The algebra of anomaly interplay

We give a general description of the interplay that can occur between local and global anomalies, in terms of (co)bordism. Mathematically, such an interplay is encoded in the non-canonical splitting of short exact sequences known to classify invertible field theories. We study various examples of the phenomenon in 2, 4, and 6 dimensions. We also describe how this understanding of anomaly interplay provides a rigorous bordism-based version of an old method for calculating global anomalies (starting from local anomalies in a related theory) due to Elitzur and Nair.


Introduction
Anomaly inflow relates fermionic anomalies to quantum field theories in one dimension higher. In the perturbative case the anomaly theory has a lagrangian description via Chern-Simons terms, while the non-perturbative generalization involves the exponentiated η-invariant [1][2][3] of Atiyah, Patodi, and Singer [4][5][6]. This idea forms the root of a more general understanding that any anomalous theory can be described by a relative field theory [7] between an extended field theory in one dimension higher, called the anomaly theory, and the trivial extended field theory [8][9][10][11]. The anomaly theory is typically a rather special type of quantum field theory, namely an invertible one. 1 In many cases the anomaly theory will also be topological. Then the anomaly theory corresponds to a map of spectra. It was proven in [11] that deformation classes of such invertible, topological theories are classified by the torsion subgroup of homotopy classes of such maps. 2 Unfortunately, perturbative anomalies due to massless chiral fermions are not of this type because the Chern-Simons anomaly theory is not strictly topological, having a mild dependence on the background metric [14]. Nonetheless, it is conjectured [11] that a broader class of physically sensible invertible theories (i.e. the reflection positive ones), not necessarily topological, are still classified up to deformation by the homotopy classes of maps between spectra -only now non-torsion elements should be included. Such non-torsion elements are required to describe perturbative fermionic anomalies, whose coefficients are in general arbitrary integers. In the case of fermionic anomalies, the torsion elements capture global anomalies (which, in the absence of local anomalies, are described by genuinely topological anomaly theories).
This formal classification of invertible field theories, and thus of anomaly theories, naturally encodes a possible interplay between global and local anomalies, which is the subject of this paper. While it has become well known that global anomalies in d spacetime dimensions are detected by the torsion subgroup of a bordism group in degree d`1, it is perhaps less widely appreciated that the local anomalies are also detected by bordism invariants, albeit in one degree higher still. The two parts are naturally combined into a dual 'cobordism group' via a short exact sequence, which coincides precisely with the group of homotopy classes of maps of spectra appearing in the classifications of Freed and Hopkins [11].
Like the universal coefficient theorem in ordinary cohomology, these short exact sequences defining the cobordism groups split, meaning the most general anomaly 'fac- 1 In fact, invertibility is not a strict requirement for an anomaly theory; a counterexample is provided by the six-dimensional N " p2, 0q superconformal field theory [9]. We shall assume invertibility of the anomaly theory in this paper. 2 Classifying such topological and invertible field theories therefore reduces to computations in stable homotopy theory (see e.g. [12,13] for examples).
tors' into its global and local parts, although crucially this splitting is not canonical. This last property allows for an interplay between the global and local anomalies of theories with different symmetries (for example between two gauge theories whose gauge groups are related by some obvious map, such as inclusion of a subgroup). Specifically, local anomalies in one theory can pullback to global anomalies in the other. (The converse is not possible.) Physically, such a pair of theories might be related to each other along an RG flow. This includes, but is not restricted to, the familiar situation whereby only a subgroup of the microscopic symmetries are linearly realised at low energies due to spontaneous symmetry breaking.
Two examples of this interplay were recently observed in U p2q vs. SU p2qˆU p1q gauge theories in four dimensions [15], defined with or without spin structures. In the spin case, a bordism computation reveals that the U p2q theory cannot suffer from global anomalies, 3 but nonetheless a U p2q theory with a perturbative anomaly can pullback to an SU p2q theory with a global anomaly. We emphasize that the idea of such an interplay is, however, far from new. For an early example of its use, Elitzur and Nair [16] showed how the global 4d SU p2q anomaly [17] can be derived from a perturbative anomaly in SU p3q. For another well-known example, Ibañez and Ross derived anomalies in discrete Z{k gauge symmetries, which are necessarily global anomalies, from local anomalies in U p1q [18,19]. It is the recent progress in classifying invertible field theories that allows one to make a precise algebraic statement of such interplay in terms of cobordism, and to see that it is a generic property of the space of anomaly field theories. One application of this formalism is therefore to provide a proper bordism-based version of Elitzur and Nair's suggestion that global anomalies can be precisely derived from perturbative anomalies in some larger group.
In §2 we explain how anomaly interplay is encoded in the non-canonical splitting of a short exact sequence that classifies invertible field theories. As a corollary we describe a method for computing global anomalies (or anomalies in subgroups) using this idea. In the rest of the paper we discuss a number of examples exhibiting the phenomenon of anomaly interplay, ascending the ladder of increasing dimension. We begin by analyzing the interplay between anomalies in U p1q and Z{2 gauge theories in two dimensions, followed by the examples of U p2q gauge theories in four dimensions that were recently analyzed in [15]. We close just as we begin, with an example of anomaly interplay between U p1q and Z{2 gauge theories, but this time in six dimensions. Other examples from recent [20,21] and not-so-recent [16,18] references are also discussed from the point of view of our formalism.

The generalities of anomaly interplay
We are concerned in this paper with anomalies that arise from integrating over massless chiral fermions in d spacetime dimensions, assuming Euclidean signature. The anomaly theory A is in this case a reflection positive invertible field theory in d`1 dimensions, but not necessarily a topological one. Specifically, its partition function is the exponentiated η-invariant associated to an appropriate pd`1q-dimensional extension of the Dirac operator, where the η-invariant is a regularised sum of the signs of the eigenvalues λ k of the Dirac operator. A possible regularization is η X " lim Ñ0`ÿ k e´ |λ k | signpλ k q{2. (2.1) When evaluated on an open pd`1q-manifold X, the phase exp 2πiη X equals the phase of the anomalous partition function living on the d-dimensional boundary BX [3]. If exp 2πiη equals unity on all closed d`1 manifolds to which the necessary structures are extended, then there is no anomaly. In the case where there are only global anomalies, the anomaly theory is strictly topological; in fact exp 2πiη becomes bordism invariant under these conditions, which is stronger than topological invariance. Such invertible field theories in n :" d`1 dimensions fit inside a formal classification in terms of maps of spectra, conjectured by Freed and Hopkins in Ref. [11]. Before we discuss the physics of anomaly interplay, we must first recap some technicalities of their conjecture. The classification of invertible field theories depends on two pieces of data; the spacetime dimension n and the symmetries of the theory. Following [11], the symmetry type pH n , ρ n q of an Euclidean quantum field theory consists of a compact Lie group H n and a homomorphism ρ n : H n Ñ Opnq (2.2) whose image (either Opnq or SOpnq Ă Opnq) constitutes the Wick-rotated spacetime symmetries, and whose kernel K defines the internal symmetries of the theory. 4 Given a fixed symmetry type pH n , ρ n q, the deformation classes r¨s def of reflection positive, invertible extended field theories in n spacetime dimensions are classified by homotopy classes of maps of spectra [11,Conjecture 8.37], Here, the source spectrum M T H is the Madsen-Tillman spectrum associated to a stabilization H of the symmetry type H n . This stabilization is a dimension-independent way of describing the symmetry type of the theory, which is technically defined to be the colimit of a sequence of compact Lie groups H m (for all m ą n) that fit inside a commutative diagram of group homomorphisms, (2.4) where all the horizontal arrows denote injections, and the squares are pullbacks. This sequence of symmetry groups can be used to construct a sequence of Madsen-Tillman spectra M T H n whose colimit is the spectrum M T H that appears in (2.3). For the mathematical definitions of all these objects, we are content to refer the reader to [11]. For our purposes, it is important to emphasize that the object M T H encodes the specific information about the symmetries of the theory.
The target spectrum Σ n`1 IZ that appears in (2.3), on the other hand, is a universal object, in particular a suspension shift of the Anderson dual of the sphere spectrum IZ. Reflection positivity is required for the corresponding Lorentzian field theory to be unitary.
The massless chiral fermion anomaly theories that we are interested in fit inside this classification, with n " d`1 and symmetry H -although we should stress that not every such deformation class in rM T H, Σ d`2 IZs can be realised as the anomaly theory from free chiral fermions. 5 In other words, if we cut a pd`1q-dimensional bulk theory with such an invertible phase, then we are not guaranteed a boundary theory of free chiral fermions in d dimensions. Gravitational Chern-Simons terms provide an obvious such counterexample when d is even.

Exact sequences of anomaly theories
The universal target spectrum IZ represents a particular generalized cohomology theory, call it H ‚ IZ . In cohomological language, a deformation class of anomaly theory, for symmetry type H and original spacetime dimension d (of the anomalous fermionic theory), is then a degree d`2 class in the H ‚ IZ cohomology of the spectrum M T H. As described in [10,11], this generalized cohomology group sits inside a short exact sequence, analogous to the universal coefficient theorem for ordinary cohomology, where π k denotes stable homotopy groups. This exact sequence will be central to our discussion, since it determines how a general anomaly is built out of both global and local parts. The exact sequence splits, though not canonically (as we discuss soon). Let us unpack this short exact sequence in a special case of particular interest, namely where fermions are defined using a spin structure and charged under gauge group G. 6 In this case, a suitable stabilization of the symmetry type is where Spin " colim nÑ8 Spinpnq, and the stable homotopy groups that appear in (2.5) coincide with bordism groups, For this reason, it makes sense to refer to the generalized cohomology theory H ‚ IZ as a cobordism theory, a terminology we will use quite generally. Now, the left factor of the exact sequence (2.5) can here be written as HompTor Ω Spin d`1 pBGq, R{Zq. (2.8) This group detects global anomalies because, as previously mentioned, the exponentiated η-invariant that appears as the phase of the fermionic partition function [3] becomes a bordism invariant when perturbative anomalies vanish [1-3, 22, 23], and the global anomalies will be of finite order. To understand how local anomalies are captured by the cobordism group in (2.5), consider now the case where Ω Spin d`1 pBGq vanishes (which means there is no possible global anomaly). The anomaly theory can then be computed on any closed pd`1qmanifold X using the Atiyah-Patodi-Singer (APS) index theorem [4][5][6] on a pd`2qmanifold Y whose boundary is X, and to which the SpinˆG structure extends. We is constant on bordism classes simply by virtue of Stokes' theorem, and so does provide a well-defined map out of Ω Spin d`2 pBGq.) Exactness of the sequence in the middle means that, when the local anomalies vanish, there may still be non-trivial global anomalies and that these correspond precisely to the image of the (injective) map from the left factor into H d`2 IZ pM T Hq. Before continuing, it is important to remark that the generalized cohomology groups H d`2 IZ pM T Hq classify only the deformation classes of anomaly theories, not (isomorphism classes) of anomaly theories themselves. Said more prosaically, the right factor of the exact sequence (2.5) can only measure rather coarse information about the local anomaly, namely the anomaly coefficients, and not the differential form Φ d`2 itself. Recall that the fermionic anomaly theory exp 2πiη is a section of the inverse determinant line bundle associated to the Dirac operator [2]. The perturbative anomaly can be calculated by taking a holonomy of the determinant line bundle around a contractible loop in the parameter space (see e.g. [24] for one account of this perspective). The fully non-perturbative factor of exp 2πiη arises more generally from the holonomy around any given loop over the parameter space, and so can capture both global and local anomalies. The complete information describing the anomaly theory is thus encoded in the holonomies of a principal line bundle 7 over the parameter space, equivalently in a principal line bundle with connection (up to isomorphism). Therefore one really needs to use a differential cohomology theory, specifically a differential refinement of H ‚ IZ pM T Hq, 8 to describe the local anomalies properly. Nonetheless, for the purpose of analysing the interplay between global and local anomalies, we find that the 'topological' theory H ‚ IZ pM T Hq will suffice.

Anomaly interplay as non-canonical splitting
We have seen that the short exact sequence (2.5) tells us how a general fermionic anomaly is put together out of global and local pieces, each of which are classified (up to deformation class) using bordism data. We now discuss the interplay between local and global anomalies in related theories.
To do so, we must first explain what exactly we mean by 'related' theories. We have seen that deformation classes of anomalies are essentially determined by two inputs, 7 The fact that fermionic anomalies can be understood via line bundles is the reason why the corresponding maps of spectra factor through a K-theory, as mentioned in footnote 5. 8 This differential refinement could be constructed using the tools set out in [25].
a spectrum M T H (given a stabilization H of the symmetry type) and a spacetime dimension. It is therefore natural to introduce "morphisms" between theories of the same dimension by specifying appropriate maps between two underlying symmetry types H 1 and H 2 . Now, maps of spectra constitute the usual notion of morphisms in the category of spectra, which is the appropriate domain for the cohomology functor H ‚ IZ . So an appropriate morphism between theories ought to correspond to a map of spectra π : M T H 1 Ñ M T H 2 . One way to construct such a map of spectra is to provide a sequence of group homomorphisms π n : pH 1 q n Ñ pH 2 q n on the underlying symmetry types, which commute with the homomorphisms pi n , ρ n q of Eq. (2.4). This induces maps between the spaces of the Madsen-Tillman spectra that commute with the structure maps, thus giving a function between spectra ergo a map of spectra.
In special cases such as (2.6) where the symmetry type factors into a product of a spacetime symmetry and an internal symmetry that is independent of the dimension n, and where H 1 and H 2 share the same spacetime symmetry, one can replace the set of homomorphisms π n by a single homomorphism π between the internal symmetry groups. This suggests a convenient abuse of notation, in which we frequently use the shorthand π : H 1 Ñ H 2 to denote the set of homomorphisms on the underlying Hstructures that give rise to the map of spectra π : M T H 1 Ñ M T H 2 .
Continuing, given a pair of symmetry types H 1 and H 2 and such a π : H 1 Ñ H 2 , there is a pullback π˚: H ‚ IZ pM T H 2 q Ñ H ‚ IZ pM T H 1 q between anomaly theories. Indeed, there is a pullback diagram for the whole short exact sequence (2.5), which is a commutative diagram. Like the universal coefficient theorem in ordinary cohomology, the short exact sequence for H ‚ IZ splits, but the splitting is not canonical. This means that, while there exist splitting maps for each row of (2.10), i.e. homomorphisms going horizontally from right to left, these splitting maps do not give rise to commutative squares. We remark that the same pullback diagram (2.10) was recently used in [26] to study gauged Wess-Zumino-Witten terms from a bordism perspective.
In many of the ensuing examples, we will be interested in the special case of (2.10) where both symmetry types H 1 and H 2 involve a spin structure, as in (2.6). Then the anomaly interplay diagram can be written in terms of spin-bordism groups as where the various bordism groups can often be straightforwardly computed using, say, the Atiyah-Hirzebruch or the Adams spectral sequence.
To illustrate how this non-canonical splitting might manifest itself, we will frequently encounter scenarios where the pullback of exact sequences takes the following form in which a non-zero element in Local 2 , corresponding to a local anomaly in the second theory, can pullback to a non-zero element in H d`2 IZ pM T H 1 q (by first following the splitting map to the left, then pulling back the anomaly theory along π˚in the middle column), which must correspond to a global anomaly in the first theory. 9 We will study many examples of this scenario in this paper, for example in §5 in which H 1 " SpinˆSU p2q (which has only global anomalies for d " 4) and H 2 " SpinˆU p2q (which has only local anomalies), with π : H 1 Ñ H 2 defining the usual embedding of SU p2q as a subgroup of U p2q.
Importantly, the 'reverse' situation in which a global anomaly pulls back to a local anomaly is not possible. Suppose the theory H 2 theory has only global anomalies, and the H 1 theory has only local anomalies, where again π : H 1 Ñ H 2 denotes a group homomorphism. There is a commutative diagram Here the anomaly pullback map π˚must be the zero map, forbidding any anomaly interplay, because π˚is a homomorphism from a finite abelian group into a free one, and thus zero. For a physics example, consider embedding H 1 " SpinˆU p1q, which has only local anomalies, as the Cartan of H 2 " SpinˆSU p2q, which has only global anomalies. Any SU p2q theory with a global anomaly necessarily pulls back to a U p1q theory that is free of local anomalies; an SU p2q doublet, say, decomposes to a pair of opposite charged particles under the Cartan U p1q subgroup. Thus, completely generally, the possibilities for pulling back a local or global anomaly are Local anomalies pullback Ý ÝÝÝÝ Ñ Local and/or global anomalies, Global anomalies pullback Ý ÝÝÝÝ Ñ Global anomalies only.
The first option corresponds to what we call 'anomaly interplay'.
A crucial step in this analysis of anomalies is the pulling back of a particular anomaly theory, which corresponds to finding the pullback map π˚: H ‚ IZ pM T H 2 q Ñ H ‚ IZ pM T H 1 q. Precisely because of anomaly interplay, this cannot simply be achieved by pulling back the local and global anomalies separately and then combining the two via a Cartesian product. Alas, we will not in this paper give a general 'formula' for this pullback map for any pair of symmetry types. However, in specific cases we often find that it can be computed using simple arguments.
Finding the anomaly pullback map turns out to be especially simple in a rather important special case, namely where 1. H 1 is a subgroup of H 2 , with π : H 1 Ñ H 2 the embedding, 2. the H 2 theory has only local anomalies and the H 1 theory has only global anomalies, as indicated by Eq. (2.12), and 3. the dimension d of the original anomalous theory is even. (2.14) Most of the examples we examine in this paper fall into this class (see § § 5, 6, and 7). Computing the pullback map between anomaly theories is simple under these conditions in large part because it becomes straightforward (in either theory) to explicitly evaluate the anomaly theory on an arbitrary closed pd`1q-manifold (with appropriate H-structure). For the theory with only local anomalies, the vanishing of Ext 1 pΩ H 1 d`1 , Zq implies Ω H 1 d`1 " 0 because, in odd degrees, such bordism groups are pure torsion. Therefore, the APS index theorem can be used to evaluate exp 2πiη X " exp 2πi ş Y Φ d`2 for any closed pd`1q-manifold X " BY , for any potentially-anomalous fermion repre-sentation R 2 of H 2 . To pullback this anomaly theory, 10 it suffices to decompose into representations of the subgroup, viz. R 2 Ñ À α R α 1 , and then evaluate exp 2πiη X on all closed pd`1q-manifolds with H 1 structure for the representations R α 1 . This is in principle straightforward given the assumption that H 1 is free of local anomalies, because in that case exp 2πiη X is constant on bordism classes, and so we need only evaluate exp 2πiη X on each generator of the (finitely-generated) bordism group Ω H 1 d`1 , for each R α 1 . This last step is often easier said than done; however, in the particular examples in § § 5, 6, and 7, we will for the most part get away with using known results.

Physics applications 3.1 Deriving global anomalies
The procedure just described for pulling back the anomaly theory can be turned around to give a new method for computing global anomalies, and the conditions for their cancellation, starting from purely local anomalies in a 'π-related' theory. We envisage this as being an important application of our rather formal treatment of anomaly interplay.
The general idea is as follows. Suppose we identify a symmetry type H 1 for which the bordism group Ω H 1 d`1 is non-vanishing torsion, giving the possibility of a global anomaly. To compute the global anomaly one must evaluate the η-invariant on each generator of Ω H 1 d`1 , which is likely a hard task. But suppose one can embed H 1 as a subgroup of some H 2 , and for simplicity let us assume that the set of conditions (2.14) hold. This induces an injection π˚: M H 1 d`1 Ñ M H 2 d`1 from a set of closed pd`1q-manifolds with H 1 structure M H 1 d`1 to a set of closed pd`1q-manifolds with H 2 structure M H 2 d`1 . Let X P M H 1 d`1 be a representative of a generator of Ω H 1 d`1 . By assumption (2.14), π˚X, whose H 1 structure is viewed as an H 2 structure, is nullbordant, i.e. the boundary of a pd`2q-manifold Y to which the H 2 structure extends. One can then evaluate on π˚X the anomaly theory for a representation R 2 , which we pick to correspond to a generator of the cobordism group HompΩ H 2 d`2 , Zq classifying local anomalies, using the APS index theorem, viz. exp 2πiηpπ˚X, Since we can view η as a function (on its first argument) from M H 2 d`1 to R, its pullback π˚η, which is the etainvariant for the H 1 theory, is defined as π˚η " η˝π˚. Therefore, exp 2πi

then there is a global anomaly in the original
H 1 theory for the set of representations R α 1 . Moreover, if the phase exp 2πi{k is of maximal order in Ω H 1 d`1 then we have identified a generator of the global anomaly, and can derive the general conditions for cancelling the global anomaly.
Although described abstractly here, we will put to work this method for deriving global anomalies in § §4-8. For example, we will see how the general condition for cancelling the 4d SU p2q global anomaly can be derived by computing only local anomalies in either U p2q ( §5) or SU p3q ( §6). In the 2d example of §4 we will use this method to evaluate the exponentiated η-invariant on a generator of the bordism group Ω Spin 3 pBZ{2q -Z{8 from scratch, by embedding Z{2 inside U p1q and thence using only perturbative anomalies.
The idea to derive global anomalies from perturbative anomalies in larger groups was first proposed as a method for analyzing global anomalies by Elitzur and Nair in [16]. 11 However, that paper preceded the modern understanding of global anomalies in terms of bordism, and certainly preceded widespread knowledge of the exact sequence (2.5) used to classify reflection positive invertible field theories, which is crucial to our arguments here. Instead, the main algebraic tool used in [16] was an exact sequence of homotopy groups (with spacetime accordingly assumed to have spherical topology 12 ). This was applied, for example, to derive the 4d SU p2q global anomaly for an SU p2q doublet, evaluated on a spacetime M -S 4 , from the perturbative anomaly for the triplet representation of SU p3q. In §6 we recast this analysis using the bordism-based version of anomaly interplay set out in this paper. This bordism-based method, which results in a condition on the anomaly polynomial reduced mod 2 (via the APS index theorem), is powerful enough to derive the full condition for cancelling the SU p2q anomaly on any manifold, and given arbitrary fermion content. (The condition is that the total number of SU p2q multiplets with isospins j P 2Z ě0`1 {2 should be even.) Moreover, the method is arguably more straightforward, with no need to consider homotopy classes of specific gauge field configurations.
From this perspective, one purpose of the present paper is to give a rigorous bordism-based version of Elitzur and Nair's method for computing global anomalies, or more generally for computing any anomalies in subgroups.
We also want to stress that, despite giving the correct results in various examples, computing homotopy groups does not offer a direct way to detect global anomalies, which are correctly detected by bordism (in one degree higher). Thus, it is perhaps not surprising that homotopical methods have been erroneously applied to postulate various 6d global anomalies, for G " SU p2q and SU p3q. 13 (A quick bordism calculation reveals that neither of these theories can in fact suffer global anomalies.)

Anomaly matching
We have described one use of anomaly interplay as a technical method for computing global anomalies from anomaly polynomials, in which the larger gauge group H 2 is a mathematical device. From a physics perspective, the maps π˚which we use to pullback anomaly theories can often be interpreted in terms of renormalization group (RG) flows. This will be the case in several of the examples we consider in the rest of this paper. When a quantum field theory flows under RG there are various ways in which the symmetry type can change. The most well-known is that some symmetries may become spontaneously broken at low energies, typically leading to massless Goldstone bosons if the broken symmetry were global, and gauge bosons acquiring mass in the case of spontaneously broken gauge symmetries. There are, however, more exotic possibilities; for example, certain theories feature a global symmetry enhancement at high energies (famously this occurs in 5d supersymmetric gauge theories, as originally conjectured in [28]). The changing symmetry type could also involve the spacetime symmetry structures; for example, coupling a Spin-SU p2q theory to certain Higgs fields can trigger an RG flow that dynamically generates a spin structure in the IR [29].
In the case of spontaneous symmetry breaking, the IR will exhibit a subgroup G IR Ă G UV of the UV symmetries. There is always an injective homomorphism π : G IR Ñ G UV defining the embedding of a subgroup, which can be used to analyze anomalies (and their interplay) along this RG flow. 14 The pull-back maps π˚in (2.10) then go in the other direction, and so determine a homomorphism from the full anomaly in G UV to the anomalies in the subgroup G IR . By the arguments above, this generically involves an interplay between local and global anomalies. In the case that (2.10) describes 't Hooft anomalies in global symmetries, in which the RG flow is between consistent quantum field theories, the constraint of anomaly matching then means that any residual anomalies must be matched by the bosonic degrees of freedom that emerge in the IR, via Wess-Zumino-Witten terms. The non-perturbative version of this mechanism, which is needed in the present context to include any global anomalies, was discussed in [30]. 13 We will discuss the shortcomings of the homotopy approach, and the absence of these 6d anomalies, in future work.
14 In rare cases there might also exist a group homomorphism going the other way, viz. π : G UV Ñ G IR . For example, if G UV is a finite abelian group, then any subgroup of IR symmetries can be realised as the image of a homomorphism. Given the pullbacks π˚go in the opposite direction, this could in principle allow one to determine a finite UV anomaly from the finite anomaly in the IR. If G is a simple group, however, then the image of any homomorphism out of G is either zero or all of G.

2d anomalies in U p1q vs. Z{2
For our first concrete example, we discuss the anomaly interplay between Z{2 and U p1q gauge theories in two dimensions, each defined using a spin structure. Let the map π : Z{2 Ñ U p1q denote the usual embedding of Z{2 Ă U p1q which maps the non-trivial element of Z{2 to e iπ . This naturally induces the map π : H 1 " SpinˆZ{2 Ñ H 2 " SpinˆU p1q between symmetry types.
To compute the short exact sequences (2.5) that capture all possible anomalies for these theories in two dimensions, we need to compute the torsion subgroup of Ω 3 and the free part of Ω 4 for each symmetry type. The relevant bordism groups for the H 1 theory are 15 Ω Spin leading to anomalies classified by the short exact sequence Unlike many of the examples of anomaly interplay that will be discussed later on, the cobordism group here detects both a global and a local anomaly. However, this particular local anomaly is already present in a fermionic system without any internal symmetry, as can be seen from the fact that Ω Spin 4 pptq -Z, a generator for which is the K3 surface. It is therefore a pure gravitational anomaly. Indeed, we know that without any extra symmetry the 4 th anomaly polynomial Φ 4 does not vanish, but is given bý A general local anomaly is thus classified by two integers, which in this case one can take to be just the U p1q anomaly coefficient A gauge and the pure gravitational anomaly coefficient A grav . Consider an arbitrary spectrum of charged fermions, consisting of a set of N L left-moving Weyl fermions with charges q 1 , . . . , q N L together with N R right- 15 We remark that this Z{8-valued global anomaly, for a 2d theory with unitary Z{2 symmetry, is related to a parity anomaly for (0+1)-dimensional Majorana fermions [31]. This can be understood in terms of the Smith isomorphism Ω Spin 3 pBZ{2q -Ω Pin2 -Z{8 [32] (see also [33]).
moving Weyls with charges r 1 , . . . , r N R . The anomaly coefficients are given by Note that in 2 dimensions, unlike in 4, conjugating a complex fermion does not flip its chirality; hence we cannot now take all the Weyls to be left-moving without loss of generality as we did in previous examples. We now study the anomaly interplay between Z{2 and U p1q gauge theories in 2d. As usual, the subgroup embedding π : Z{2 Ñ U p1q induces a map of spectra π : M T H 1 Ñ M T H 2 as well as a pullback diagram for the anomaly theories, which encodes an anomaly interplay between a U p1q local anomaly and a global anomaly in the Z{2 theory through the pullback π˚. The gravitational anomaly, corresponding to the second factor in Ω Spin 4 pBU p1qq -ZˆZ, maps to itself under π˚, playing no role in the interplay. This will always be the case for pure gravitational anomalies whenever we consider a pair H 1 and H 2 of symmetry types that differ only in their internal symmetry groups.
To work out how the pullback acts on the first factor, we consider the generic fermion spectrum coupled to a 2d U p1q gauge field described above, and begin by making some simple arithmetic observations. To wit, let the N L left-moving Weyl fermions split into N e L fermions with even charges 2k i , i " 1, . . . , N e L and N o L fermions with odd charges 2l i`1 , i " 1, . . . , N o L . Similarly, let the N R right-moving Weyl fermions divide into N e R fermions with even charges 2k 1 The gravitational anomaly cancellation condition requires that the index N L´NR vanishes. In terms of our variables, this reads The U p1q gauge anomaly cancels when A gauge " 0, which in these variables translates to the condition The second condition immediately implies that the index for the oddly-charged Weyl fermions must be a multiple of 4, viz.
Now consider a Z{2 subgroup of this U p1q gauge group. It acts on a fermion with charge q as p´1q q . In other words, fermions with even U p1q charges decouple from this Z{2 gauge field, and only the oddly-charged fermions can contribute to the Z{8 global anomaly encoded in (4.2). Since the anomaly pullback map π˚is a homomorphism it maps zero to zero, and thus maps any anomaly-free spectrum to another. So if a single left-moving, Z{2-charged Weyl fermion contributes an anomaly equal to ν mod 8, then our considerations in the previous paragraph imply that 4ν " 0 mod 8. Thus, the elementary algebra above tells us that either ν " 0, 2 or 4 mod 8. To see which of these maps is the correct one, we should evaluate the η-invariant on a generator of the appropriate bordism group for a single Z{2-charged Weyl fermion. The novelty here is that, by exploiting the interplay with U p1q anomalies, we can do so simply by integrating an anomaly polynomial, following the arguments set out in §3.1. We thereby derive the conditions for global anomaly cancellation in the Z{2 theory from local anomaly cancellation in U p1q. We now turn to this calculation.

Computing the η-invariant via anomaly interplay
One choice for the generator of Ω Spin holonomy zero otherwise. 17 Now consider a cylinder T 2ˆr 0, 1s, with t a coordinate on the unit interval, and glue together its two ends to form a three-dimensional mapping torus M 3 by identifying pθ, χ, 1q " pθ´2χ, χ, 0q, (4.9) as well as imposing the anti-periodic spin structure along the new cycle parametrized by t. 18 One cannot realise this mapping torus as the boundary of a 4-manifold with both the Z{2 gauge bundle and the spin structure extended; in other words, this mapping torus is in a nontrivial bordism class of Ω Spin 3 pBZ{2q -Z{8. In fact, it may be taken as a generator of the Z{8.
If one embeds the Z{2 background inside a flat U p1q connection, then one must be able to extend all structures to a 4-manifold bounded by the mapping torus M 3 because Ω Spin 3 pBU p1qq " 0. Moreover, computing the exponentiated η-invariant associated with such a U p1q connection must equal that for the original Z{2 background, by pullback. This will ultimately allow us to derive the Z{8-valued global anomaly in the discrete gauge theory from the local U p1q anomaly, which can be done using the APS index theorem (and hence by simply integrating differential forms over the bounding 4-manifold).
Our task is thus to embed the Z{2 connection inside U p1q, and extend both the U p1q connection and the spin structure from the mapping torus described above to a 4-manifold that it bounds. An appropriate flat U p1q connection on M 3 with nontrivial holonomy (equal to´1) only along cycles that wrap the θ direction is simply We now extend the mapping torus to a 4-manifold X 4 by filling in a 2-disc D 2 , with radial coordinate r P r0, 1s, that is bounded by the cycle wrapping the θ coordinate. A suitable extension of the U p1q connection to X 4 is then Note that one cannot simply extend the connection as Apr, θ, χ, tq " rdθ{2 to the whole of X 4 because it is incompatible with the construction of the mapping torus, since Apr, θ, χ, 1q " rdθ{2 is not gauge equivalent (in the bulk) to Apr, θ´2χ, χ, 0q " pr{2qdθ´rdχ. (Using Eq. (4.11), we have Apr, θ, χ, 1q " rpdθ{2´dχq`dχ " Apr, θ2 χ, χ, 0q, which does respect the gluing condition.) With this U p1q connection on X 4 in hand, which reduces to the desired connection on the M 3 boundary, we are now in a position to evaluate the exponentiated η-invariant exp 2πiηpM 3 q. Firstly, it is straightforward to show that Choosing a single Weyl fermion with odd charge under U p1q, corresponding to a fermion charged under the original Z{2 (we assume an uncharged fermion of opposite handedness cancels the gravitational anomaly discussed above), the anomaly polynomial is and so the original Z{2-charged single Weyl fermion is associated with a mod 4 global anomaly. Going back to our anomaly pullback map π˚, we learn that the charged Weyl has an anomaly ν " 2 mod 8. (4.14) Equivalently, the mod 8 anomaly counts the number of left-moving Majorana-Weyl fermions minus the right-moving ones. This conclusion, which we have derived from a purely local U p1q anomaly using the interplay diagram (4.5), agrees with other examples of Z{8 anomalies in fermionic system with a unitary Z{2 symmetry [34][35][36], 19 where a Majorana-Weyl fermion that couples to the Z{2 symmetry contributes the finest anomaly of 1 mod 8. So, to summarize, the anomaly pullback map π˚is here given by In contrast to all the other examples we discuss below, the pullback is not surjective. Despite this failure to surject, the most general anomaly cancellation conditions can still be inferred from the pullback simply because theories with a unitary Z{2 symmetry in two dimensions allow the existence of a Majorana-Weyl fermion, a pair of which combines to give a fundamental Weyl fermion of the U p1q gauge theory. 5 4d anomalies in U p2q vs. SU p2q

The spin case
For our next example, we move up two dimensions and revisit the anomaly interplay between U p2q and SU p2q gauge theories in d " 4, each defined using a spin structure [15] (see also [38]). The map π : SU p2q Ñ U p2q will denote the usual embedding of SU p2q Ă U p2q as the subgroup of 2-by-2 unitary matrices that have determinant one.
As before, we need to compute the torsion subgroup of Ω 5 and the free part of Ω 6 , for each G, in order to compute the short exact sequences (2.5). Computations using spectral sequences (see Refs. [21,38,39] For SU p2q the vanishing of the right factor accords with there being no local anomaly, but there is the Z{2-valued global anomaly discovered by Witten [17]. This global anomaly cancels if and only if there is an even number of states with SU p2q isospins in the set 2Z ě0`1 {2. For G " U p2q, the vanishing of Ω 5 means that there is no global anomaly, even though π 4 pU p2qq " Z{2, associated with a homotopically non-trivial large gauge transformation in the SU p2q subgroup. 20 However there are local anomalies, and we next describe explicitly how these are also detected (up to deformation class) by bordism, only now it is the free part of the bordism group in degree d`2 " 6.
The local anomaly for G " U p2q is classified by three integers which are linear combinations of the three anomaly coefficients: (i) the cubic U p1q anomaly A cub , (ii) the mixed U p1qˆSU p2q 2 anomaly A mix , and (iii) the U p1q-gravitational anomaly A grav . Supposing there are N j fermions transforming in isospin-j representations of U p2q, with U p1q charges tq j,α u " 2j (mod 2) 21 for α " 1, . . . , N j , these anomaly coefficients can be written where T pjq " 2 3 jpj`1qp2j`1q is the SU p2q Dynkin index. A general element of the group HompΩ Spin 6 pBU p2qq, Zq is given by the anomaly polynomial Φ 6 "Â tr ρ exppF{2πq| 6 , whereÂ is the Dirac genus of the tangent bundle, F is the U p2q field strength, and ρ is the representation specified above. By expanding F in terms of the U p1q field strength f and SU p2q field strength F , and expandingÂ in terms of the first Pontryagin class p 1 , we can express the anomaly polynomial as re viewed as generators and the trace is taken in the fundamental representation of SU p2q, and the anomaly coefficients are given in Eq. (5.4). One might naïvely try to identify HompΩ Spin 6 pBU p2qq, Zq -Z 3 with Ze 1 'Ze 2 'Ze 3 , where the three integers are given precisely by the three anomaly coefficients. However this is not quite correct, because the anomaly coefficients are not independent integers. 22 Firstly, the difference between the cubic U p1q anomaly and the U p1q-gravitational anomaly is always a multiple of 6, because we can write q 3 j,α´q j,α " pq j,α´1 qq j,α pq j,α`1 q 21 This 'isospin-charge relation' that links the U p1q charge to the SU p2q isospin is a consequence of the Z{2 quotient in the definition of U p2q " pSU p2qˆU p1qq{Z{2. 22 Another way to see this is to observe that e i are products of rational characteristic classes p 1 , ch 1 , ch 3 in H ‚ pBSpinˆBU p2q; Qq -HompΩ Spin 6 pBU p2qq, Zq b Q. So each of them is not necessarily integral and cannot represent a homomorphism from Ω Spin 6 pBU p2qq to Z. Only combinations of them that arise from the anomaly polynomial Φ 6 are guaranteed to be integral by virtue of the Atiyah-Singer index theorem.
for each charge. Thus, there is a constraint A cub´Agrav " 6r P 6Z. Furthermore, the mixed U p1q-gravitational anomaly A grav must be an even integer 2t, because q j,α " 2j mod 2. So Φ 6 can be written as and thus HompΩ Spin 6 pBU p2qq, Zq should be identified with the subgroup an element of which is labelled by the following three linear combinations of the anomaly coefficients, For example, a fermion transforming under U p2q as an SU p2q singlet with U p1q charge q " 2 corresponds to a local anomaly in the deformation class p1, 0, 1q P H 6 IZ pSpinÛ p2qq -Z 3 .
The commutative diagram (5.3) will encode a non-trivial anomaly interplay if π˚is not the zero map, meaning that a local U p2q anomaly can pullback to a global anomaly in SU p2q. To see that this is the case, it suffices to consider a fermion ψ transforming in the U p2q representation ψ " p2, qq, q P 2Z`1, (5.8) i.e. as a doublet under the SU p2q subgroup. The vanishing of Ω Spin 5 pBU p2qq implies that any closed 5-manifold X equipped with spin structure and a U p2q gauge bundle is the boundary of a 6-manifold Y to which these structures extend. Hence on X " BY the anomaly theory for this representation evaluates to exp`2πi ş Y Φ 6˘b y the APS index theorem.
The U p2q representation (5.8) is associated with a trio of local anomalies, corresponding to the element ppq 3´q q{3, q, qq P HompΩ Spin 6 pBU p1qq, Zq -H 6 IZ pSpinˆU p2qq. To pullback this element to H 6 IZ pSpinˆSU p2qq -Z{2 along π˚, we decompose (5.8) into representations of the subgroup, here simply p2, qq Ñ 2, and evaluate the anomaly theory for a single SU p2q doublet on an arbitrary closed 5-manifold. Because the free part of the 6 th bordism group of SU p2q vanishes (i.e. because there are no local anomalies), it is sufficient to evaluate the exponentiated η-invariant on the generator of the bordism group Ω Spin 5 pBSU p2qq. Of course, we know that a suitable generator of Ω Spin 5 pBSU p2qq is the mapping torus X " MˆS 1 with SU p2q instanton number one through the M factor, on which exp p2πiη MˆS 1 q "´1 for the doublet representation, as originally computed by Witten [17]. Thus, we know that the pullback map π˚between anomaly theories is such that Z 3 Q ppq 3´q q{3, q, qq Þ Ñ 1 P Z{2, which already tells us that π˚is a surjection. Performing a similar calculation for an arbitrary U p2q fermion representation, one can deduce that the anomaly pullback map is As a result, the splitting maps in (5.3) do not give rise to commuting squares, on either the left or right, which is the mathematical statement of this anomaly interplay.
In the account just given, we deduced what the anomaly pullback map was by first decomposing representations of the locally-anomalous theory into representations of SU p2q, then invoking a known result for exp 2πiη evaluated on a 5-manifold in the non-trivial bordism class of Ω Spin 5 pBSU p2qq -Z{2. Now, as demonstrated in §4 for a 2d example, one can in fact turn the argument around to derive the conditions for global anomaly cancellation in the SU p2q theory from local anomaly cancellation in U p2q.
The first step is to view a representative of the generator of Ω Spin 5 pBSU p2qq -Z{2, which we can take to be the mapping torus X " MˆS 1 equipped with an SU p2q bundle with instanton number one, as a manifold with a SpinˆU p2q structure by embedding the SU p2q connection inside U p2q. The result must be nullbordant as a U p2q bundle because Ω Spin 5 pBU p2qq " 0. Indeed, this U p2q bundle can be extended to a six-manifold Y " MˆD 2 , where D 2 is a hemisphere bounded by the S 1 in X (so that BY " MˆS 1 ) and through which the U p1q field strength f has half-integer magnetic flux (see footnote 11 of [15]). We then evaluate the anomaly theory for the p2, qq representation using the APS index theorem for this choice of bounding manifold Y , giving remembering that q is necessarily odd. By pullback, this must equal the evaluation of the SU p2q anomaly theory for the doublet representation on the original mapping torus, which therefore suffers from a Z{2-valued global anomaly. By repeating the exercise for other representations, one learns that in general the anomaly pullback map π˚is given by (5.9). Hence, the SU p2q anomaly vanishes if and only if for the U p2q local anomaly. Now, T pjq is odd if and only j P 2Z ě0`1 {2, and for these half-integral isospins q must be odd, so this equation becomes [15] ÿ jP2Z ě0`1 {2 1 " 0 mod 2 (5.12) which, upon decomposing to irreps of SU p2q, is precisely the condition for SU p2q anomaly cancellation. This amounts to a rigorous derivation of the SU p2q anomaly constraint using the APS index theorem for nullbordant manifolds in U p2q, which is arguably easier than computing η on non-trivial bordism classes in the original theory.
Because the most general anomaly in the SU p2q theory is classified by H 6 IZ pSpinŜ U p2qq -Z{2, there can be no further conditions for SU p2q anomaly cancellation.

Non-spin generalization: the 'w 2 w 3 anomaly'
In Ref. [29] it was shown that if one instead defines a 4d SU p2q gauge theory by extending the spacetime symmetry group Spinp4q non-trivially by SU p2q, then there is a new Z{2-valued global anomaly for fermions in the isospin representations 4r`3{2, r P Z ě0 . In particular, the symmetry structure is in which the Z{2 quotient identifies the central element´1 P SU p2q with p´1q F in Spin, which we abbreviate to H 1 " Spin-SU p2q.
The relevant bordism group capturing global anomalies (there are no local anomalies) is Ω A suitable generator for the first Z{2 factor is given by the Dold manifold D " pCP 2ˆS1 q{pZ{2q (where the Z{2 quotient identifies complex conjugation on CP 2 with the antipodal map on the circle), equipped with a certain Spin-SU p2q structure. This provides a mapping torus on which one can detect the 'new SU p2q anomaly' of Ref. [29], where the mapping torus is glued using a combined diffeomorphism and gauge transformation. A cobordism invariant that evaluates to 1 mod 2 on the Dold manifold D is provided by w 2 Y w 3 (henceforth just denoted w 2 w 3 ), the cup product of the sec-ond and third Stiefel-Whitney classes of the tangent bundle, sometimes referred to as the 'de Rham invariant'. This cobordism invariant can be obtained as the exponentiated η-invariant for a single fermion in the isospin j representation of SU p2q for any j P 4Z ě0`3 {2.
The second Z{2 factor in the bordism group (5.13), which is generated by the mapping torus X " S 4ˆS1 with SU p2q instanton number one through the S 4 factor, captures the original SU p2q anomaly of Witten. As we have seen above, a cobordism invariant that evaluates to 1 mod 2 on X is the exponentiated η-invariant for a fermion in the doublet representation of SU p2q.
In analogy with the anomaly interplay between SpinˆSU p2q and SpinˆU p2q studied in the previous Section, we here embed the symmetry type H 1 " Spin-SU p2q in the symmetry type H 2 " Spin-U p2q, whose relevant bordism groups are given by is again generated by the Dold manifold D, now with a certain Spin-U p2q structure. The embedding H 1 Ñ H 2 induces the anomaly interplay diagram The pullback π˚maps the local anomaly factor Z 3 into the second Z{2 factor of Ω Spin-SU p2q 5 exactly as in the interplay between SU p2q and U p2q. The role of the first Z{2 factor is in a sense more subtle, but in another sense rather trivial. While the same Dold manifold described above can be taken as a generator of the Z{2 factor in the bordism group, the corresponding element in the cobordism group can in fact never be realised as a fermionic global anomaly in the Spin-U p2q theory. This can be seen from the fact that the conditions for local anomaly cancellation in the Spin-U p2q theory preclude a non-vanishing 'new SU p2q anomaly', as shown in Ref. [15]. In other words, evaluating the exponentiated η-invariant for any set of chiral fermions coupled to the Spin-U p2q connection can never give a non-trivial phase on D when the local anomalies cancel. This is the first concrete example we have seen of a non-trivial invertible phase that appears in the cobordism group, but which cannot be realised as a chiral fermion anomaly. Despite this, it is possible to couple the Spin-U p2q theory directly to a topological quantum field theory (TQFT) whose partition function is given by p´1q w 2 w 3 on closed 5-manifolds, thereby generating an invertible phase in the non-trivial element of Z{2 Ă H 6 IZ pSpin-U p2qq which reproduces the same anomaly. (In a sense this is a 'pure-gravitational' anomaly, being computed from characteristic classes of the tangent bundle.) The anomaly pullback map π˚acts as the identity map on the first Z{2 factor, mapping the cohomology class w 2 w 3 to itself. The crucial difference is that in the Spin-SU p2q theory (unlike for Spin-U p2q), the anomaly theory p´1q w 2 w 3 can be reproduced by the exponentiated η-invariant, say for a single isospin-3/2 fermion coupled to a particular Spin-SU p2q connection.

A remark concerning the 5d SU p2q anomaly
The fact that Ω Spin 6 pBSU p2qq -Z{2 corresponds to a Z{2-valued global anomaly in SU p2q gauge theory in 5d, which is generated by a 'symplectic Majorana fermion' multiplet [40]. The bordism group Ω 6 pBU p2qq, however, is torsion-free, ruling out a global anomaly in the U p2q version. In analogy with the 4d case, one might again wonder what has become of the 5d SU p2q global anomaly when SU p2q is embedded as a subgroup in U p2q, and whether there is a similar interplay with local anomalies. The story here is in fact much more mundane; Ω 7 pBU p2qq vanishes, so there are no anomalies whatsoever in a 5d U p2q gauge theory. The resolution to this little puzzle is simply that one cannot embed the symplectic Majorana multiplet, responsible for the 5d SU p2q global anomaly, into representations of U p2q; the isospin-charge relation in U p2q means any SU p2q doublet must have non-zero U p1q charge, and so cannot be Majorana.

4d SU p2q anomaly from SU p3q
Elitzur and Nair showed how the SU p2q global anomaly associated with the doublet representation can be computed from a local anomaly not in U p2q, but in SU p3q [16]. To our knowledge, this was the first instance in which a local anomaly was essentially 'pulled back' to derive a global anomaly, implicitly exploiting the possible interplay between the two. Their method was based on the homotopy groups of SU p2q and SU p3q, which fit inside a long exact sequence together with the homotopy groups of SU p3q{SU p2q -S 5 , and embedding SU p2q gauge field configurations inside SU p3q.
The bordism-based formalism which we use here is significantly more powerful. Because SU p2q has no local anomalies in 4d, cobordism invariance allows one to analyse the most general possible global anomaly theory by performing only a handful of computations, accounting for all possible SU p2q bundles over all possible mapping tori. Moreover because the local anomaly for SU p3q has a single generator, we will in fact only need to do a single computation, namely 'pushing forward' the SU p2q single instanton mapping torus and then evaluating the SU p3q local anomaly for the triplet using the APS index theorem. This will be enough to recover complete information about the SU p2q global anomaly for arbitrary representations on arbitrary closed manifolds.
To deduce the anomaly interplay map for the subgroup embedding π : SU p2q Ñ SU p3q (in which SU p2q is embedded in, say, the upper-left 2-by-2 block of SU p3q), we need the bordism groups for BSU p2q which were already given above (5.1), and we need Ω Spin 5 pBSU p3qq -0, Ω Spin 6 pBSU p3qq -Z (6.1) for SU p3q. We then have the following pair of short exact sequences The top line encodes the Z{2-valued global anomaly in SU p2q, and the bottom line encodes the SU p3q perturbative anomaly, a generator for which is a single fermion in the fundamental 3 representation.
In general, the SU p3q anomaly coefficient A SU p3q is obtained by summing over the contributions from some number npa 1 , a 2 q of left-handed fermion multiplets in each SU p3q irreducible representation, indicated by the Dynkin labels pa 1 , a 2 q, viz.
where the individual anomaly coefficients are (c.f. page 72 of [41]) A pa 1 , a 2 q " 1 120 pa 1´a2 qp2a 1`a2`3 qpa 1`2 a 2`3 qpa 1`1 qpa 2`1 qpa 1`a2`2 q. (6.4) (One can verify that, with this normalization, the anomaly coefficient for the fundamental representation p1, 0q equals one.) We now ask whether there is an anomaly interplay, in other words can local anomalies in SU p3q pullback to the non-trivial SU p2q anomaly? To answer this question, it is sufficient to decompose the triplet representation 3 Ñ 2 ' 1 into representations of SU p2q and compute the anomaly theory. Since there is an unpaired SU p2q doublet (and an irrelevant SU p2q singlet) this theory has a global SU p2q anomaly. This is enough to completely fix the anomaly pullback map π-to be the non-trivial homomorphism from Z to Z{2, thus π˚: H 6 IZ pSpinˆSU p3qq -Z Ñ H 6 IZ pSpinˆSU p2qq -Z{2 : Now, we can once again turn this argument on its head, pretending for a moment that we do not know the conditions for anomaly cancellation in SU p2q, and we can derive them from the perturbative SU p3q anomaly. The argument would be as follows. Firstly, to compute a global anomaly for the SU p2q doublet representation, one views the mapping torus equipped with a single SU p2q instanton as a manifold with a SpinŜ U p3q structure by embedding the SU p2q connection inside SU p3q. This is exactly as in the previous section. This mapping torus, which is nullbordant in Ω Spin 5 pBSU p3qq " 0, can be extended to a bounding 6-manifold Y with SU p3q connection (in fact using precisely the same U p2q connection of the previous section and embedding U p2q in SU p3q). Thus, for any SU p3q representation we can evaluate exp 2πiη on the pushed forward mapping torus by integrating the SU p3q anomaly polynomial on Y . When we do this for the triplet we compute exp 2πiη MˆS 1 "´1, and because 3 Ñ 2 ' 1 under SU p3q Ñ SU p2q this must coincide with the anomaly theory for the SU p2q doublet representation evaluated on the original mapping torus, by pullback (the singlet appearing in the decomposition does not couple to the SU p2q connection, and so plays no role here). So we learn that the SU p2q doublet has a Z{2-valued global anomaly, thus providing us with a suitable generator of the anomaly group H 6 IZ pSpinˆSU p2qq, and moreover that the pullback map from the local SU p3q anomaly must be (6.5).
One can then deduce the more general condition for SU p2q anomaly cancellation, by studying the equivalent condition for anomaly cancellation in the SU p3q theory, Now, the irreducible representation pn, 0q of SU p3q decomposes into irreducible representations of SU p2q as When we decompose an SU p3q representation R :" pn´1, 0q ' p0, n´2q, the SU p2q irreducible representations 1, 2, . . . , n´1 (here labelled by their dimensions) thus appear in pairs, and so cannot contribute to a mod 2 global anomaly once the anomaly is pulled back to SU p2q; only the irreducible representation n remains unpaired and so possibly contributes to the SU p2q anomaly. Using the anomaly interplay map (6.5), a left-handed fermion in the representation n of SU p2q therefore contributes to the global SU p2q anomaly if and only if the SU p3q anomaly coefficient for the representation R is odd. Using (6.4), this anomaly coefficient is which is odd if and only if the dimension of the SU p2q representation n " 2 mod 4. Equivalently, in terms of isospin j " pn´1q{2, a left-handed fermion in the isospin j representation of SUp2q contributes nontrivially to the global anomaly if and only if j P 2Z ě0`1 {2. We thus reproduce the general condition for SU p2q anomaly cancellation by evaluating a single local anomaly in SU p3q using the APS index theorem (for the triplet representation), supplemented by basic representation theory arguments.

4d discrete gauge anomalies
For our next examples we turn to anomalies in 4d theories with discrete internal symmetries. This story goes back to pioneering work by Ibañez and Ross [18,19] on the (necessarily global) anomalies that can afflict a discrete Z{k gauge theory, which they derived by embedding Z{k in U p1q. Much more recently, these discrete gauge anomalies were rigorously derived by computing η-invariants by Hsieh [20], providing an intrinsic description of the global anomaly that does not rely on any microscopic completion in a U p1q gauge symmetry.
In the context of the present paper, the version of this story as told by Ibañez and Ross can be understood as an instance of anomaly interplay. In this Section we recast the relation between the local U p1q anomalies and the global Z{k anomalies from the bordism perspective, by using the calculations of Hsieh [20] to write down the precise pullback map between the anomaly theories. This pullback map from U p1q anomalies to Z{k anomalies turn out to be surjective, meaning that one can derive necessary and sufficient conditions for cancelling the discrete gauge anomalies starting from the local U p1q anomalies. In this sense, we suggest that an anomaly interplay reminiscent of Ibañez and Ross's original method does in fact give both necessary and sufficient conditions for a discrete gauge symmetry to be anomaly-free. 23 Assuming a fermionic theory without time-reversal symmetry, an internal discrete symmetry group K " ker ρ n " Z{k accommodates two possible symmetry types pH n , ρ n q when k " 2m is even. These are where the Z{2 quotient identifies the central element p´1q F P Spinpnq with the order-2 central element in Z{2m (that is, the element m¨q where q is a generator for Z{2m). We will consider global anomalies for both these symmetry types, as was analyzed in [20].
In the former case we will study the interplay with local anomalies in a SpinˆU p1q theory, while for the latter we consider the interplay with local anomalies in a theory with Spin c structure.
We are content to restrict to the case where k " 2m " 2 n is a power of two with n ě 2, thereby streamlining the algebra somewhat, because it is for these cases that the story with the 'twisted' symmetry type (7.2) is most interesting. For an exhaustive treatment of the Z{k global anomalies applicable for any integer k, we refer the reader to §2 of [20]. with the latter condition precluding any local anomalies as we expect given there are no gauge transformations connected to the identity (and there are no pure gravitational anomalies). We compute these bordism groups using the Adams sequence in Appendix A.3, noting that Ω 5 was computed by other means in [20,42] and partial results appear also in [43]. For U p1q on the other hand, we have the bordism groups Ω Spin 5 pBU p1qq " 0, Ω Spin 6 pBU p1qq -ZˆZ, (see e.g. Section 3.3 of [21] for Ω 5 , and e.g. §3.1.5 of [44] for Ω 6 .) The two factors of Z appearing in HompΩ Spin 6 pBU p1qq, Zq correspond to the cubic U p1q anomaly A cub and the combination pA cub´Agrav q{6 between the mixed U p1q-gravitational anomaly and the cubic U p1q anomaly.
The reasoning is similar to that used in §5 for the U p2q local anomalies, as follows.
An element of HompΩ Spin 6 pBU p1qq, Zq is the 6 th degree anomaly polynomial where we think of e 1 :" 1 3!ˆf 2π˙3 and e 2 :" p 1 24ˆf 2πȧ s generators. In this way, HompΩ Spin 6 pBU p1qq, Zq is seen as a subgroup of Ze 1 ' Ze 2 . However, it cannot be the whole of Ze 1 ' Ze 2 because, even though both A cub and A grav are integers, they are not independent and so cannot span the whole lattice. For a set of left-handed Weyl fermions with U p1q charges Q i P Z, i " 1, . . . , N , the two local anomaly coefficients are Their difference can be rewritten as ř N i"1 pQ i´1 qQ i pQ i`1 q " 6r, a multiple of 6. Writing also A grav " s P Z, the anomaly polynomial can now be written as Φ 6 " p6r`sqe 1`s e 2 " spe 1`e2 q`rp6e 1 q. (7.7) We should therefore identify ZˆZ -HompΩ Spin 6 pBU p1qq, Zq with an element of which is labelled by the following pair of independent integers, pr, sq "ˆ1 6 pA cub´Agrav q , A grav˙. (7.8) For example, a single left-handed Weyl fermion with unit charge corresponds to a local anomaly in the deformation class p0, 1q. As usual, we abuse notation and define the embedding π : Z{2 n Ñ U p1q : q mod 2 n Þ Ñ expp2πiq{2 n q. This gives rise to a map of spectra π : M T H 1 Ñ M T H 2 and thus a pull-back diagram for the anomaly theories pertaining to the case d " 4, As long as the map π˚is non-zero, this diagram encodes a non-trivial anomaly interplay. We will in fact see that π˚is surjective, allowing the complete conditions for global anomaly cancellation in the Z{2 n theory to be derived from local anomalies in U p1q,à la Ibañez and Ross.
To study this interplay, first consider a single Weyl fermion in a general representation of U p1q, specified by an integer charge Q P Z. The anomaly theory corresponds to the element ppQ 3´Q q{6, Qq P ZˆZ -HompΩ Spin 6 pBU p1qq, Zq. To pull this back, decompose into representations of the discrete subgroup Z{2 n , simply Q Þ Ñ q " Q mod 2 n , and evaluate the global anomaly for this representation by computing the exponentiated η-invariant on generators of the bordism group Ω Spin 5 pBpZ{2 n qq -Z{2 nˆZ {2 n´2 .
Using the results of [20], the evaluations of the exponentiated η-invariant for the charge q representation of Z{2 n on two independent generators X and Y of Ω Spin 5 pBpZ{2 n qq are given by , q˙˙( 7.10) and expp2πiηpq, Y qq " expˆ2 πi k{4ˆq 2`k 2`3 k`2 12 q 3˙: " expˆ2 πi k{4 ν k{4ˆq 3´q 6 , q˙˙, where it is convenient to write these and the following formulae in terms of k " 2 n , and where we define the functions ν k pr, sq :" 1 6 pk`1qpk`2qp6r`sq, ν k{4 pr, sq :" 1 2 pν k pr, sq`sq .

pSpinˆZ{2 n q{pZ{2q
We now consider the non-trivial extension of the discrete symmetry Z{2 n by Spin, resulting in the stable structure group H 1 " pSpinˆZ{2 n q{pZ{2q as defined in (7.2), and which we henceforth abbreviate by Spin-Z{2 n . The relevant bordism groups for " 0. (7.15) Again, the latter condition precludes any local anomalies as expected since there are no gauge transformations connected to the identity, and no purely gravitational anomalies in 4d. We compute these bordism groups using the Adams sequence in Appendix A.4, noting that Ω 5 was computed by other means in [20]. On the other hand, the corresponding non-trivial extension of U p1q by Spin results in the structure group H 2 " Spin c , with the relevant bordism groups [45]  The different factor of 12 instead of 6 relative to Eq. (7.8) comes from the fact that fermions can only carry odd charges. (Equivalently, the Spin c connection can have half-integral normalization relative to an ordinary U p1q connection.) As before, the embedding π : Z{2 n Ñ U p1q : q mod k Þ Ñ expp2πiq{2 n q gives rise to a map of spectra π : M T H 1 Ñ M T H 2 and thus a pullback diagram for the anomaly theories pertaining to the case d " 4, This diagram encodes a non-trivial anomaly interplay if the map π˚is non-zero. Again this will be the case. Indeed π˚will again be a surjection. This means that the most general conditions for global anomaly cancellation in the Spin-Z{2 n theory can be derived by pulling back a local anomaly in Spin c .
To study this interplay, first consider a single Weyl fermion in a general representation of U p1q, specified by a charge Q P 2Z`1 which now must be an odd integer. The anomaly theory corresponds to the element ppQ 3´Q q{12, Qq P ZˆZ -HompΩ Spin c 6 , Zq.
To pull this back, decompose into representation of the discrete subgroup Z{2 n , simply Q Þ Ñ q " Q mod 2 n , and evaluate the global anomaly for this representation by computing the exponentiated η-invariant on generators of the bordism group Ω Spin-Z{2 n 5 -Z{2 n`2ˆZ {2 n´2 . Using the results of [20], the evaluations of the exponentiated ηinvariant for the charge q representation of Z{2 n on two independent generatorsX and Y of Ω Spin-Z{2 n 5 are given by expp2πiηpq,Xqq " expˆ2 πi 4k 1 12`p k 2`k`2 qq 3´p k`6qq˘: , q˙˙(

(7.21)
It is easy to see that both µ 4k ppQ 3´Q q{12, Qq and µ k{4 ppQ 3´Q q{12, Qq are integers whenever k " 2 n ą 4 and Q P 2Z`1. Moreover, for two Spin c charges Q 1 and Q 2 that are congruent modulo k, the corresponding µ 4k are congruent modulo 4k, while the corresponding µ k{4 are congruent modulo k{4. Therefore, a general deformation class of Spin c local anomaly ppQ 3´Q q{12, Qq P

Example: Standard Model and the topological superconductor
A particularly interesting special case of the Spin-Z{2 n anomaly interplay is when n " 2. This case is most straightforward to analyse because there is only one independent global anomaly corresponding to 24 Ω Spin-Z{4 5 -Z{16, (7.24) which is generated by µ 16 pr,sq " 22r`s (withr ands as defined above). Any fermion coupled to a Spin-Z{4 structure must have charge q i equal to˘1 mod 4 and thus q 3 i´q i " 0, contributing 0 tor and˘1 tos. Thus the global anomaly in fact reduces to µ 16 pr,sq "s, which vanishes if and only if µ 16 " n`´n´" 0 mod 16, (7.25) where n˘denotes the number of Weyl fermions with charge˘1 mod 4. 24 This bordism group can also be given a lower-dimensional interpretation thanks to a Smith isomorphism Ω Spin-Z{4 5 -Ω Pin4 [32]. From the physics perspective, one can relate each 4d Weyl fermion with Spin-Z{4 charge to a 3d Pin`Majorana fermion on a domain wall, offering an alternative way to understand this 4d global anomaly in terms of a 3d 'topological superconductor'. This Z{16-valued global anomaly was studied in Ref. [21,32] due to a connection with the Standard Model (SM) of particle physics. The key observation was that there is a linear combination of SM global U p1q symmetries under which every SM fermion has a charge equal to 1 mod 4, which if gauged could be used to define the SM using a Spin-Z{4 structure. Specifically, the U p1q charges correspond to the linear combination X "´2Y`5pB´Lq, where B´L denotes the difference between baryon number and lepton number, and Y denotes global hypercharge.
When the SM is augmented by a trio of right-handed neutrinos (which offers the simplest route to explaining the origin of neutrino oscillation data) there are n`" 16 Weyl fermions within each generation, meaning that condition (7.25) is satisfied and there is no global anomaly. 25 As a consequence, the SM fermions can be gapped in groups of 16 by including relevant operators that preserve the Spin-Z{4 symmetry [47], at least in specific supersymmetric extensions for which the low-energy dynamics is explicitly calculable. This example of 'symmetric mass generation' in 4d is analogous to the Fidkowski-Kitaev mechanism for gapping 1d Majorana fermions in multiples of 8 [31].

Going between discrete groups
We have seen above how the pullback property of the anomaly interplay map can be used to determine the global anomaly in a theory with Spin-Z{2 n structure, using a local anomaly calculation in a theory with Spin c structure. A simpler exercise is to deduce how global anomalies map to themselves between theories with different discrete gauge symmetries, which can be used to put constraints on global anomalies. Similar ideas were discussed in §3 of Ref. [20].
As an example, consider the interplay between a 4d theory with symmetry structure H 1 " Spin-Z{4, as just discussed in §7.2.1, and a 4d theory with symmetry structure H 2 " Spin-Z{8. In the latter, there is a unitary symmetry operator U obeying U 4 " p´1q F . We embed H 1 as a subgroup of H 2 as π : H 1 Ñ H 2 : V Þ Ñ U 2 , where V is the order-4 element in the Z{4 factor of H 1 that squares to p´1q F . Given this embedding, a fermion with charge 1 mod 8 with respect to H 2 has charge 1 mod 4 with respect to H 1 . Suppose that we wish to calculate the global anomaly in the Spin-Z{8 theory, and that we already know that a charge 1 mod 4 fermion in the Spin-Z{4 theory with structure contributes a global anomaly equal to 1 mod 16 (either by direct calculation, e.g. as above, or by exploiting the Smith isomorphism Ω Spin-Z{4 5 -Ω Pin4 -Z{16 and knowing that a 3d Pin`Majorana fermion has a mod 16 parity anomaly). The embedding π therefore gives rise to the interplay diagram Without any direct calculation, one can immediately deduce that the most refined global anomaly in the H 2 theory must be of order p " 16m with m a positive integer. If it were not so, one could have a set of fewer than 16 fermions each with charge 1 mod 8 that is anomaly-free with respect to the Spin-Z{8 structure, but that is known to have non-vanishing mod 16 anomaly with respect to H 1 . This is a contradiction because the pullback π˚is a homomorphism and must map anomaly-free Spin-Z{8 fermion content to anomaly-free Spin-Z{4 content. Indeed, we have seen earlier that Ω Spin-Z{8 5 -Z{32ˆZ{2 and the most refined anomaly is of order 32 which is a multiple of 16.

6d anomalies in U p1q vs. Z{2
For our final example we turn to six spacetime dimensions and consider a similar setup to the 2d example discussed in §4, in which a theory defined with a spin structure and a unitary Z{2 symmetry is embedded inside one with a U p1q symmetry. As before, the internal symmetries do not intertwine with the spacetime symmetry, so the symmetry types are given by H 1 " SpinˆZ{2 and H 2 " SpinˆU p1q. The analysis bears similarities with the 2d example but differs in important details.
In six dimensions, the relevant bordism groups that appear in the short exact sequence classifying anomalies for the H 1 theory are 26 Ω Spin 7 pBZ{2q -Z{16, Ω Spin 8 pBZ{2q -ZˆZ, (8.1) leading to anomalies classified by the short exact sequence As we saw in the 2d example (and in contrast with the 4d examples of §7), there are local anomalies even though the gauge group is discrete. They must be purely 26 We remark that the Z{16-valued global anomaly that we here discuss, for 6d Weyl fermions with a unitary Z{2 symmetry, ought to be related to a parity anomaly in 4+1 dimensions, as suggested by the existence of a Smith isomorphism Ω Spin 7 pBZ{2q -Ω Pin6 -Z{16 [32].
gravitational since Ω Spin 8 pptq -ZˆZ implies that they appear even in the absence of any gauge bundle. The group HompΩ Spin 8 pptq, Zq -ZˆZ that detects local anomalies is generated by two integer-valued bordism invariants, which we can take to be the degree 8 anomaly polynomial where the latter can be deduced from the Atiyah-Hirzebruch spectral sequence. As above, one of the Z factors in HompΩ Spin 8 pBU p1qq, Zq is generated by the signature, which is not realised via chiral fermion anomalies. A general local anomaly due to free chiral fermions is classified by the other three integers, which we can take to be (i) the U p1q anomaly coefficient A gauge , (ii) the mixed U p1q-gravitational anomaly coefficient A mix , and (iii) the pure gravitational anomaly coefficient A grav . For an arbitrary fermion content with N L left-handed Weyl fermions of charges q 1 , . . . , q N L and N R right-handed 27 The classification of the deformation classes of anomaly theory should nonetheless include both factors of Z; one can couple to an invertible theory whose exponentiated action is the 7d gravitational Chern-Simons term corresponding to the signature, even though this does not arise as the bulk anomaly theory for any set of massless chiral fermions in six dimensions. We emphasize the asymmetry to the corresponding situation in 2d, in which the signature and the anomaly polynomial are both proportional to the same characteristic class p 1 . Thus, pure gravitational anomalies in 2d, unlike in 6d, can always be realised via chiral fermions.
Weyl fermions of charges r 1 , . . . , r N R , these anomaly coefficients are given by Again, we cannot take all the Weyl fermions to be left-handed because in 6 dimensions, as in 2 dimensions, conjugating a complex fermion does not flip its chirality. We now study the anomaly interplay between Z{2 and U p1q gauge theories in 6d. As usual, the subgroup embedding π : Z{2 Ñ U p1q : 1 mod 2 Þ Ñ e iπ induces a map of spectra π : M T H 1 Ñ M T H 2 as well as a pullback diagram for the anomaly theories, which encodes an anomaly interplay between a U p1q local anomaly and a global anomaly in the Z{2 theory through the pullback π˚. Since the difference between the pair of symmetry types H 1 and H 2 does not involve spacetime symmetry, the pure gravitational anomalies maps to themselvess under π˚, playing no role in the interplay. To see how the pullback acts on the remaining two factors of Z in H 8 IZ pSpinˆU p1qq, we consider a generic fermion spectrum coupled to a 6d U p1q gauge field, the charges of which we parametrize as in §4. The gravitational anomaly and the mixed U p1qgravitational anomaly cancellation conditions, A grav " A mix " 0, are given by Eqns. (4.6) and (4.7), respectively. In addition to these two conditions (which are of the same form as in the 2d case), we also require that the quartic U p1q anomaly vanishes, and hence that the index for the oddly-charged Weyl fermions must be a multiple of 16, viz.
The Z{2 subgroup of this U p1q gauge group only acts non-trivially on fermions with odd charge. Only these odd-charged fermions contribute to both the Z{16 global anomaly encoded in (8.2) and the gravitational anomaly, while the even-charged fermions can be used to soak up the gravitational anomaly. Since π˚is a homomorphism it maps zero to zero, and thus maps any anomaly-free spectrum to another. So if a single lefthanded, Z{2-charged Weyl fermion contributes an anomaly equal to ν mod 16, then our considerations in the previous paragraph imply that 16ν " 0 mod 16. Thus, it is permissible on the ground of anomaly interplay that a single left-handed odd Weyl fermion contributes the most refined anomaly of ν " 1 mod 16.
This is indeed the case, as can be seen by explicit computation of the exponentiated η-invariant on a generator of the bordism group Ω Spin 7 pBZ{2q. To wit, for a left-handed odd-charged Weyl fermion together with a right-handed even-charged Weyl fermion (thus cancelling any local anomaly, making the exponentiated η-invariant a cobordism invariant), one finds expp2πiη X q " expp2πi{16q where X is RP 7 equipped with a Z{2 gauge bundle whose first Stiefel-Whitney class equals the generator of H 1 pRP 7 ; Z{2q, which is a generator of Ω Spin 7 pBZ{2q (c.f. §3.6 of Ref. [32]). 28 To conclude our analysis of 6d anomalies, the anomaly pullback map π˚is here given by π˚: H 8 IZ pSpinˆU p1qq -Z 4 Ñ H 8 IZ pSpinˆZ{2q -Z{16ˆZ 2 pA gauge , A mix , A grav , kq Þ Ñ pA gauge mod 16, A grav , kq . Note that, unlike in the 2d case, the pullback map π˚is surjective. This is permissible because in 6 dimensions there is no chiral basis for the gamma matrices where all elements are real. Hence, a Weyl fermion cannot be divided further into two real chiral fermions as in 2 dimensions. David Tong's Simons Investigator Award. We are supported by the STFC consolidated grant ST/P000681/1.

A Some bordism calculations
In this Appendix, we sketch how bordism groups are calculated using the Adams spectral sequence [48,49], and then present the calculations of some bordism groups mentioned in the main text. For a more detailed introduction to practical calculations using the Adams spectral sequence, we recommend Ref. [13]. All cohomology rings have coefficients in Z{2 unless otherwise stated.

A.1 Using the Adams spectral sequence
We will use the Adams spectral sequence to obtain the 2-completion`Ω H t´s˘2 of a given bordism group Ω H t´s that we wish to calculate. Recall that the 2-completion of the group of integers Z is the 2-adic group Z 2 , the 2-completion of the cyclic group Z{2 n is Z{2 n , while the 2-completion of Z{m when m is odd is the trivial group. Therefore, if there is no odd torsion involved, the whole bordism group can be obtained from its 2-completion.
The for t´s ă 8, where Ap1q is the subalgebra of A 2 generated by the Steenrod squares Sq 1 and Sq 2 . Fortunately, this simplification occurs in all of the examples we consider in this Appendix. For example, when H " SpinˆG is a product of the stable spin group and an internal symmetry group G, we have M T H " M Spin^BG`, whered enotes a disjoint basepoint. In this case Ω H t´s is the pt´sq th spin bordism group of BG, often denoted by Ω Spin t´s pBGq. The next step is to compute the Ap1q-module structure of H ‚ pX H q, and plot the corresponding graded extension in an Adams chart. The Adams chart for H ‚ pX H q is a visual representation of Ext s,t Ap1q pH ‚ pX H q, Z{2q on the pt´s, sq-plane, in which a dot represents a generator of Z{2. There can be some non-trivial relations between generators of E s,t 2 . The ones that we will encounter in the sequel are a) multiplication 29 by an element h 0 P Ext 1,1 Ap1q pZ{2, Z{2q, which is represented by a vertical line segment, and b) multiplication by an element h 1 P Ext 1,2 Ap1q pZ{2, Z{2q, which is represented by a line segment of slope 1. Being a spectral sequence, each page E r is a bi-graded complex of abelian groups, equipped with group homomorphisms or 'differentials' Pictorially, in the Adams chart each of these differentials is represented by an arrow that goes back one column to the left, and up by r rows. These differentials commute with multiplication by h 0 . Turning to the next page of the Adams sequence, an element E s,t r`1 is obtained by taking the homology of the complex E r at E s,t r . As one continues to turn the pages, the elements of the Adams charts stabilise to what we will denote by E s,t 8 .
The existence of the Adams spectral sequence means that there is a filtration of Ω H t˘2 given by`Ω This information can be formally assembled to obtain`Ω H t˘2 , namely by taking the inverse limit`Ω Finally, it is worth remarking that the extension problem just described can sometimes be non-trivial in a controlled way, determined by the module structure on the E 2 page of the Adams chart. For example, a multiplication by h 0 on the E 2 page records multiplication by 2 between F s,t 8 and F s`1,t`1

8
. Thus, an infinite h 0 -tower in the t column implies F 0,t 8 L F s,t`s 8 -Z{2 s for all s, which gives F 0,t 8 -Z 2 . Similarly, if there is a truncated h 0 -tower of length m, we get F 0,t 8 L F s,t`s 8 -Z{2 s for s ď m and remains Z{2 m when s ą m; it can be easily seen tha F 0,t 8 " Z{2 m in this case. In most cases considered here, these are all the non-trivial extensions that appear. So when the bordism groups can be fully calculated at the 2-completion we can read off the results from the Adams chart directly, whence an infinite h 0 -tower gives the free Abelian group Z, and a truncated h 0 -tower of length m gives the torsion group Z{2 m . There can also be non-trivial extensions that do not arise from the structure of the E 2 page. These are called exotic extensions.

A.2 Calculation of Ω
Spin-U p2q 6 It was calculated in Ref. [15] that the associated Madsen-Tillmann spectrum for the symmetry type H " Spin-U p2q is M T H " M Spin^Σ´5M SOp3q^M U p1q. The second page of the Adams spectral sequence is then given by The cohomology ring here is where w 1 i P H i pBSOp3qq are the universal Stiefel-Whitney classes for SOp3q while w 2 2 P H 2 pBSOp2qq is the second universal Stiefel-Whitney class for SOp2q. Here U P H 3 pM SOp3qq and V P H 2 pM SOp2qq are the Thom classes for M SOp3q and M SOp2q, respectively. The Ap1q-module structure of this ring, up to degree 10, was also calculated in Ref. [15]. This module structure is represented by Fig. 1. This was computed by applying the 'Wu formula' for the action of the Steenrod squares on Stiefel-Whitney classes, together with the relations Sq 2 U " w 1 2 U , Sq 2 V " w 2 2 , and Sq 1 U " Sq 1 V " 0. Since there are only 5 generators in H 11 pM SOp3q^M SOp2qq and 3 generators in H 12 pM SOp3q^M SOp2qq, the diagram in Fig. 1 in fact represents the module H ‚ pM SOp3q^M SOp2qq up to degree 12, so it can be used to calculate the Adams chart (and thence the bordism groups) up to degree t´s " 7, which is reproduced in Fig. 2. The 6 th bordism group can then be read off directly from the chart, giving To compute the bordism groups for the symmetry type SpinˆBZ{2 m , our first task is to work out the Ap1q-module structure of H ‚ pBZ{2 m q. It is known that where the generator a is in degree 1, and the generator b is in degree 2 [50] (see also Theorem 6.19 of [51]). The generator b can be defined as follows. The short exact sequence 0 ÝÑ Z{2 ÝÑ Z{2 m`1 ÝÑ Z{2 m ÝÑ 0 (A.10) of coefficient groups induces the long exact sequence in cohomology . . . Ñ H i pX; Z{2q Ñ H i pX; Z{2 m`1 q Ñ H i pX; Z{2 m q βm Ý Ý Ñ H i`1 pX; Z{2q Ñ . . . , (A.11) for any topological space X (here we temporarily restore the explicit Z{2 coefficients for emphasis). The connecting homomorphism β m is the m th power Bockstein homomorphism. One can then define b " β m pa 1 q, where a 1 P H 1 pBZ{2 m ; Z{2 m q is a canonical choice of generator.
The Ap1q-module structure of H ‚ pBZ{2 m q is then given in Fig. 5, where the dashed lines represent the m th power Bockstein β m . Hence, as an Ap1q-module, we can write H ‚ pBZ{2 m q as where the Ap1q-module M together with its corresponding Adams chart, and the Adams chart for the Ap1q-module Z{2, are shown in Figs. 3 and 4, respectively. We can then combine these to construct the Adams chart for H ‚ pBZ{2 m q, shown in Fig. 6 for m " 3 as an example. By the May-Milgram Theorem [52], the only non-trivial differentials are those denoted by d m , which are induced by the Bockstein homomorphism on the classes with even t´s. From the Adams chart it is clear that Ω Spin 6 pBZ{2 m q " 0. The resulting spin bordism groups in degrees d ď 6 are given in Table 1. Note that there are non-trivial extensions in degree 5, whose corresponding bordism group has been calculated by other means in Refs. [20,42].     where 2ξ is twice the sign representation. Now, we have to work out the Ap1q-module structure of H ‚ pBpZ{2 m qq 2ξ q, which turns out to be the same as that of H ‚ pBZ{2 m q but with the two bottom cells removed, as shown in Fig. 7. Hence, as an Ap1q-module  where the Ap1q-module M is defined as before (see Figure 3a). Using the Adams charts for M given in Fig. 3b, we can see that the second page of the Adams chart for H ‚ pBpZ{2 m qq 2ξ q must be given by Fig. 8. As in the previous case of H ‚ pBZ{2 m q, the only non-trivial differentials are d m on the m th page, which are induced from the m th power Bockstein homomorphism [12,52]. So the non-trivial differentials act only on the even t´s columns. As an example, these are shown for m " 2 in Fig. 9. This can be easily generalised to m ě 2, with results shown in Table 2. The 5 th bordism group was calculated by a different method in Ref. [20].   Table 2. The bordism groups with Spin-Z{2 m`1 structure in degrees d ď 6.