator of H 2 pB 2 Z{2; Z{2q (see Appendix A.1). However, once pulled back to an orientable 5-manifold X, Wu's relation [77] tells us that Sq 2 Sq 1 u 2 " w 2 pT Xq Y Sq 1 u 2 , so it is not an independent bordism invariant, and we end up with four invariants, as expected from the bordism group computation.

The anomaly interplay
Given an appropriate map π between two spectra M T H and M T H 1 , there is an induced pullback map between the corresponding cobordism theories, π˚: IZ pM T Hq, that can be used to relate anomalies between the two theories. This idea of 'anomaly interplay' has recently been used, for example, to relate local anomalies in 4d Up2q gauge theory to Witten's SUp2q anomaly [78], to study anomalies in Z{k symmetries in 2d [79,80] with applications to bootstrapping conformal field theories, and to derive anomalies in non-abelian finite group symmetries in 4d [31]. Physically, the idea of 'pulling back anomalies' from one symmetry to another is not new, but goes back to Elitzur and Nair's analysis of global anomalies [81], following Witten [82]. In all these cases the symmetry type takes the form tSpin or SOuˆG r0s , where G r0s is a 0-form symmetry. In that case, a map of spectra π is induced by any group homomorphism π : G Ñ G 1 .
In the present case, where we have theories with 1-form global symmetries, it is straightforward to adapt this notion of anomaly interplay (which is just pullback in cobordism). Again, the crucial fact we use is that the nerves associated with the 1-form symmetries are themselves just ordinary (topological) groups, between which we can define group homomorphisms. Letting π : Z{2 Ñ Up1q : p1,´1q Ñ p1, e iπ q denote the subgroup embedding, there is an associated map between symmetry types, π : SOˆ|G 1 | Ñ SOˆ|G| and an induced pullback map between the cobordism theories, Note that, since H ‚ IZ is a contravariant functor, the map between anomaly theories goes in the opposite direction to the subgroup embedding that we started with. Moreover, there is a pullback diagram for the whole short exact sequence characterizing H 6 IZ , which encodes the notion of 'anomaly interplay': Since all the generators of the cobordism groups can be represented by characteristic classes, we can represent the pullback π˚by its action on the characteristic classes. The non-trivial action of π˚is encoded in Trivially, π˚maps w 2 Y τ m and w 2 Y w 3 to themselves. The most interesting pullback relation is Eq. (4.17), which says that a Z-valued local anomaly pulls back to a Z{2-valued global anomaly. This is somewhat analogous to the interplay studied in [23,78] between Up2q local anomalies and SUp2q global anomalies for 4d 0-form symmetries, where the maps there corresponded to pulling back exponentiated η-invariants.
To see how the interplay works in this example involving 1-form symmetries, which is arguably simpler than the story for chiral fermion anomalies, we follow the general methodology set out in Ref. [23]. To wit, we start with a 5-manifold M 5 representative of a class in Ω SO 5 pB 2 Z{2 eˆB 2 Up1q m q that is dual to u 2 τ m (i.e. on which u 2 τ m evaluates to 1 mod 2). We choose factor, and θ P r0, 2πq parametrizes S 1 θ . The important thing is the flux relation Recalling τ m " rdB m {2πs Z , integrating gives xx 2 Yτ m , rM 5 sy " k mod 2. When k is odd, M 5 is not nullbordant, and the background fields cannot be simultaneously extended to any 6-manifold that M 5 bounds. Now, to show that this global anomaly is the pullback under π˚of the local anomaly τ e Y τ m , we simply use the subgroup embedding π : Z{2 Ñ Up1q to embed the Z{2 2connection in an Up1q 2-connection B e . One can take The 5-manifold (4.19) equipped with these structures B e and B m can be regarded as the pushforward in bordism of the M 5 that we started with, π˚M 5 . This is nullbordant in Ω SO 5 pB 2 Up1q eˆB 2 Up1q m q; it can be realised as the boundary of a six-manifold e is one half of a 3-sphere S 3 that is bounded by S 2 e , to which the electric 2-form connection (4.21) can now be extended with ş S 3 dBe 2π " 2n. 16 Using Stokes' theorem, we thus have obtaining the same phase from τ e τ m as we did from u 2 τ m . On the other hand, we only need to see that the characteristic class τ e reduces to the class Sq 1 u 2 when we restrict to a flat 2-bundle with mod 2 2-holonomy, in order to show that w 2 Y τ e pulls back to w 2 Y Sq 1 u 2 . To see this most clearly, it is best to use the language of cochains instead of differential forms. By identifying Up1q with R{Z, we represent the Up1q 2-form gauge field B e by a real 2-cochain b e . If the gauge field is flat, δb e must be trivial as a cochain valued in R{Z. In other words, δb e is an integral 3-cochain, whose cohomology class can be identified with τ e . To use this flat Up1q 2-cochain to describe a Z{2 2-form gauge field, we further impose that it must have mod 2 2-holonomy, i.e. b e must be half-integral-valued, not just any real cochain. 16 We emphasize that B e is no longer a flat connection when extended into the S 3 bulk, even though it restricts to a flat connection on the boundary of D 3 e .
Thenb e :" 2b e is an integral cochain whose mod 2 reduction defines a cohomology class in H 2 pM 5 ; Z{2q that coincides with u 2 . As Sq 1 u 2 " β 2 pu 2 q, where β 2 is the Bockstein homomorphism in the long exact sequence for cohomology induced by the short exact sequence 0 Ñ Z{2 Ñ Z{4 Ñ Z{2 Ñ 0, it can be represented by the mod 2 reduction of 1 2 δb e . But this is exactly the same as δb e which represents τ e . Therefore, the mod 2 reduction of τ e is Sq 1 u 2 when we embed the Z{2 1-form symmetry inside the Up1q 1-form symmetry.
There will be no further examples of anomaly interplay in the present paper; in particular, we do not consider any examples with non-trivially fibred 2-groups (although see footnote 28 for some comments along these lines).

QED anomalies revisited
The Maxwell examples in the previous Section exhibited only 1-form global symmetries. In this Section, we move on to theories with both 0-form and 1-form global symmetries that are fused together in a non-trivial 2-group structure.
Quantum electrodynamics (QED) in 4d with certain fermion content furnishes us with such a theory. Here both the 0-form and 1-form symmetry groups are Up1q, and the 2-group structure is non-trivial when there is a mixed 't Hooft anomaly between the global and gauged 0-form Up1q currents, as was discovered in [45]. There is a further possible 't Hooft anomaly in the 2-group structure that comes from the usual cubic Up1q anomaly for the 0-form symmetry, but, rather than being a Z-valued local anomaly, the 2-group structure transmutes this cubic anomaly to a discrete, Z{m-valued global anomaly [45], with m given by the modulus of the integral Postnikov class of the 2-group. After reviewing the physics arguments for these statements, we derive them from the cobordism perspective. Our spectral sequence calculations reproduce the order of the finite Z{m anomaly.

From the physics perspective
Consider a system of Weyl fermions coupled to a Up1q a gauge group with dynamical gauge field a. Assuming that a fermion with unit charge is present, the electric 1form symmetry is broken completely. This leaves only the magnetic Up1q r1s 1-form symmetry, with 2-form current j m " ‹ f 2π as in (4.2), where f " da. Given enough Weyl fermions, one can find a global Up1q r0s 0-form symmetry that does not suffer from the ABJ anomaly. It is possible, however, that upon coupling a background gauge field A to this global 0-form symmetry, there is an operator-valued mixed anomaly that is captured by the anomaly polynomial term where F " dA is the field strength for the background gauge field A. It describes, via the usual descent procedure, the shift in the effective action under the background gauge transformation A Þ Ñ A`dλ p0q , with λ p0q a 2π-periodic scalar, by It was realised in Ref.
[45] that we should not interpret this term as an anomaly, but rather as a non-trivial 2-group structure.
To see why this is the case, we first couple a 2-form background gauge field B p2q , which satisfies the usual normalisation condition on closed 3-manifolds, to the magnetic 1-form symmetry via the coupling Then the potential anomaly (5.2) can be cancelled by modifying the background gauge transformation for B p2q from an ordinary 1-form gauge transformation B p2q Þ Ñ B p2qd λ p1q , that is independent of the 0-form gauge transformation of A, to a 2-group gauge transformation provided that we identifyκ with´κ{2. 17 Here the gauge transformation parameter λ p1q is a properly normalised Up1q gauge field. This modified transformation mixes the magnetic Up1q r1s 1-form symmetry with the Up1q r0s 0-form symmetry, and encodes the 2-group structure at the level of the background fields. In our quadruplet notation, the 2-group is G " pUp1q, Up1q, 0,κq, (5.6) with the Postnikov classκ P H 3 pBUp1q; Up1qq -Z, where we use the fact that H 3 pBUp1q; Up1qq with continuous Up1q coefficients is isomorphic to H 4 pBUp1q; Zq -Z via the universal coefficient theorem. The nerve of such a 2-group is the extension The non-trivial 2-group structure can also be seen without turning on the background gauge fields explicitly. If we write j for the 1-form current associated with Up1q r0s , then the Ward identity takes the non-trivial form The 2-group symmetry Up1q r0sˆκ Up1q r1s can still suffer from an 't Hooft anomaly. Recall that, whenκ " 0, the anomaly polynomial for our system of Weyl fermions is where c 1 pF q " F {2π is the first Chern class of the Up1q r0s bundle, and p 1 pRq " 1 8π 2 Tr RR is the first Pontryagin class of the tangent bundle, where R is the curvature 2-form. Before we continue, it is important to discuss the role of these anomaly coefficients in the (co)bordism context, still for the caseκ " 0. Naïvely, one might think that the cubic anomaly coefficient A 3 and the mixed Up1q-gravitational anomaly coefficient A mixed are the two integers that classify anomalies according to cobordism, i.e. that A 3 and A mixed can be chosen as generators of the group Hom´Ω Spin 6 pBUp1qq , Z¯-ZˆZ. However, this is not the correct identification because A 3 and A mixed are not quite independent. It can be shown that α 1 :" c 1 pF q 3 and are independent integral cohomology classes. 18 In terms of these basis generators, we can write Φ 6 as Again, by the Atiyah-Singer index theorem, Φ 6 must be integral. Since α 1 and α 2 are integral basis generators, we can deduce that the two integers pr, sq that label Hom´Ω Spin 6 pBUp1qq , Z¯are pr, sq "ˆ1 6 pA 3´Amixed q , A mixed˙. (5.12) Continuing, let's switch the Postnikov classκ back on, and couple the background 2-form gauge field B p2q to the Up1q r1s 1-form symmetry. We are free to add a Green-Schwarz counter-term to the action. Under the 2-group transformation (5.5), the effective action shifts if the anomaly coefficients A 3 and A mixed are non-zero, by Since one may choose the counterterm coefficient n to be any integer, one sees that A 3 is only well-defined modulo 6κ [45]. It follows that the integer r " 1 6 pA 3´Amixed q that we claim classifies the anomaly is not really valued in Z, being well-defined only modulo |κ|. This corresponds to a global anomaly in the 2-group symmetry, that is valued in the cyclic group Z{m, m " |κ|. We emphasize that, even though A 3 is well-defined modulo 6κ (as was derived in [45]), the discrete group that classifies the global anomaly has order |κ| (and not 6|κ|). The mixed gravitational anomaly remains a Z-valued local anomaly.

From the bordism perspective
In this Section we show how the 't Hooft anomalies afflicting the 2-group global symmetry, that we have just described, can be precisely understood using cobordism. To do so, we need to compute the generalized cohomology groups pB|G|q, for each value of the Postnikov clasŝ κ. These abelian groups detect and classify all possible anomalies for this 2-group symmetry type.
To compute these bordism groups, we follow the general strategy outlined in §3.2. We first apply the cohomological Serre spectral sequence to the fibration KpZ, 3q Ñ B|G| Ñ KpZ, 2q (5.17) to calculate the cohomology of B|G| in relevant low degrees. Then we will convert the result into homology groups, which are fed into the Atiyah-Hirzebruch spectral sequence for the fibration pt Ñ B|G| Ñ B|G|, for which the second page is E 2 p,q " H p`B |G|; Ω Spin q pptq˘, to compute the spin bordism. So, to begin, the E 2 page of the cohomological Serre spectral sequence for the fibration (5.17) is given by Using H ‚ pKpZ, 3q; Zq -tZ, 0, 0, Z, 0, 0, Z{2, 0, Z{3, Z{2, . . .u, given in Table 3   The differentials α, β, and γ shown on the left-hand diagram of Fig. 2 are linear in the Postnikov classκ P Z (see the Appendix of Ref. [66]). More precisely, we write H 3 pKpZ, 3q; Zq -HompZ r1s , Zq -Z using the universal coefficient theorem and the fact that KpZ, 3q " B 2 Up1q r1s " B 3 Z r1s .
Here, we include the superscript to emphasise that this Z comes from our Up1q 1form symmetry part. From this, we can write the entry E 0, 3 4 in the Serre spectral sequence as HompZ r1s , Zq. Then, the differential α is given by 'contraction' (adopting the terminology of [66]) with the Postnikov class α : HompZ r1s , Zq Ñ H 4 pBUp1q r0s ; Zq where we make use of the fact that the Postnikov class is a cohomology clasŝ κ P H 3 pBUp1q r0s , Up1q r1s q -H 4 pBUp1q r0s ; Z r1s q .
Similar arguments apply for the differentials β and γ. Thus, α, β, and γ map 1 to˘κ, resulting in the E 5 page as shown, where m :" |κ|. We can then read off the integral cohomology groups to be H ‚ pB|G|; Zq -tZ, 0, Z, 0, Z{m, 0, epZ{2, Z{mq, 0, epZ{6, Z{mq, . . .u (5.20) where the notation epA, Bq denotes an extension of A by B, viz. a group that fits in the short exact sequence B ãÑ epA, Bq ↠ A. We compute the mod 2 cohomology by the same method in Appendix A.2. Heuristically, one can also argue for the form of α and β as follows (c.f. Appendix B.6 of Ref. [24]). Let us start with a representative of a generator of the cohomology group H 3 pKpZ, 3q, Zq -Z, and ask what it becomes when KpZ, 3q is the fibre of B|G|. Given the normalisation condition (5.3), the 3-form represents the generator of H 3 pKpZ, 3q; Zq -Z. This is true when KpZ, 3q stands on its own. However, when we pass to the 2-group G and take KpZ, 3q to be the fibre of B|G|, thenh is not gauge-invariant under (5.5) for a general Postnikov class, and cannot be a representative of any cohomology class for B|G|. We can remedy this by modifying the definition ofh to h :"h´κ 2π A^F.

(5.22)
The trade off is that the gauge-invariant h is no longer closed; instead dh "´κ where c 1 :" F 2π is the first Chern class of the U p1q r0s bundle, and c 1 Y c 1 can be taken as the generator of H 4 pKpZ, 2q; Zq. The 2-group relation (5.23) implies that both the differentials α and β in Fig. 2 map 1 to´κ. 19 From this cohomological starting point, we can proceed to compute the spin bordism groups. We find it is helpful to split the discussion into the cases where m is even or odd, for which the bordism group calculations are tackled using different tricks. As a warm up, we first consider the simplest case where m is zero, corresponding to a 0-form and 1-form symmetry that do not mix.

Zero Postnikov class
We first consider the trivial toric 2-group where B|G| is a simply a product space KpZ, 2qˆKpZ, 3q. As B|G| is a product, we can use the Künneth theorem to determine the homology groups of KpZ, 2qˆKpZ, 3q from the homology groups of each factor, given by Eq. (A.2) and Table 3 in Appendix A.1. We obtain H ‚ pKpZ, 2qˆKpZ, 3q; Zq -tZ, 0, Z, Z, Z, ZˆZ{2, Z, ZˆZ{6, ZˆZ{2, . . .u (5.24) and construct the second page of the AHSS as shown in Fig. 3, with non-trivial differentials on the E 2 page indicated by coloured arrows.
The differentials on the zeroth and the first rows are the composition Ą Sq 2˝ρ 2 and Ą Sq 2 , respectively, where ρ 2 is reduction modulo 2. To compute the action of these differentials, we need to know how the Steenrod squares act on the mod 2 cohomology ring of the product KpZ, 2qˆKpZ, 3q. From the mod 2 cohomology rings of KpZ, 2q and KpZ, 3q given by (A.3) and (A.4), we obtain H ‚ pKpZ, 2qˆKpZ, 3q; Z{2q -Z{2rc 1 , τ 3 , Sq 2 τ 3 , Sq 4 Sq 2 τ 3 , . . .s, (5.25) where c 1 and τ 3 are the unique generators in degree 2 and 3, respectively. In our diagram, the red maps correspond to Sq 2 c 1 " c 2 1 , the blue maps to Sq 2 τ 3 being a gener- ator of H ‚ pKpZ, 3q; Z{2q, and the magenta maps to Sq 2 pc 1 τ 3 q " c 2 1 τ 3 . The differential depicted in the E 3 page from E 3 7,0 in the diagram must be non-trivial by comparing Ω Spin 6 pKpZ, 2qˆKpZ, 3qq with the result computed with the Adams spectral sequence. We can then read off the bordism groups in lower degrees, which we collect below in Table 1. We piece together the cobordism group which classifies anomalies for this symmetry type. The pair of Z-valued local anomalies anomalies just corrresponds to the usual cubic c 3 1 and mixed gravitational c 1 p 1 anomalies associated with the Up1q 0-form symmetry. The presence of the 1-form symmetry here plays no role in anomaly cancellation for this dimension.

Even Postnikov class
Now we turn to the case where m " |κ| is a non-zero even integer. Continuing from the cohomology calculation above, summarised in Eq. (5.20), there are two options for the extension epZ{2, Z{mq; either the trivial extension Z{2ˆZ{m or the non-trivial one epZ{2, Z{mq -Z{p2mq, which are non-isomorphic. By comparing the mod 2 cohomology calculated from applying the universal coefficient theorem to (5.20) and the one calculated directly from the Serre spectral sequence, one can show that the correct extension is the direct product Z{2ˆZ{m. The integral homology for B|G| is then H ‚ pB|G|; Zq -tZ, 0, Z, Z{m, 0, Z{mˆZ{2, 0, epZ{6, Z{mq, . . .u (5.27) and the mod 2 homology is The E 2 page of the AHSS is then given by Fig. 4. We observe that there are nonvanishing differentials already on the E 2 page (which will not be the case when we turn to the case of odd m). These non-trivial E 2 differentials make the spectral sequence easier to compute, as follows. The differentials α and β on the second page are duals Ą Sq 2 of the Steenrod square Sq 2 , while the differential γ is a mod 2 reduction followed by the dual of Sq 2 [83]. The Steenrod square dual to β sends a unique generator in H 2 pB|G|; Z{2q to a unique generator in H 4 pB|G|; Z{2q (c.f. Appendix A.2), so β must be the non-trivial map from Z{2 to Z{2, killing off both factors. Similarly, the Steenrod square dual to α acts on the unique generator of H 3 pB|G|; Z{2q that comes from the generator τ 3 P H 3 pKpZ, 3q; Z{2q of the fibre, which we will also label by τ 3 . The image is a generator of H 5 pB|G|; Z{2q that comes from Sq 2 τ 3 of H 5 pKpZ, 3q; Z{2q. It generates the Z{2 factor in the mod 2 cohomology that is a reduction from the Z{2 factor in the integral cohomology, and not the Z{m factor since it is the Z{2 factor that arise from the fibre's contribution. Therefore, both α and γ must be non-trivial, with ker γ -Z{m, as indicated in the E 3 page in Fig. 4.
Before continuing, we pause here to emphasize the importance of using (spin) cobordism, rather than just cohomology, to study 't Hooft anomalies in these theories. At the level of cohomology, the existence of a non-trivial Z{2-valued cohomology class Sq 2 τ 3 P H 5 pKpZ, 3q; Z{2q might suggest there is a non-trivial SPT phase on 5-manifolds M 5 , or equivalently a non-trivial anomaly theory for the corresponding 4d theory, with partition function (5.29) where Sq 2 τ 3 is pulled back to M 5 . But, one can use Wu's relation to trade the Steenrod square operation for a Stiefel-Whitney class [77], viz.
If we now restrict to M 5 being a spin manifold, then w 2 pT M 5 q is trivial and we immediately learn that this SPT phase is trivial. Of course, this 'trivialisation' of the SPT phase corresponding to the cohomology class in H 5 pKpZ, 3q; Z{2q is automatically captured by the spectral sequence computation for spin bordism, by the non-triviality of the map γ on turning from the second to the third page of the AHSS. Continuing with the bordism computation, we now find that the entries in the range p`q ď 5 stabilise on this E 3 page, whence we can read off the spin bordism groups for 2-group symmetry G " Up1q r0sˆκ Up1q r1s with even Postnikov classκ up to degree-5, which are listed in Table 1. There is still an undetermined group extension in Ω Spin 3 pB|G|q but we will argue from the physics point of view in Section 6 that it must be the non-trivial extension Z{p2mq. In summary, our results are those given in Table 1.

Odd Postnikov class
We now turn to the case where m is odd, for which the spin bordism computation turns out to be rather more difficult, technically. Firstly, when m is odd, the extension epZ{2, Z{mq is unambiguously Z{mˆZ{2 (which is, for odd m, isomorphic to Z{p2mq). The integral homology of B|G| is as written in Eq. (5.27), and the mod 2 homology  Table 1. Spin bordism groups for the twisted 2-group symmetry G " Up1q r0sˆκ Up1q r1s .
Here, Tors denotes an undetermined pure torsion part. For comparison, we also include the bordism groups in the caseκ " 0, which corresponds to a direct product of 0-form and 1-form symmetries. One can thus see how, for both 2d and 4d quantum field theories, local anomalies in the untwisted case morph into global anomalies when the Postnikov classκ is turned on, thanks to a Green-Schwarz-like mechanism.
groups are now by the universal coefficient theorem. Now we feed these results into the AHSS for the trivial point fibration pt Ñ B|G| Ñ B|G|. The E 2 has no non-trivial differentials in the range of interest. The E 3 page is shown in Fig. 5, with a single potentially non-trivial differential in the range of interest, d 3 : E 3 5,0 Ñ E 3 2,2 . If this differential were trivial, then Ω Spin 5 pB|G|q would equal Z{mˆZ{2, and Ω Spin Ω Spin 5 pB|G|q would correspond to the global anomaly that we described in the previous Section, which we know to be valued in Z{m as in (5.15). The extra factor of Z{2 would presumably correspond to a further global anomaly that we have not seen so far by physics arguments. On the other hand, if this differential were the non-trivial map, then Ω Spin 5 pB|G|q would equal Z{m, agreeing precisely with the physics account, and Ω Spin 4 pB|G|q would just equal Z. It turns out that this all-important d 3 differential is non-trivial. But being a differential on the third page, there are no straightforwardly-applicable formulae (analogous to the formulae in terms of Steenrod squares that are available on the second page) that we can use to compute it directly. Indeed, our usual AHSS plus ASS techniques are not sufficient to constrain this differential. This differential can nonetheless be evaluated using different arguments, 20 which we present in Appendix C. The gist of the argument is as follows.
The central character is a long exact sequence in bordism groups, 21 analogous to the Gysin long exact sequence in ordinary homology, of the form¨¨Ñ Here V Ñ B|G| is the pullback bundle (which is rank 2) of the tautological line bundle L Ñ BUp1q r0s along the quotient map q : B|G| Ñ BUp1q r0s , SpV q Ñ B|G| is the associated sphere bundle of V , and pB|G|q V´2 is the Thom spectrum of the virtual bundle. This long exact sequence in bordism is extremely powerful: by proving that the groups Ω Spin 5 pSpV qq and Ω Spin 3 ppB|G|q V´2 q both lack free and 2-torsion summands, one learns that the group Ω Spin 5 pB|G|q in the middle, which is our bordism group of interest, also lacks 2-torsion. The Z{2 factor discussed above is therefore absent (from which we learn that it must be killed by the differential d 3 which is therefore non-trivial).
Putting things together, we can thus extract all the spin bordism groups for 2-group symmetry G " Up1q r0sˆκ Up1q r1s , with odd Postnikov classκ, up to degree-5.

The free part
When m is non-trivial, regardless of its parity, we can also show that the free part of Ω Spin 6 pB|G|q is given by Z coming from the entry E r 2,4 . First, observe that the free part of the Ω Spin 6 pB|G|q, if there is one at all, can only come from this entry as any other entry from the diagonal p`q " 6 is either trivial or pure torsion. Next, we need to show that E 8 2,4 is non-trivial. The only differential that could kill it along the way is d 5 : E 5 7,0 Ñ E 5 2,4 . But this differential must be trivial because E 5 7,0 is pure torsion (as E 2 7,0 , given by H 7 pB|G|; Zq -epZ{6, Z{mq in (5.27), is pure torsion), and HompG, Zq " 0 if G is pure torsion. Therefore, there is one free factor in the diagonal E 8 p,q with p`q " 6, and we obtain Hom´Ω Spin 6 pB|G|q, Z¯-Z. (5.33) From these results, we piece together the cobordism group that classifies anomalies for this global 2-group symmetry, valid for even and odd m ‰ 0: In terms of the anomaly coefficients, the space of anomaly theories is classified by pr, sq "ˆ1 6 pA 3´Amixed q mod m, A mixed˙P Z{mˆZ , m " |κ| , (5.35) in agreement with the results of the previous Subsection. To reiterate, the local anomaly associated with c 3 1 (i.e. whose coefficient is the sum of U p1q r0s charges cubed) becomes a global anomaly when the 0-form and 1-form symmetries are fused into a 2-group defined so that the mixed 't Hooft anomaly (" f^F^F ) vanishes. The mixed gravitational anomaly associated with p 1 c 1 remains a Z-valued local anomaly.
The result for the spin bordism group in degree-3 in Table 1 is also interesting, implying the presence of an novel global anomaly structure for the corresponding symmetry in two dimensions. We discuss this from a physics perspective in §6.

Abelian 2-group enhancement in two dimensions
In this Section, we consider 2d avatars of the 4d 2-group anomalies that we have been discussing. In this lower-dimensional version we will find some interesting differences.
In §5.2, we computed the spin bordism groups for the 2-group symmetry G " Up1q r0sˆκ Up1q r1s . In particular, we obtained where m " |κ|. With the fourth bordism group given by Z, we can put things together to get the cobordism group This is similar to the result for H 6 IZ that was pertinent to anomalies in 4d, except for the fact that the global anomaly is here 'twice as fine' as the Z{m anomaly in 4d. 22 That extra division by 2 will correspond to a subtle new global anomaly associated with the spin structure, which will be our main interest in this Section.
Before we discuss the global Z{p2mq anomaly in more depth, let us first discuss the physics interpretation of a 2-group symmetry G in two dimensions, which is rather different to the 4d case. Recall that in 4d, the 1-form symmetry Up1q r1s was identified with the magnetic 1-form symmetry, with 2-form current proportional to ‹f . But in general d ě 3 spacetime dimensions, the magnetic symmetry is a pd´3q-form symmetry, and so there is no such symmetry in 2d. There is, however, a trivially conserved '1-form symmetry' Up1q r1s top whose 2-form current is simply j top " vol 2 , the volume form. This symmetry does not act on any line operators in the theory and so one should not think of it as a physical 1-form symmetry -but it will play a role in what follows. Now suppose there is a global 0-form symmetry G r0s with background gauge field A with curvature F . If we first cancel the pure gravitational anomaly by adding neutral fermions, the local anomalies in G r0s are captured by the degree-4 anomaly polynomial The usual 't Hooftian interpretation of Φ 4 ‰ 0 would be that, if one tries to gauge G r0s , the anomaly 'breaks' the symmetry G r0s in the quantum theory. However, Sharpe reinterprets Φ 4 ‰ 0 as indicating a weaker 2-group symmetry structure in the 2d quantum theory [44], corresponding to the extension BUp1q ãÑ G ↠ G r0s . In other words, in 2d one can invoke the auxiliary Up1q 1-form symmetry to trade an anomaly in a 0-form symmetry for a 2-group structure. This is of course just a simpler version of the 4d situation described in §5.1, in which a mixed 't Hooft anomaly corresponding to a term " f^F^F Ą Φ 6 was re-interpreted as signalling a weaker 2-group symmetry. But in this Section we will see that, in the case that one needs a spin structure to define the field theory, it is not always possible to completely absorb the 't Hooft anomaly associated with Φ 4 by a well-defined 2-group structure. Rather, there can be a residual Z{2-valued anomaly left over, which one should interpret as an 't Hooft 22 The Z-valued local anomaly here is simply the gravitational anomaly associated with 1 24 p 1 pRqTr R 1 Ă Φ 4 in the degree-4 anomaly polynomial, which can always be cancelled by adding neutral fermions and so plays no further role in our discussion.
anomaly in the 2-group symmetry itself.

'Spin structure anomalies' in two dimensions
To see how this works, let us now be more precise and set G r0s " Up1q r0s . Letting tq i u denote the Up1q r0s charges of a set of (say, left-handed) chiral fermions, the anomaly polynomial is where c 1 " F {2π is the first Chern class of the Up1q r0s bundle. By inflow, one can use the associated Chern-Simons form I 3 " A 2 8π 2 A^F to compute the variation of the effective action under the background gauge transformation A Ñ A`dλ p0q , Now, let us try to implement the philosophy above, and trivialise this anomaly by promoting Up1q r0s to a 2-group global symmetry by fusing with Up1q r1s top . We turn on a background 2-form gauge field B p2q that couples to the trivially conserved current j top " vol 2 , via the coupling because ‹vol 2 " 1. We consider the Up1q r0s and Up1q r1s top symmetries to be fused via the toric 2-group structure G " Up1q r0sˆκ U p1q r1s top , whereκ P Z is the Postnikov class. This prescribes the by-now-familiar transformation (5.5) on the background gauge fields, under which We see that the local anomaly associated with the F^F term in Φ 4 is completely removed iffκ But, since the Postnikov classκ is neccessarily integral, this is possible only when A 2 is an even integer. Thus, there is an order 2 anomaly remaining when A 2 is odd. This chimes perfectly with our computation of the bordism group Ω Spin 3 pB|G|q in §5.2, that we quoted at the beginning of this Section, which implies that there is in general a mod 2|κ| global anomaly. If one imagines fixing the 2-group structure, in other words fixing an integerκ, then one can vary the fermion content and ask whether there is a global 't Hooft anomaly for the 2-group G. We can think of the coupling term (6.5) as a Green-Schwarz counterterm, which effectively shifts the anomaly coefficient by and so the residual anomaly is clearly This is the Z{p2mq-valued global anomaly in the 2-group symmetry that is detected by Ω Spin 3 pB|G|q. Even if we choose the minimal value forκ, a mod 2 anomaly persists iff 10) where N o is the number of fermions with odd Up1q r0s charges. One can offer a different perspective on this anomaly by thinking about a generator for Ω Spin 3 pB|G|q, which makes it more transparent how this mod 2 anomaly is related to the requirement of a spin structure. If we look back at the AHSS in Fig. 5, we see that the factor of Z{2 that ends up in Ω Spin 3 pB|G|q comes from the E 2 2,1 element, which stabilizes straight away from the E 2 page. Since E 2 2,1 " H 2´B |G|; Ω Spin 1 pptq¯, this suggests a generator for this Z{2 factor can be taken to be a mapping torus with c 1 pF q P 2Z`1 on the S 2 factor, corresponding to an odd-charged monopole, and the non-bounding spin structure on the S 1 , corresponding to a non-trivial element of Ω Spin 1 pptq -Z{2. This would suggest that the transformation by p´1q F , which counts fermion zero modes, is anomalous on S 2 with an odd-charged monopole configuration for the background gauge field. 23 We can of course see this from an elementary calculation. For our set of left-handed chiral fermions with charges tq i u, the index of the Dirac operator on S 2 , which recall is Indpi { D 2 q " n L´nR the number of (LH minus RH) zero modes of the Dirac operator, 23 We do not believe, however, that this anomaly can be detected from the torus (in this case, just a circle) Hilbert space, using the methods set out in Ref. [85]. This is because a system with one Weyl fermion with unit Up1q charge (together with a Weyl fermion of opposite chirality to cancel the gravitational anomaly) will have an even number of Majorana zero modes, which allows construction of a Z{2-graded Hilbert space even in the sector twisted by´1 P Up1q. (The failure to construct a Z{2-graded Hilbert space is one sign of anomalies involving the spin structure, as studied in [85].) is equal to by the 2d Atiyah-Singer index theorem. Thus, the total number of zero modes N :" n L`nR , which is congruent to n L´nR mod 2, satisfies Choosing c 1 pF q to be odd on S 2 , as we are free to do, we see that p´1q F , which counts these zero modes, flips the sign of the partition function when ÿ i q i " 1 mod 2 ùñ N o " 1 mod 2. (6.14) For such fermion content, the 2-group symmetry and p´1q F are therefore equivalently anomalous. (From the 2-group perspective, the anomalous transformation corresponds simply to the element e iπ P Up1q r0s . 24 ) This should be contrasted with the situation in four dimensions that we examined in §5.1. In that case, the corresponding formula for the number of fermion zero modes, on say But, for a spin 4-manifold, the Atiyah-Singer index theorem implies that c 2 1 is an even integer, and moreover cancelling the local mixed gravitational anomaly ùñ ř i q i " 0 ùñ ř i q 2 i " 0 mod 2. The index and therefore the number of zero modes is then necessarily even, meaning there is no analogous anomaly in p´1q F . This is why the spin bordism calculation exposes 'only' a mod |κ| anomaly in the 4d case, but the finer mod 2|κ| in 2d. 25 Example: the 3-4-5-0 model. To furnish an explicit example of a theory featuring this irremovable anomaly, we consider the so-called "3-4-5-0 model"; that is, a Up1q gauge theory in 2d with two left-moving Weyl fermions ψ 1 , ψ 2 with charges 3 and 4, and two right-moving Weyl fermions χ 1 , χ 2 with charges 5 and 0. The theory is free of both gauge and gravitational anomalies, and so we can ask about its global symmetries. There is a somewhat trivial Up1q χ 2 global symmetry rotating the neutral fermion χ 2 24 The existence of this global Z{2-valued anomaly, despite the fact the anomalous Up1q gauge transformation is clearly connected to the identity, further evidences the fact that global anomalies do not require the existence of 'large gauge transformations' captured by, say, π d pGq (when restricting the spacetime topology to be a sphere). This disagreement between homotopy-and bordism-based criteria for global anomalies was discussed in Ref. [25]. In this case, the global anomaly arises because an otherwise local anomaly is only incompletely dealt with by the Green-Schwarz mechanism. 25 A more mundane way to understand this difference is therefore simply as a consequence of the normalisation of the various characteristic classes on spin manifolds of the relevant dimension.
(with charge 1). Naïvely, there are two further Up1q global symmetries, call them Up1q 1 and Up1q 2 , that act on the remaining fermions, whose charges Q 1 and Q 2 can be chosen independently in one basis as follows: One immediately sees that Q 1 coincides with the gauge charge. Thus, the Up1q 1 transformation can be undone by a gauge transformation. Thus, the correct global symmetry is a product Up1q 2ˆU p1q χ 2 with charges given by: The corresponding 't Hooft anomaly for each Up1q factor of the global symmetry is odd as desired.

Non-spin generalisation
We have argued that the order 2 anomaly just described, for a 2d theory with 2-group symmetry G " Up1q r0sˆκ Up1q r1s , is intrinsically related to the requirement of a spin structure and the use of spin bordism. More broadly, all the 2-group anomalies that we study are sensitive to the full choice of tangential structure -this is especially the case when computing the order of a finite global anomaly.
To develop this idea further, we consider a theory defined with the same 2-group structure, but now using a Spin c structure. Recall that in d dimensions the group Spin c pdq is rSpinpdqˆUp1qs{Z{2, defined by identifying the element p´1q F P Spinpdq with the order 2 element of a Up1q symmetry, that we here take to be an auxiliary global symmetry used to define spinors; using a Spin c rather than a Spin structure allows us to define our fermionic theory on non-spin manifolds (indeed, all orientable manifolds up to and including dimension 4 admit a Spin c structure).
Having already computed the integral homology (5.27) of B|G| in §5.2, and using the well-known result for the Spin c -bordism groups of a point [86] (see the first row in Table 2), it is straightforward to use the AHSS  Table 2. Spin c bordism groups for a point [86], and for 2-group symmetry G " Up1q r0sˆκ Up1q r1s . to compute the Spin c bordism groups of interest. The E 2 page is shown in Fig. 6. Things could not be much simpler; in the region of interest, all differentials are vanishing and all the elements on E 2 up to the p`q " 6 diagonal stabilize to the last page. The bordism groups are recorded in the second row of Table 2. There is an extension problem that we cannot resolve using the AHSS for Ω 5 , but it is not relevant to our discussion here. It is curious that Ω Spin c 3 pB|G|q -Z{m compared to Ω Spin 3 pB|G|q -Z{p2mq. One might naïvely expect the same classification because the anomaly coefficient A 2 is shifted by 2nκ for any n P Z from the 2-group transformation of the Green-Schwarz counter-term in ş M 2 B p2q . The puzzle can be resolved by correctly identifying the four integers in Ω Spin c 4 pBUp1qq -Z 4 that properly classify all perturbative anomalies, before turning on the 2-group structure.
Let F be the field strength for the background gauge field of the Up1q 0-form global symmetry as before, and let G denote the field strength associated to the Spin c connection. 26 Consider N left-moving fermions with charges tq i u under the ordinary Up1q global symmetry and with charges tg i u under the Spin c connection. There is no constraint on the q i , but the g i must all be odd integers. The anomaly polynomial associated to this matter content is given by where p 1 pRq is the first Pontryagin class of the tangent bundle. The four anomaly coefficients that accompany each term in Φ 4 are not strictly independent from one another, so they cannot be taken as labels for each factor of Z in Ω Spin 4 pBUp1qq. To extract a set of correct integer labels, we first choose an integral basis for the anomaly polynomial in terms of linear combinations of c 1 pF q 2 , c 1 pF qc 1 pGq, c 1 pGq 2 , and p 1 pRq. One option is (6.17) We know that α 4 is integral because ş M 4 p 1 pRq{3 is just the signature of M 4 which is integral (and takes a minimal value of 1, e.g. on M 4 " CP 2 ) for any orientable 4-manifold M 4 . To see the integrality of α 3 , we apply the Atiyah-Singer index theorem for a Spin c connection coupled to a single left-moving fermion with q " 0 and g " 1.
Next, applying the same index theorem to a single left-moving fermion with q " 1 and g " 1 tells us that α 2 is an integer. Finally, the class α 1 is integral by definition, and indeed it takes a minimal value of 1. To see this, take M 4 " CP 2 with F such that ş F {2π " 1 on a CP 1 subspace. Hence c 1 pF q is the generator x of H 2 pCP 2 ; Zq, so c 1 pF q 2 " x 2 " 1 P H 4 pCP 2 ; Zq generates the top cohomology, which evaluates to 1 on the fundamental class rCP 2 s.
In terms of these basis elements, the anomaly polynomial can be written as Φ 4 " ř 4 i"1 n i α i , where the variables n i are and our arguments above tell us that each n i must be an integer. This is clearly the case for n 2 and n 3 . To see that n 1 is an integer, recall that g i are all odd. So when q i is odd, pq i´gi q is even, and vice versa. So their product must be even. Similarly, to see that n 4 is an integer, we observe that g i´1 and g i`1 are both even and differ by 2, so one of them must be divisible by 4. Hence their product is divisible by 8. When we turn the 2-group structure back on, the shift in anomaly coefficients induced by the Green-Schwarz S Ą in ş M 2 B p2q counter-term amounts to a shift It is now obvious that the local anomaly labelled by the integer n 1 is truncated down to a mod |κ| anomaly. If one first takes care to cancel the anomalies involving the Spin c connection, which amounts to choosing charges tg i u such that n 2 , n 3 , and n 4 vanish, then we find that the 2-group structure can indeed be used to completely capture the remaining 't Hooft anomaly in the ordinary Up1q p0q 0-form global symmetry, in the sense proposed by Sharpe [44]. This should be contrasted with the residual order 2 anomaly (6.7) that cannot be cancelled in this way for the theory defined with a spin structure. 27

Conclusion and outlook
In this paper, we explored anomalies in 2-group global symmetries G in field theories in d " 4 and d " 2 dimensions using the cobordism classification. We compute the bordism groups Ω H d`1 pB|G|q and Ω H d`2 pB|G|q of the classifying space B|G| of the nerve of G, with the tangential structure H " SO or Spin. The torsion part of the former group gives us the global anomaly while the free part of the latter gives the perturbative anomaly. Throughout the work, we focus on 2-groups G whose 0-form and 1-form symmetry parts are compact, connected abelian groups ergo tori, which we refer to as 'toric 2-groups'.
As a warm-up example, we first looked at Maxwell's theory in 4d, on a general orientable manifold (not necessarily spin). The 2-group symmetry in this theory contains only the 1-form symmetry part, being a product of the Up1q electric 1-form symmetry and the Up1q magnetic 1-form symmetry. The familiar mixed anomaly [35] between these two Up1q factor can be given in terms of a bordism invariant in the free part of Ω 6 . There are also three global anomalies, probed by the torsion subgroup of Ω 5 , which can only be seen on non-spin manifolds and can be interpreted as mixed anomalies between the 1-form symmetries and gravity. We also studied anomalies when one of the Up1q factor breaks explicitly to its cyclic subgroup due to the presence of a charged scalar, and related them to anomalies in pure Maxwell's theory via anomaly interplay.
When both the 0-form and 1-form symmetry constituents of G are Up1q, with the Postnikov class given byκ P Z, we used a variety of techniques to prove that the relevant spin bordism groups are Ω Spin 5 pB|G|q -Z{|κ|, Hom´Ω Spin 6 pB|G|q, Z¯-Z . (7.1) The free part of the sixth bordism group signals the existence of a perturbative anomaly in 4d, which is just the mixed anomaly between the Up1q 0-form symmetry and gravity.
On the other hand, the perturbative cubic anomaly for Up1q 0-form symmetry that would be there wereκ to vanish now transmutes into a discrete global anomaly captured by the torsion part of the 5th bordism group. These cobordism computations are in perfect agreement with the physics argument put forward by Córdova, Dumitrescu, and Intriligator [45]. We next discussed the fate of anomalies for a Up1q 0-form symmetry in 2d, for both spin and non-spin manifolds. We revisited the construction, originally due to Sharpe [44], whereby this 0-form symmetry is enhanced to a 2-group symmetry by mixing it with a Up1q 1-form symmetry whose current is given by the top form on the underlying manifold. When the anomaly coefficient A 2 for the pure Up1q anomaly is even, we found that, by choosing the Postnikov class of the enhanced 2-group to bê κ " A 2 {2, the anomaly can be fully absorbed away. On the other hand, in the presence of the spin structure, when A 2 is odd and restricting to spin-manifolds, the most one can do is reduce the anomaly down to an order 2 discrete anomaly, that we intrepret as an anomaly in the spin structure.
There are many future directions to pursue that are natural extensions of this work. One obvious alley is to look at more general classes of 2-group beyond our 'toric' assumption, e.g. 2-groups whose 0-form symmetry is a non-abelian Lie group, and/or 2-groups whose 0-form or 1-form symmetry is a finite group [39, 46, 47, 49-52, 54, 55, 88]. One can rigorously analyse anomalies in these 2-group symmetries using the cobordism classification. Doing so involves an extension of the mathematical machinery developed in this paper, and would offer a non-trivial check on other methods of seeing the anomalies that have been explored recently in the literature [89]. One can also study how these anomalies relate to the anomalies studied in this paper through a generalised version of anomaly interplay. 28 Another avenue to pursue concerns a higher-dimensional generalisation of the 2d 2-groups discussed in § 6, as follows. We saw that one can cancel (almost) all pure Up1q anomalies in 2d by enhancing the Up1q 0-form symmetry to a 2-group by making use of the top-form Up1q current. One then wonders whether there is an analogous story for 'top-group' symmetry structures in higher dimensions, e.g. 4-group symmetry in 4d, in which a 'topological 3-form symmetry' with j 4 " vol 4 is fused with the ordinary 0-form symmetry to kill the cubic anomaly. Moreover, the remnant order 2 anomaly in the 2d case is a direct result of the non-trivial role played by the spin structure. It remains to be seen whether such an interplay between 'top-group' symmetries and spin-structure anomalies persists in higher dimensions.
For all these anomalies, there is of course a dual interpretation concerning symmetry protected topological (SPT) phases of matter that are protected by these generalised symmetry types, which are rigorously classified by cobordism. Furthermore, it is conceivable that by extending the bordism computations to more generalised symmetries one could uncover novel anomalies/phases, and suggest corresponding new constraints on the dynamics of theories with such symmetry.  A.2 Mod 2 cohomology ring of the classifying space of an abelian 2-group In this Subsection, we use the Serre spectral sequence to calculate the mod 2 cohomlogy ring of B|G| when G is the abelian 2-group Up1q r0sˆκ Up1q r1s studied in §5.2 with the Postnikov classκ.
The cohomological Serre spectral sequence induced by the fibration KpZ, 3q Ñ B|G| Ñ KpZ, 2q is The entries on the E 2 page can be computed from the results given in §A.1. There is no non-trivial differentials on the E 2 or E 3 pages in the range of degrees we are interested in, and the E 2 entries in this range propagate to the E 4 page, shown in Fig. 7. As explained in §5.2, the differentials α and β on the E 4 page are given by contraction with the Postnikov classκ. Thus, they are non-trivial if and only ifκ is odd. In this case, the resulting E 5 page is shown on the right-hand side of Fig. 7. Whenκ is even, the entries stabilise already on the The entries in the range of interest stabilise on this page. On the other hand, whenκ is even, the entries stabilise already on the E 4 page.

B Additional bordism calculations B.1 Maxwell
Here we compute the bordism groups for the symmetry type relevant to 'free Maxwell theory', i.e. Up1q gauge group with no dynamical matter, which has intact electric and magnetic Up1q 1-form global symmetries, discussed in §4.
Since Up1q -S 1 -BZ, the classifying space for a U p1q 1-form symmetry B 2 Up1q -KpZ, 3q is an Eilenberg-Maclane space whose cohomology is known. Using the Künneth theorem we find the integral homology of B|G| " B 2 Up1qˆB 2 Up1q, as recorded in Table 5. We also need the oriented bordism groups of a point [93], which are recorded in Table 6.   Table 6. Oriented bordism groups for a point [93].
We next write down the AHSS for oriented bordism associated to the trivial fibration of |G| by a point, for which the second page is given by This is shown in Fig. 8. We want to read off the fifth and sixth bordism groups, which receive contributions from the p`q " 5 (blue) and p`q " 6 (orange) diagonals respectively. For the sixth bordism group, there is a single factor E 2 6,0 " Z to consider. The only possible differential is d 5 : E 5 6,0 Ñ E 5 0,5 -Z{2, whose kernel is obviously isomorphic to Z, meaning that E 8 6,0 " Z. Thus, For the fifth bordism group, the E 2 5,0 " pZ{2q 2 element stabilizes to the last page, because the only possible differential is d 4 : E 4 5,0 Ñ E 4 0,4 : Z{2 Þ Ñ Z which is therefore the zero map. For the E 2 0,5 element, it is again the d 5 : E 5 6,0 Ñ E 5 0,5 map that is important. We can in fact deduce that this differential must be zero simply because it hits the zeroth column which records the bordism groups of a point, and since there is a canonical split Ω SO ‚ pXq -Ω SO ‚ pptq ' r Ω SO ‚ pXq for X connected, where the second factor is the reduced bordism of X, the E 0,5 factor must stabilize to the last page. 29 29 Equivalently, we could have used the AHSS for reduced bordism to get the same result. This also makes it obvious that there can be no non-trivial extension, and so we can read off Ω SO 5`p B 2 Up1qq 2˘" pZ{2q 3 .

(B.4)
It is easy to read off the lower-degree bordism groups too, and we summarize the results in Table 7. Table 7. Oriented bordism groups up to degree 6, for a pair of Up1q 1-form symmetries.

B.2 Scalar QED with charge-2 boson
Next we compute the bordism groups for relevant to QED with a charge-2 boson, which has the effect of breaking the Up1q e 1-form symmetry down to a discrete Z{2 e subgroup. Using the Künneth theorem we find the integral homology of B 2 Z{2ˆB 2 Up1q, as recorded in Table 8. Table 8. Integral and mod 2 homology groups of B 2 Z{2 and B 2 Z{2ˆB 2 Up1q. Recall the homology of B 2 Up1q is recorded in the first two lines of Table 5.
We next write down the AHSS for oriented bordism associated to the trivial fibration pt Ñ B 2 Z{2ˆB 2 Up1q Ñ B 2 Z{2ˆB 2 Up1q, for which the second page is given by (see Fig. 9), We want to read off Ω 5 and Ω 6 from the stabilization of elements on the p`q " 5 and p`q " 6 diagonals respectively. For the fifth bordism group, very similar arguments to those of B.1 tell us that E 8 5,0 " pZ{2q 3 and E 8 0,5 " Z{2, and the canonical split Ω SO ‚ pXq -Ω SO ‚ pptq ' r Ω SO ‚ pXq again tells us that the extension problem is trivial, giving: For Ω 6 , there is a potentially non-vanishing differential on the fifth page, d 5 : E 7,0 Ñ E 2,5 : pZ{2q 2ˆZ {3ˆZ{4 Þ Ñ Z{2. Without knowing this differential, one cannot read off Ω 6 -the most we can conclude is that it equals Z{2, Z{4, or Z{2ˆZ{2. In any case, we have that Ω SO 6 pB 2 Z{2ˆB 2 Up1qq is pure torsion, which is all that's relevant for discussing the corresponding anomalies in 4d.

C Computation of a key differential
This Appendix is written by Arun Debray.
In this appendix, we finish the computation from §5.2.3, showing that if G is a 2-group with 0-and 1-form parts both Up1q and k-invariant equal to m times the generator of H 4 pBUp1q; Zq -Z, where m is odd, then Ω Spin 5 pB|G|q -Z{m. After the calculation in §5.2.3, the only ambiguity is in the 2-torsion, which could be either 0 or Z{2, depending on the value of a d 3 in the Atiyah-Hirzebruch spectral sequence in Fig. 5. This differential resisted several of our standard techniques to address it, as did an analogous differential in the Adams spectral sequence computing the 2-completion of Ω Spin 5 pB|G|q. Here we finish the computation of the 2-torsion subgroup of Ω Spin 5 pB|G|q in a different way, resolving the differentials only implicitly. Specifically, we fit Ω Spin ‚ pB|G|q into a long exact sequence (C.2) and compute enough other terms in the long exact sequence to determine Ω Spin 5 pB|G|q. This long exact sequence is a version of the Gysin sequence for spin bordism, and one of its homomorphisms has a geometric interpretation as a Smith homomorphism, a map between bordism groups defined by sending a manifold M to a smooth representative of the Poincaré dual to a characteristic class of M . The general version of (C.2) was written down in [32] to study Smith homomorphisms and their applications to anomalies of quantum field theories; see there and [94,95] for more on Smith homomorphisms and their role in physics.
Theorem C.1 ( [32]). Let V Ñ X be a vector bundle of rank r over a CW complex X, π : SpV q Ñ X be the sphere bundle of V , X V´r be the Thom spectrum of the rank-zero virtual vector bundle V´R r , and E ‚ be a generalised homology theory. Then there is a long exact sequencë¨¨Ý Ñ E n pSpV qq πÝ Ñ E n pXq ψ ÝÑ E n´r pX V´r q ÝÑ E n´1 pSpV qq ÝÑ . . . (C.1) The map ψ is the Smith homomorphism, but we will not need that fact; see [32] for more information.
Proof. We will show that the Thom space ThpX, V q of V is the homotopy cofibre of π, so there is the Puppe long exact sequence¨¨¨Ñ E n pSpV qq Ñ E n pXq Ñ r E n pThpX, V qq Ñ E n´1 pSpV qq Ñ¨¨¨. Once this is established, (C.1) follows: X V´r » Σ´rΣ 8 ThpX, V q, and Σ 8 induces an isomorphism for any generalised homology theory and Σ´r shifts generalised homology down in grading by r, yielding (C.1). 30 The homotopy cofibre of an inclusion A ãÑ X of CW complexes such that the image of A is a union of cells is the quotient X{A; in general, one can compute the homotopy cofibre of a map f : A Ñ X between CW complexes by changing f and X by a homotopy so that f is indeed a cellular inclusion. For π : SpV q Ñ X from the theorem statement, replace π with the inclusion i : SpV q ãÑ DpV q of SpV q into the disc bundle DpV q of V Ñ X; the map π 1 : DpV q Ñ X is a homotopy equivalence and π 1˝i " π, and the CW structures on X and the standard CW structure on D n can be used to put CW structures on SpV q and DpV q such that the image of i is a union of cells. Thus the homotopy cofibre of π : SpV q Ñ V is the quotient DpV q{SpV q, which is by definition ThpX, V q.
We are interested in E ‚ " Ω Spin ‚ , and specifically in 2-torsion. To avoid worrying about torsion for other primes, we localise at 2, meaning we tensor with the ring Z p2q of rational numbers whose denominators in lowest terms are odd. If A is a finitely generated abelian group, so that A is a direct sum of a free abelian group Z r and cyclic groups Z{p e i i of prime-power order, localising at 2 has the effect of replacing Z r with pZ p2q q r , preserving all 2 e i -torsion, and sending all odd-prime-power torsion to zero. Since we are trying to determine the 2-torsion subgroup of Ω Spin 5 pB|G|q, which is a finitely generated abelian group, we lose no relevant information by localising at 2. Moreover, since Z p2q is a flat ring, tensoring with Z p2q preserves long exact sequences (i.e. 2-localisation is exact), so we can still apply (C.1) after 2-localising.
For any 2-group H, the classifying space B|H| has a CW structure (e.g. because it is the geometric realization of a simplicial space), so we can invoke Theorem C.1 for 30 The long exact sequence (C.1), and indeed this whole appendix, could have been done with ThpX, V q, but once we apply bordism, it is conceptually helpful to replace ThpX, V q by X V´rr Ω Spin k pThpX, V qq is a bordism group of pk´rq-dimensional manifolds, and Ω Spin k pX V´r q is a bordism group of k-manifolds. The degree shift must occur somewhere, and we believe our choice is clearest.
X " B|G|. Quotienting G by its 1-form symmetry defines a map to the 0-form part of G, which on classifying spaces induces a map q : B|G| Ñ BUp1q. Let V Ñ B|G| be the pullback of the tautological complex line bundle L Ñ BUp1q by q. Then Theorem C.1, together with exactness of 2-localisation, gives us a long exact sequencë¨¨Ñ The sphere bundle of a complex line bundle is the associated principal Up1q-bundle; for L Ñ BUp1q, this is the tautological bundle EUp1q Ñ BUp1q, meaning SpLq is contractible. In general, a homotopy pullback in which one of the two legs is contractible is the homotopy fibre (denoted hofibp-q) of the other leg, meaning the map SpV q Ñ B|G| can be identified with the canonical map hofibpqq Ñ B|G|. Recall that q arose from a short exact sequence of topological groups by applying the classifying space functor; specifically, G " |Up1qr1s|, H " |G|, and K " Up1q, and q " Bp. The classifying space functor turns short exact sequences of groups into fibre sequences of spaces, meaning that for (C.4) the homotopy fibre of Bp is homotopy equivalent to BG, and this homotopy equivalence identifies the canonical map hofibpBpq Ñ BH with Bi : BG Ñ BH. For our specific choices of G, H, and K, this tells us that up to homotopy equivalence, the map SpV q Ñ B|G| is the map B|Up1qr1s| » KpZ, 3q Ñ B|G| as claimed in the theorem statement.
Proof. We check both of these using the Adams spectral sequence; the relevant differentials and extension questions are trivial for degree reasons, so these computations are straightforward analogues of Beaudry and Campbell's computations in Ref. [96]. We direct the reader to [96] for more on our proof strategy and notation. The Adams spectral sequence computes 2-completed spin bordism, not 2-localised spin bordism, but because Ω Spin 5 pKpZ, 3qq and Ω Spin 3 ppB|G|q V´2 q are finitely generated abelian groups, this distinction is not important: the 2-localisation of a finitely generated abelian group A vanishes if and only if the 2-completion of A vanishes, if and only if A lacks both free and 2-torsion summands.
For KpZ, 3q, we need as input the Ap1q-module structure on H ‚ pKpZ, 3q; Z{2q in degrees 6 and below. This was computed by Serre [90, §10]: H ‚ pKpZ, 3q; Z{2q is a polynomial algebra on generators τ 3 in degree 3, Sq 2 τ 3 in degree 5, and other generators in degrees too high to matter to us. Using this, one can compute that if Q denotes the Ap1q-module Ap1q{pSq 1 , Sq 2 Sq 3 q, then there is an isomorphism of Ap1q-modules where P is concentrated in degrees 7 and above, hence is irrelevant for our computation. We draw this decomposition in Fig. 10, left. Liulevicius [97,Theorem 3] computes Ext Ap1q pZ{2q and Beaudry-Campbell [96, Figure 29] compute Ext Ap1q p Q q, so we can draw the E 2 -page of the Adams spectral sequence in Fig. 10, right. The E 2 -page vanishes in topological degree 5, i.e. for t´s " 5, so the E 8 -page must also vanish in that degree, and we conclude Ω Spin 5 pKpZ, 3qq b Z p2q " 0. For pB|G|q V´2 , using the description of the Ap1q-module structure on H ‚ pB|G|; Z{2q in low degrees from §A.2 and the way Stiefel-Whitney classes twist the Steenrod squares of a Thom spectrum (see [96, §3.3]), we obtain an isomorphism of Ap1q-modules where Cη :" Σ´2 r H ‚ pCP 2 ; Z{2q and P 1 is concentrated in degrees 5 and above, hence is irrelevant to our computation. We draw this isomorphism in Fig. 11, left. Ext Ap1q pCηq is computed in [96, Example 4.5.6 and Figure 22], so we can draw the E 2 -page of the Adams spectral sequence in Fig. 11, right. The E 2 -page is empty in topological degree 3, so the E 8 -page is also empty, and we conclude Ω Spin   Figure 11. Left: the Ap1q-module structure on H ‚ ppB|G|q V´2 ; Z{2q in degrees 4 and below, where V is the complex line bundle associated to the map q : B|G| Ñ BUp1q given by quotienting G by its 1-form symmetry. Right: the E 2 " E 8 -page of the Adams spectral sequence computing the 2-completion of Ω Spin ‚ ppB|G|q V´2 q. There is nothing in topological degree 3, so the 2-completion of Ω Spin