Global anomalies in the Standard Model(s) and Beyond

We analyse global anomalies and related constraints in the Standard Model (SM) and various Beyond the Standard Model (BSM) theories. We begin by considering four distinct, but equally valid, versions of the SM, in which the gauge group is taken to be $G=G_{\text{SM}}/\Gamma_n$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $\Gamma_n$ isomorphic to $\mathbb{Z}/n$ where $n\in\left\{1,2,3,6\right\}$. In addition to deriving constraints on the hypercharges of fields transforming in arbitrary representations of the $SU(3)\times SU(2)$ factor, we study the possibility of global anomalies in theories with these gauge groups by computing the bordism groups $\Omega^{\text{Spin}}_5(BG)$ using the Atiyah-Hirzebruch spectral sequence. In two cases we show that there are no global anomalies beyond the Witten anomaly, while in the other cases we show that there are no global anomalies at all, illustrating the subtle interplay between local and global anomalies. While freedom from global anomalies has been previously shown for the specific fermion content of the SM by embedding the SM in an anomaly-free $SU(5)$ GUT, our results here remain true when the SM fermion content is extended arbitrarily. Going beyond the SM gauge groups, we show that there are no new global anomalies in extensions of the (usual) SM gauge group by $U(1)^m$ for any integer $m$, which correspond to phenomenologically well-motivated BSM theories featuring multiple $Z^\prime$ bosons. Nor do we find any new global anomalies in various grand unified theories, including Pati-Salam and trinification models. We do, however, show that there is a potential global anomaly in a five-dimensional grand unified theory based on $SO(18)$. We also consider global anomalies in a family of theories with gauge group $SU(N)\times Sp(M)\times U(1)$, which share the phase structure of the SM for certain $(N, M)$.


Introduction
The Standard Model (SM) has been tremendously successful in explaining all the data collected from collider physics experiments such as at the LHC, with the gauge, flavour, and Higgs sectors having been tested at the per mille, per cent, and ten per cent levels respectively [1]. However, despite its successes, there are a number of unsolved problems in the SM. Some of these are experimental or observational in origin, such as the inability to account for the dark matter and dark energy that are observed by astrophysicists and cosmologists, while other problems appear to be more theoretical or aesthetic, such as the inability to describe physics beyond the Planck scale, and the (two) hierarchy problems associated with the two super-renormalisable operators in the SM lagrangian. It is clear that in order to offer a complete description of Nature, one must go beyond the Standard Model (BSM). In order to be a consistent quantum field theory, any BSM theory that we construct (as well as the SM itself) must not suffer from any anomalies associated with its gauge group.
In fact, before we consider going beyond the SM, it is important to emphasise that there is not even an unique SM, but many possible Standard Models, all of which are consistent with the same experimental data. The experimentally-observed SM gauge bosons and their interactions, together with the representations of the SM fermion fields, tell us that the Lie algebra of the SM gauge group is sup3q ' sup2q ' up1q. The four gauge groups G " G SM Γ n , G SM " SU p3qˆSU p2qˆU p1q, Γ n -Z{n, n P t1, 2, 3, 6u , (1.1) all share this Lie algebra and have representations corresponding to the SM fermions, 1 and any one of these may be the gauge group of the SM. 2 Thus, in addition to the various deficiencies in the SM that necessitate its extension, there is also an ambiguity in the SM. The potential physical distinctions between the four options in Eq. (1.1) were studied recently in Ref. [2], and amount to different periodicities of the θ angle associated with the hypercharge factor, and different spectra of Wilson lines in the theory. Perhaps unsurprisingly, all of these effects have a topological flavour.
Another possible distinction, which is also topological in origin but which was not discussed in Ref. [2], is that some of these options might not in fact be consistent after closer inspection, in the sense that they might suffer from anomalies. Of course, since the four groups in Eq. (1.1) share the same Lie algebra the conditions for local anomaly cancellation will be the same, and thus all these SMs are free of local anomalies, as is well known. However, this does not rule out the possibility of more subtle global anomalies in the SMs associated with the topology of the gauge group, analogous to (but much more general in scope than) the SU p2q anomaly discovered by Witten [3], which might render some of the SM variants recorded in Eq. (1.1) inconsistent. Our first goal in this paper is to investigate the possible global anomalies for each choice of discrete quotient in (1.1), for arbitrary fermion content.
To do so, we exploit the relation that arises in the absence of local gauge anomalies between the potential anomaly of the partition function (which arises in the phase) of a chiral gauge theory and the exponentiated η-invariant [4] (which is a regularized sum of positive eigenvalues minus negative eigenvalues) associated to an extension of the Dirac operator to a five-manifold that bounds spacetime. This relation, which was first suggested in Ref. [5], follows from a set of mathematical results due to Dai and Freed [6], which we briefly review in §2 (for a more detailed discussion, see [7][8][9]). To wit, one may show (via a vast generalisation of Witten's original 'mapping torus' argument [3]) that if exp 2πiη " 1 on all closed five-manifolds that are equipped with a spin structure and a map to BG, 3 then there will be no anomalies on spacetimes which bound (in the sense that the requisite spin and gauge structures can be extended). Since exp 2πiη is invariant under bordism in the case that local anomalies vanish, this is guaranteed to be the case when the group Ω Spin 5 pBGq (of equivalence classes under bordism of five-manifolds equipped with a spin structure and a map to BG) vanishes. 4 In this paper we begin by applying this criterion for global anomaly cancellation to the four versions of the SM given by Eq. (1.1). The computations we report in this paper build upon those of Ref. [10], which used the Atiyah-Hirzebruch spectral sequence to compute Ω Spin dď5 pBGq for a number of simple gauge groups G including SU pnq, P SU pnq, U Spp2kq, and SOpnq, as well as for U p1q. From there it was argued in Ref. [10] that there are no global anomalies in the SMs, by exploiting the (perhaps fortuitous) fact that the particular fermion content of the SM can be embedded in an anomaly-free grand unified theory (GUT) with G " SU p5q (which breaks down to G SM {Γ 6 as we go below the GUT scale). Alternative derivations of this result can be found in Refs. [11,12]. It turns out that this guarantees that all 4 versions of the SM in Eq. (1.1) are anomaly-free for the SM fermion content, or any other fermion representations that form representations of SU p5q.
We analyse the global anomalies in theories with one of the SM gauge groups by computing each Ω Spin 5 pBGq for the four gauge groups listed in Eq. (1.1) directly. At least in 3 out of the 4 cases (those in which n P t1, 2, 3u), we can do this by first noting that the 3 To see why BG is relevant, note that a gauge field is defined by a connection on a principal G-bundle over a spacetime manifold Σ, and every such bundle corresponds to a map Σ Ñ BG; for global anomalies, the connection plays no role, and we have a one-to-one correspondence between G-bundles (without connection) and homotopy classes of maps Σ Ñ BG. 4 In fact, there are reasons to believe that the vanishing of Ω Spin 5 pBGq is sufficient for the vanishing of global anomalies not only on spacetimes that bound, but also on those that do not -we discuss this at the end of §2.
gauge group can be written as a product (for example, G SM {Γ 2 -U p2qˆSU p3q). Next, we extend the methods of Ref. [10] to treat gauge groups which are products, by exploiting the fact that BpGˆHq " BGˆBH, 5 and using a Künneth formula in (co)homology. The 4th case, in which G " G SM {Γ 6 , succumbs to a slightly more sophisticated attack, which we describe in §4.5.
Our results for the four possible connected SM gauge groups can be applied, unlike those of Ref. [10], to any BSM theories with one of the SM gauge groups but with different fermion content (that do not necessarily fit inside any GUT with a simple gauge group). While one might have expected, given the much more general nature of the anomaly cancellation condition imposed, more constraints to appear beyond those required to cancel the familiar SU p2q global anomaly discovered by Witten, one finds that in fact that the opposite happens: in some cases there are actually fewer constraints, due to a subtle interplay between global and local anomalies, which we describe in §4. 6. This is related to the more mundane fact that for the gauge groups featuring quotients by Γ n‰1 there are non-trivial constraints on the hypercharges of fermions depending on their representation. We give these constraints in §4.1.
We then turn our attention to global anomalies in a number of well-motivated BSM theories, which we analyse using the same bordism-based criteria. We demonstrate our methods in a wide variety of BSM examples, in the hope that readers can adapt the methods to analyse their own favourite models. In particular, we consider theories in which the SM gauge group is extended by products with arbitrary U p1q factors, as well as a number of GUTs including Pati-Salam models, trinification models, and a five-dimensional theory based on SOp18q.
One might a priori expect all bets to be off when one goes beyond the SM, and that the possibility of Ω Spin 5 pBGq being non-trivial might provide a variety of extra constraints on the fermion content of BSM models for the cancellation of new global anomalies. Interestingly, we will find that this is largely not the case. In all the four-dimensional examples we considered, we find that Ω Spin 5 pBGq detects no new anomalies beyond the Z{2-valued anomalies associated with SU p2q (or more generally Spprq) factors in the gauge group. While we essentially arrive at a large collection of null results, we hope that the absence of any potential new anomalies in all of our examples will at least provide some assurance for the more conscientious BSM model-builders, who worry that their models might suffer from secret global anomalies.
We remark that in spacetime dimensions lower (or indeed higher) than four there are, however, potentially lots of new anomalies in theories with these gauge groups. We catalogue the relevant bordism groups in lower dimensions for the gauge groups we consider alongside the results of importance to the (B)SM case, in case they might be of interest to others (for example, in the condensed matter community). For ease of reference, all our bordism group results are collated across Tables 1, 3, and 4.
In higher dimensions, we find that there are potential global anomalies in five-dimensional theories with the gauge group SOp18q, by computing that Ω Spin 6 pBSOp18qq does not vanish. Thus, any GUT based on such a gauge group (such as that proposed in Ref. [14]) must be carefully checked for global anomalies.
The outline of the rest of this paper is as follows. In §2 we review the so-called 'Dai-Freed theorem', and the arguments that underlie the bordism-based criterion for global anomalies that we use. In §3 we review the algebraic machinery of spectral sequences which we use to compute the bordism groups of interest to us. We then summarise and interpret our computations pertaining to global anomalies in the SMs in §4. In §5, we generalise the SM results to a 2-parameter family of theories that contains the SM, with gauge group SU pN qˆSppM qˆU p1q for N, M P Z. We present the details of our computations for BSM theories in §6. Finally, we find that there are no global anomalies in a BSM theory in which the SM fermions are defined using a spin c structure, allowing also for arbitrary additional fermion content, by showing that Ω Spin c 5 pBGq " 0 for each choice of G in Eq. (1.1). Such a theory can be defined on all orientable four-manifolds (not only those that are spin), but requires an additional U p1q symmetry be gauged such as B´L.
Note added: Ref. [15], which has subsequently appeared, confirms some of the bordism group calculations in this paper using the Adams spectral sequence.

Bordism and global anomalies
Both the local gauge anomalies first discovered by Adler, Bell, and Jackiw (ABJ) [16,17] and the global anomalies first discovered by Witten [3] may arise in chiral gauge theories due to subtleties in defining the Dirac operator. To see how, and to motivate the more general bordism-based criterion for anomaly cancellation that we employ, it is helpful to first review some basic facts about chiral fermions, for which we largely follow the discussion in Ref. [7]. Other helpful references for this discussion are Refs. [8,9,18] (written with physicists in mind) and the original mathematical paper by Dai and Freed on which much of the discussion rests [6].
Firstly, we recall that defining a chiral gauge theory requires that any spacetime manifold be equipped with certain geometric structures. The important structures for our purposes are • A form of spin structure to define fermions, • A principal G-bundle to define gauge fields, • A Dirac operator which couples fermions to gauge fields, whose determinant is a well-defined function on the background data if the theory is to be non-anomalous.
We work in four spacetime dimensions from the beginning, since that is the case of relevance to the particle physics applications we are interested in; however, all the material we review in this Section generalises straightforwardly to other numbers of dimensions. We always assume spacetime is euclideanised, and thus consider spacetime to be a smooth, compact, four-manifold Σ. At times it will be helpful to suppose Σ is equipped with a (riemannian) metric, but this shall not be especially important to our arguments.
In most of this paper, we assume that spacetime is orientable and that fermions are defined using an honest spin structure. It is possible, however, that fermions may be defined on an orientable spacetime using 'weaker' structures if there are gauge symmetries present, as is typically the case in particle physics. For example, the presence of a U p1q gauge symmetry allows one to define fermions using only a spin c structure; note that all orientable four-manifolds are spin c , but not all orientable four-manifolds are spin. In §7, we consider this possibility. In the presence of a larger gauge symmetry, such as SU p2q, one could get away with only a spin-SU p2q structure to define fermions [19], and so on. 6 In a time-reversal symmetric theory, 7 one could consider defining the theory also on unorientable spacetimes, in which case a form of pin structure could be used to define fermions. We describe how fermions can be defined using these various 'spin structures' in Appendix A for reference; we also invite the reader to consult Appendix A of Ref. [7]. Throughout the main body of this paper, however, we assume that spacetime is orientable and equipped with a spin structure.
Defining gauge fields for some gauge group G requires the existence of a principal Gbundle over Σ. As we wrote before, the classifying space BG of the Lie group G has the property that the homotopy classes of maps from a space X to BG are in one-toone correspondence with the set of (isomorphism classes of) principal G bundles over M . 8 Thus, we consider orientable spacetimes Σ equipped with a map f : Σ Ñ BG, in addition to a spin structure. We moreover insist that a gauge theory be defined on all manifolds admitting these structures, leading to a very broad notion of whether there is an 'anomaly' 6 A new kind of global anomaly has been recently discovered by Wang, Wen, and Witten [19] for an SU p2q gauge theory formulated on all manifolds admitting such a spin-SU p2q structure. They show that such a theory is anomalous if there is an odd number of fermion multiplets in spin 4r`3{2 representations of SU p2q (where r P Z). Of course, the more familiar SU p2q global anomaly arises when the theory is defined on all spin manifolds, in which case there is an anomaly when n L´nR " 1 mod 2, where n L (n R ) is the number of left-handed (right-handed) SU p2q doublets [3]. 7 We note that the SM is not time-reversal symmetric, since CP is explicitly broken by the phases appearing in the CKM and PMNS matrices, and in theory also by a non-zero QCD θ angle. Thus, in this paper we only consider theories with one of the SM gauge groups to be defined on orientable spacetimes. 8 The classifying space BG is the quotient of a weakly contractible space EG by a proper free action of G. Any principal G-bundle over M is the pullback bundle f˚EG along a map f : M Ñ BG.
in the theory. Ultimately, these requirements are necessary to guarantee that the theory be consistent with locality.

Fermionic partition functions
One may define fermions and gauge fields on four-manifolds equipped with the given geometric structures. In a renormalisable four-dimensional chiral gauge theory, one couples the two via the lagrangianψi { Dψ, where i { D is an hermitian Dirac operator. We are now in a position to see how both the local and global anomalies can emerge in such a gauge theory.
The heart of the trouble in both kinds of anomaly lies in performing the functional integration over fermions. The result is a partition function Z ψ rAs, which we consider to be a function of the background gauge field and also any other background fields or data such as a metric on spacetime. 9 Formally, Z ψ rAs is defined to be the determinant of the Dirac operator, 10 assumed to be appropriately regularized. The partition function Z ψ rAs of a non-anomalous quantum field theory is a kosher C-valued function on the space of background data. For the case of coupling to background gauge fields, this means that Z ψ rAs must be a well-defined function on the space of connections on principal G-bundles modulo gauge transformations.
If this is not the case, G-invariance is anomalous, and since it is a gauge symmetry, the theory is not well-defined. This viewpoint sets the traditional ideas of local and global gauge anomalies in a more general context: in the case of a local anomaly, one has that Z ψ rAs ‰ Z ψ rA g s even for a gauge transformation A Ñ A g with g infinitesimally close to the identity; for the original SU p2q global anomaly [3], one finds Z ψ rAs "´Z ψ rA U s where the group element U pxq corresponds to a gauge transformation in the non-trivial class of π 4 pSU p2qq. The partition 'function' of an anomalous theory is thus at best a section of a complex line bundle over the space of background data, called the determinant line bundle. Moreover, the modulus |Z ψ | of the partition function cannot suffer from anomalies, 11 and 9 Sometimes, we use 'A' to denote the background gauge field, while at others time we use 'A' to collectively denote all the background fields/data. Which of the two meanings is implied in a given instance ought to be clear from the context. 10 More generally, Z ψ rAs will be the Pfaffian of the Dirac operator. We essentially ignore this subtlety for the purpose of this discussion, by assuming fermions to be complex or pseudo-real. 11 To see why, note that for any set of chiral fermions ψ, one can define a conjugate set r ψ that transforms as the complex conjugate of ψ under all symmetries, and with an action that is the complex conjugate of the action for ψ. Thus, the functional integration over r ψ yields preciselyZ ψ , the complex conjugate of (2.1). Hence, for the combined system, the partition functon is Z ψZψ " |Z ψ | 2 . But given the complex the anomaly must come purely from the phase of Z ψ .
With this realisation, one might first try to simply define the fermionic partition function to be equal to its modulus, and so construct an anomaly-free theory by fiat. But the modulus |Z ψ | on its own is not a smooth function of the background data A, just as |w| is not a smooth function of the real or imaginary parts of a complex number w. The partition function must, however, depend smoothly on the background data, which includes gauge fields and metrics, otherwise correlation functions involving the stress-energy tensor and/or currents coupled to the gauge field would not be well-defined. Thus, one cannot evade anomalies in such a way, and one must instead consider carefully when Z ψ is well-defined, and when it is not.
A set of mathematical results due to Dai and Freed [6] allow one to construct a candidate partition function, which is necessarily smooth on the space of background data, with which to properly analyse anomalies. For brevity's sake, we refer collectively to these results as the Dai-Freed theorem. For an account written with physicists in mind, see Ref. [18]. The Dai-Freed theorem implies that a putative partition function Z ψ rAs that is smooth in A can always be defined when the four-dimensional spacetime Σ is the boundary of a five-manifold X, viz. Σ " BX (as depicted in Fig. 1), to which the theory (and thus the spin structure and map to BG) must be extended. The five-manifold X must approach a 'cylinder' p´τ 0 , 0sˆΣ near the boundary Σ, where the local coordinate τ P p´τ 0 , 0s parametrises the fifth dimension. Moreover, the Dirac operator is extended to define a five-dimensional Dirac operator on X which we denote by i { D X , which near the boundary conjugate set of fermions one can always write down mass terms for the set of fermions ψ, for which a Pauli-Villars regulator (which respects the symmetries of the lagrangian) is always available. Hence |Z ψ | 2 , and thus |Z ψ |, cannot suffer from any anomalies. 12 Special boundary conditions must be chosen to ensure that the operator i { D X is hermitian throughout X. These are often referred to as '(generalised) APS boundary conditions', and we will not discuss them further, but rather refer the reader to e.g. Refs. [7,18], in addition to the original papers of Atiyah, Patodi, and Singer [4,20,21].

Schematically, the Dai-Freed definition of the putative partition function is then
where we have split the phase into two distinct contributions, which we will define shortly. Importantly, Dai and Freed showed that this construction varies smoothly with the background data.
The two contributions to the phase, as separated out in Eq. (2.2), correspond to local and global anomalies. The first contribution to the phase of (2.2) is easier to understand. It is the integral of the anomaly polynomial I 0 pF q over the extended five-manifold X, which is a polynomial in the curvature F of the connection A defined such that whereÂpRq is theÂ genus (sometimes referred to as the 'Dirac genus'), with R the Riemann tensor. The bar and subscript '6' indicates that one should take only the six-form terms on the right-hand-side. This contribution to the phase is not necessarily invariant even under infinitesimal gauge transformations. Rather, its variation can be computed using Eq. (2.3), and requiring that this variation vanish after being integrated reproduces the familiar formulae for the cancellation of local anomalies (including gravitational and mixed gaugegravitational anomalies). This type of anomaly is sometimes referred to as the perturbative anomaly, because one can derive it perturbatively by expanding the path integral around the zero background fields in flat spacetime.
The second contribution comes from the fermions on X, which one can think of as a kind of regulator for the system on Σ. The η-invariant is defined as the following sum over eigenvalues λ of the Dirac operator which must of course be regularized. 13 This η-invariant was introduced by Atiyah, Patodi, and Singer (APS) in their generalisation of the Atiyah-Singer index theorem to manifolds with boundary [4,20,21]. It shall be useful in what follows to recall that the η-invariant possesses an important 'gluing' property, as follows: if two manifolds with boundary Y 1 and Y 2 are glued along a common boundary to give a manifold Y 1 Y Y 2 , then the exponentiated as illustrated in Fig. 2.

Global anomalies and the η-invariant
In order for (2.2) to describe an intrinsically four-dimensional theory on Σ, this putative definition for the fermionic partition function must be independent of the choice of fivemanifold X and the extension to X of whatever structures are necessary to define the theory on Σ. Any dependence on X invariably leads to ambiguities and inconsistencies with locality and/or smoothness in the four-dimensional theory. Such inconsistencies are precisely what we call anomalies.
It is worth mentioning here that, if the condition for anomaly cancellation is not satisfied, we can no longer use Eq. (2.2) as the partition function for our theory on the four-manifold Σ. Nonetheless, even in this context (2.2) remains a useful equation, because it precisely quantifies the anomalies in terms of anomaly inflow. Heuristically speaking, it tells us that we can make sense of an anomalous fermionic theory if it arises as a boundary degree of freedom of another theory in one dimension higher, where the anomalies at the boundary are precisely cancelled by the contribution from the bulk. This is captured solely by the ηinvariant when there is no local anomaly, justifying our moniker of 'global' anomalies. This fact lies at the heart of our current understanding of topological insulators in condensed matter physics.
Let us return to our search for a criterion for anomaly-freedom. The putative partition function (2.2) is independent of the choice of five-manifold X if and only if exp p´2πiηXq " expˆ2πi for all closed five-manifoldsX. To see this, consider a duplicate of our fermionic theory on Σ but extended to a different five-manifold X 1 . Let´X 1 denote this five-manifold with its orientation reversed. It is then possible to glue the original system defined on pX, Σq to that on p´X 1 ,´Σq along the mutual four-boundary Σ. The result is a fermionic theory on a closed five-manifoldX " X Y p´X 1 q, as illustrated in Fig. 3. Since the two systems Figure 3: Gluing of two manifolds X and X 1 with a shared boundary Σ into a closed manifold X " X Y p´X 1 q.
have the same fermionic theory on Σ, the moduli of the path integrals cancel, and the path integral of the combined system is the pure phase Using the linearity property of integrals, together with the above gluing property for the η-invariant, we can rewrite the fermionic partition function on the closed five-manifoldX as ZX " expp´2πiηXq expˆ´2πi which is trivial if and only if the condition (2.6) is satisfied. The triviality of ZX for any closed five-manifoldX implies that Z X " Z X 1 for any pair of five-manifolds which share the same boundary theory Σ.
Thus, in the absence of local anomalies, i.e. when I 0 pF q " 0, any residual global anomalies necessarily vanish, and the partition function describes an intrinsically fourdimensional theory, when exp p´2πiηXq " 1 for all closed five-manifoldsX (that admit a spin structure and a map to BG). Witten's mapping torus argument [3], by which the original SU p2q global anomaly was first detected (for a fixed spacetime Σ " S 4 ), is equivalent to insisting that exp p´2πiηXq " 1 onX " S 1ˆS4 .
Moreover, when local anomalies cancel, such that I 0 pF q " 0, it follows from the APS index theorem that expp2πiηq is a bordism invariant. 14 By 'bordism' we mean (unless explicitly stated otherwise) the equivalence relation on compact p-manifolds equipped with a spin structure and a map to BG such that two manifolds are deemed equivalent if their disjoint union is the boundary of some compact pp`1q-manifold with the structures extended appropriately. By 'bordism invariant', we mean a well-defined homomorphism on the equivalence classes under bordism (or just bordism classes), which form an abelian group Ω Spin p pBGq. This means that expp2πiηq " 1 on any five-manifold that is null-bordant. Hence, when I 0 pF q " 0 the η-invariant defines a homomorphism from the fifth spin bordism group to the phase of the partition function, or, in other words expp2πiηq P Hom´Ω Spin 5 pBGq, U p1q¯. (2.8) The group Hom pΩ Spin 5 pBGq, U p1qq clearly vanishes if Ω Spin 5 pBGq " 0. The vanishing of Ω Spin 5 pBGq is in fact not only sufficient but also necessary for vanishing of Hom pΩ Spin 5 pBGq, U p1qq, at least when Ω Spin 5 pBGq is a finitely generated abelian group (as is the case for all the examples we examine here), which means it can be written as To see that this is the case, note that for each summand there exist non-trivial maps to U p1q -for example, one can send n P Z{p to expp2πin{pq, or can send k P Z to exppπikq. Thus, as long as Ω Spin 5 pBGq ‰ 0, the set of homomorphisms from the 5th spin bordism group to U p1q is non-empty.
The exponentiated η-invariant is necessarily trivial when Ω Spin then Eq. (2.6) implies there is a well-defined fermionic partition function which is independent of the choice of five-manifold X, and thus defines a sensible local quantum field theory.
In summary, the following precise statement, which follows from the Dai-Freed theorem, forms the basis of what follows: The path integral for a d-dimensional gauge theory with gauge group G with arbitrary matter content can be consistently formulated on null-bordant spacetime manifolds of dimension d using the Dai-Freed prescription if I 0 " 0 and Ω Spin d`1 pBGq " 0.
Two caveats are warranted here. Firstly, we still don't have a definition for spacetimes Σ that are not null-bordant. Such spacetimes appear regardless of the gauge group, 15 being generated by a K3 surface [23]. In general, locality forces such spacetimes to appear in the theory, and so one needs a general prescription for the fermionic partition function evaluated on spacetimes in non-trivial bordism classes, which goes beyond the original Dai-Freed theorem.
The second caveat is that, even if the Dai-Freed prescription cannot be made to work, it is still possible that some other suitable definition of the path integral might be found in cases where the condition (2.10) is violated.
In fact, recent developments in the mathematical field of topological field theory give hints that these two caveats can safely be struck out. Those developments suggest that an anomalous theory should be viewed as a special case of a relative field theory [24], namely a natural transformation between an extended field theory in one higher spacetime dimension (defined as a functor from some higher bordism category to some linear category) to the trivial field theory with the same dimension. Thus, part of the data of an anomalous field theory is a non-anomalous, non-trivial quantum field theory in one dimension higher. If there are no such theories, then there can be no anomalies. The putative theory in one dimension higher is, in many cases (but see Refs. [24,25]), both topological and invertible, meaning that it can be described by a classical topological action. It turns out that such actions can be classified by some Abelian group A corresponding to some (generalized) differential cohomology theory. The group is characterised by an exact sequence of Abelian groups B Ñ A Ñ C, where C corresponds here to the local anomaly and B to the global anomaly. In the case of ordinary differential cohomology (in which we have not bordism classes of manifolds with spin, but rather homology classes corresponding to smooth singular simplices), the group B is just the group H 5 pBG, U p1qq -HompH 5 pBGq, U p1qq and so it is tempting to conjecture that the corresponding group here is indeed HompΩ Spin 5 pBGq, U p1qq. Moreover, in the ordinary differential cohomology case, the exact sequence B Ñ A Ñ C extends to a short exact sequence 0 Ñ B Ñ A Ñ C Ñ 0, so that A " 0 iff. B " C " 0. If the same is true here, then we have a complete characterisation of the anomaly cancellation conditions, whose global part is HompΩ Spin 5 pBGq, U p1qq " 0.

Methodology
It remains to explain how we actually compute a bordism group of the form Ω Spin 5 pBGq, for a specific G. As is so often the case in algebraic topology, one is faced with a calculation that is seemingly impossible, no matter how simple the choice of G, but which turns out to be possible for almost any G, provided one knows enough tricks. The main tricks in the case at hand are the Atiyah-Hirzebruch spectral sequence [26] (see Refs. [27,28] for introductions to spectral sequences) and the use of cohomology operations (see Ref. [29]). We follow, essentially verbatim, the method set out in Ref. [10], but we feel it might be helpful to readers to give a more pedestrian description, as follows.
Spectral sequences are an important calculational tool in algebraic topology. So, what is a spectral sequence? In essence, a spectral sequence is a collection of abelian groups E r p,q indexed by three non-negative integers r, p, and q, together with a collection of group homomorphisms between them. Perhaps more appealingly, one can picture a spectral sequence to be a 'book' consisting of (infinitely) many pages, labelled by a 'page number' r, with a two-dimensional array of abelian groups E r p,q on each page. There are maps (called 'boundary maps' or 'differentials') between the groups within a given page of the form 16 which endows the groups E r p,q on the corresponding 'diagonals' of a given page with the structure of a chain complex. The first few pages are illustrated schematically in Fig. 4. Moreover, one passes from one page to the next by 'taking the homology' with respect to the differentials, specifically As we keep 'turning the pages' in this way, the abelian group appearing in any given pp, qq position will eventually stabilise (because there are only a finite number of differentials going 'in' and 'out' for any pp, qq). It is conventional to refer to the 'last page', after which all entries of the AHSS have stabilised, as E 8 p,q . Important topological information will be contained in this last page. For example, the Serre spectral sequence can be used to compute the (co)homology groups of a topological space X appearing as the total space in a fibration F Ñ X Ñ B, from the (co)homology of the two spaces F and B, where we take B to be simply connected. For the Serre spectral sequence, we can in fact ignore the first page, and begin at the second page, whose entries are given by the peculiar formula E 2 p,q " H p pB; H q pF ; Aqq; in words, the homology groups of the base space with coefficients valued in the homology groups of the fibre (for some coefficient group A). We then proceed to turn the pages using the differentials (3.1), until we get to the last page at which all the entries have stabilised. Then the nth homology group of the total space X can be pieced together for each n, using H n pX; Aq " À p E 8 p,n´p , in others words, by taking the direct sum of all the groups on the nth diagonal of the last page of the Serre spectral sequence. 17 The Atiyah-Hirzebruch spectral sequence (AHSS) is a generalisation of the Serre spectral sequence just described, in which ordinary (co)homology is replaced by generalised (co)homology. The bordism groups Ω Spin 5 pBGq that we want to compute to classify global anomalies are examples of generalised homology groups, and so the AHSS provides an appropriate tool for our computation, if we can fit BG into a useful fibration Given such a fibration, the AHSS is then constructed in a similar fashion to the Serre spectral sequence. We begin at the second page, whose entries are now the homology groups If the singular homology groups H p pB; Zq are free (i.e. do not contain torsion) then this simplifies to If this is not the case, then the universal coefficient theorem (in homology) must be used to calculate (3.4). This second page comes equipped with differentials as specified in Eq. (3.1), and if the differentials are known we can turn to the next page. If we are able to continue turning pages until all the entries with p`q " 5 are stabilised, then we can use these entries to extract Ω Spin 5 pBGq. Analogous to the example of the Serre spectral sequence, it shall be the case in all the examples we consider that Ω Spin 5 pBGq shall simply be the direct sum of the entries E 8 p,q with p`q " 5. 18 The simplest fibration involving BG, which we shall employ most frequently, is the trivial one in which BG is fibred over itself, such that the fibre is a point which we denote by pt, i.e. we consider pt ÝÑ BG ÝÑ BG.
In this case, computing the elements (3.5) of the second page of the AHSS requires two ingredients: (i) the singular homology groups of the classifying space, H p pBG; Zq, and (ii) the bordism groups (preserving the spin structure) equipped with maps to a point; in other words, simply the equivalence classes (under bordism) of spin five-manifolds. Fortunately for us, these bordism groups are well known in low dimensions [30]: n 0 1 2 3 4 5 6 7 8 9 10 The other ingredients we need are the homology groups of the classifying space of any gauge group G we want to consider. As we have advertised above, we will consider many examples where G is a product and our strategy here will be to build up the homology groups of such groups from the homology groups of their factors. We shall make frequent use of the fact that BpGˆHq " BGˆBH, (3.8) which follows from the definition of the classifying space of a group (see, for example, Chapter 16, §5 of [31]). Thence, we shall use the Künneth theorem to compute the homology of the product space BGˆBH with coefficients in Z. In the absence of torsion, 19 this is simply The classifying spaces (and their homology rings) for some elementary groups are wellknown; for example, BU p1q " CP 8 , with and BSU p2q " HP 8 , with While the homology groups for these two examples are known in all degrees, it is often enough for our purposes to know the groups H p pBG; Zq in sufficiently low dimensions; for instance, the result H p pBSU pnq; Zq " tZ, 0, 0, 0, Z, . . . u (3.13) (for n ą 1) shall be useful for our consideration of gauge theories relevant to particle physics.
Unfortunately for our purposes, results are usually quoted for cohomology groups of classifying spaces, not least because of their starring role in the theory of characteristic classes. But one can obtain the homology groups using some universal coefficient theorem.

Turning the pages
We have now proposed how to obtain all the ingredients with which to write down the second page of the AHSS associated with the fibration (3.6); but we do not yet know how to turn to the next page of the AHSS, which requires knowledge of the differential maps introduced in Eq. (3.1). One thing we know for certain is that the differentials are group homomorphisms, and in many cases this shall turn out to be enough to deduce the image and/or kernel of many differentials unambiguously; for example, we make frequent use of 19 If there is torsion, the correct statement of the Künneth theorem is that there is a short exact sequence Tor pHmpBG;Zq, HnpBH;Zqq Ñ 0, (3.9) and that this sequence splits (although not canonically).
the fact that HompZ{n, Zq -0. Similarly, for any pair of finite integers n and m, we may use the fact that HompZ{n, Z{mq -Z{gcdpn, mq.
However, simple algebraic arguments like this will seldom be enough to determine all the differentials in the AHSS. Fortunately, we can make use of the fact that some of the differentials on the second page E 2 p,q are known for the case of the spin bordism groups Ω Spin q . In particular, we have that the differential is the composition of the (homology) dual of the Steenrod square and followed by reduction modulo 2 [32,33], and that the differential is the dual of the Steenrod square [32,33]. The Steenrod square, Sq 2 , is an operation on mod 2 cohomology classes, Sq 2 : H n pX; Z{2q Ñ H n`2 pX; Z{2q, whose particular action on the generators of H n are known for the classifying spaces of Lie groups, thanks to Borel and Serre [34]. We will make regular use of their results in what follows. We note here for future reference that Sq 2 is an example of more general Steenrod squares, Sq k : H n pX; Z{2q Ñ H n`k pX; Z{2q which are operations on mod 2 cohomology rings satisfying the following properties Moreover, the Steenrod squares, being natural transformations of cohomology functors, have the property that they commute with the map f˚: H ‚ pY ; Z{2q Ñ H ‚ pX; Z{2q induced on cohomology by a map f : X Ñ Y . Thus we have f˚Sq k Y " Sq k X f˚. By virtue of this naturality, the Steenrod squares' action on H ‚ pBG 1ˆB G 2 ; Z{2q, which we denote by Sq k for clarity, are fully determined by their action on H ‚ pBG 1 ; Z{2q and H ‚ pBG 2 ; Z{2q, denoted by Sq k 1 and Sq k 2 . To see this, consider a projection π i : BG 1B G 2 Ñ BG i , with i " 1, 2. Let c i P H ‚ pBG i ; Z{2q be a generator. By naturality we have Sq k pπ˚c i q " π˚pSq k i c i q. But since πi c i is naturally identified with c i through the Künneth theorem for cohomology, this gets simplified to With help from Cartan's formula (3.16), the Steenrod squares' action on any generator of H ‚ pBG 1ˆB G 2 ; Z{2q can be subsequently worked out.

Global anomalies in the Standard Model(s)
Now that we have laid the groundwork and described the computational tools we use to identify potential global anomalies, we are ready to report our computations. We begin with a gauge theory of indisputable importance to particle physics phenomenology, namely the Standard Model(s). Our results for the SM gauge groups are summarised in Table 1.
The Standard Model (SM) of particle physics is a four-dimensional gauge theory, with gauge group Here, the Z{6 quotient in the case of Γ 6 is generated by the element where ω is the generator of the Z{3 centre of SU p3q (with ω 3 " 1 P SU p3q), and η is the generator of the Z{2 centre of SU p2q (with η 2 " 1 P SU p2q). The Γ 3 quotient in (4.1) is generated by ξ 2 , and the Γ 2 quotient by ξ 3 . The fermion content of the SM consists of quarks and leptons, which are chiral fermions transforming in the following representations of G where here all the fields indicated are left-handed.
We compute the fifth bordism group (preserving spin structure) for all four groups listed in Eq. (4.1), and so identify potential global anomalies in these theories. Recall that in Refs. [10,11], it was argued that there are no global anomalies in the SM with any of these four gauge groups, by fitting all four possibilities inside an SU p5q GUT which is easily shown to be anomaly-free (since the computation of the bordism group for SU pnq is straightforward). What we shall prove is a more general result, since it shall apply to gauge theories with one of these four gauge groups, but with arbitrary fermion content. Thus, the results we find shall apply immediately to any BSM theories in which the gauge group is that of the SM, but in which there are additional chiral fermion fields.

Hypercharge constraints
Before we start computing bordism groups, it is important to point out that if we extend the SM by adding extra fermions, one must make sure that such fermions transform in bona fide representations of whichever gauge group from Eq. (4.1) is being considered. In the cases where G " G SM {Γ n with n P t2, 3, 6u there are constraints on the possible hypercharges fermions can take, depending on their representation under the SU p3qˆSU p2q factor of G SM . Since the derivations of these constraints involve a digression into representation theory, we relegate them to Appendix D. In this Section we simply record what these constraints are -specifically, see Eqns (4.5, 4.8, 4.10). (Needless to say, the SM fermion representations satisfy these constraints.) The Γ 2 quotient case Given the Z{2 quotient in the case G " G SM {Γ 2 is generated by ξ 3 , where ξ is given in Eq. (4.2), we can write this particular quotient of the SM gauge group as In addition to its use in deriving the hypercharge constraints, writing the gauge group in this way (i.e. as a product) is crucial to our strategy for computing its bordism groups, in §4.3. Focussing on the U p2q " pSU p2qˆU p1qq {pZ{2q factor of G, a representation of U p2q corresponds to a representation of SU p2qˆU p1q, which in this subsection we denote by pj, qq where j denotes the isospin-j representation of SU p2q (which has dimension 2j`1) and q P Z is the integer-normalised U p1q charge, with some restrictions imposed.
To see how these constraints arise, let us first consider a field ψ transforming in the representation p 1 2 , qq, i.e. in the fundamental representation of SU p2q, since this is the simplest case. This means that ψ Þ Ñ ψ 1 " exp piqθq σ¨ψ under the action of the U p2q group element corresponding to pσ, exp iθq P SU p2qˆU p1q. For this to be a kosher representation of U p2q, one must identify the action of p1, exp iπq and p´1, 1q, which gives us the constraint exp iqπ "´1. Therefore, any SU p2q doublet must have hypercharge i.e. an odd integer. 20 This is the case in the SM, where the doublet representations Q and L carry hypercharges 1 and´3 respectively, using an integer normalisation in which the 20 Similar restrictions on U p1q charges appear in the context of defining fermions on manifolds that are not necessarily spin, by using the U p1q gauge symmetry to define a spin c structure. In that context, such charge restrictions depend on the representations of fermions under the Lorentz group, and are thus referred to as 'spin-charge relations' [35]. We consider these spin-charge relations more in §7.
smallest charge (that belonging to Q) is set to one.
If one wishes to add additional electroweak doublets, choosing the gauge group (4.3), one must ensure they too have odd hypercharges.
If one adds additional BSM fields transforming in larger representations of SU p2q, there are similar constraints on their hypercharges if they are to embed in representations of U p2q. To wit, for a field transforming in the pj, qq representation, the hypercharge must satisfy q " 2j mod 2. (4.5) In other words, the charge must be even for all integer isospin representations (including, of course, any SU p2q singlets), and odd for all half-integer isospin representations. For the proof of this general statement, we refer the reader to Appendix D.
The Γ 3 quotient case Given the Z{3 quotient in the case G " G SM {Γ 3 is generated by the element ξ 2 , we can write this variant of the SM gauge group in the more useful form In this case, we obtain hypercharge constraints on any fields transforming non-trivially under SU p3q, by requiring that they embed in representations of U p3q.
Consider the simplest case of a field ψ transforming in the fundamental triplet representation of SU p3q (a.k.a. a quark) and with charge q under U p1q. Under the action of exppiqθqg P U p3q, for some g P SU p3q, we have that ψ Þ Ñ ψ 1 " exppiqθqg¨ψ. To be a bona fide representation of U p3q means that pexp 2πi{3 , 1 3 q and p1, ω " e 2πi{3 1 3 q are identified in SU p3qˆU p1q, giving the constraint e 2qπi{3 " e 2πi{3 . Hence, any colour triplet must have hypercharge q " 1 mod 3. (4.7) The SM quark fields Q, U , and D have hypercharges`1,`4, and´2 respectively, all of which are indeed equal to 1 mod 3.
One might consider adding fermions in other representations of SU p3q, and for each representation there is a corresponding hypercharge constraint. Irreducible representations of SU p3q correspond to Young diagrams with two rows, and so can be labelled by a pair integers pλ 1 , λ 2 q corresponding to the number of boxes in each of the two rows, with λ 1 ě λ 2 ě 0. In Appendix D, we prove that the hypercharge q of a field transforming in the pλ 1 , λ 2 q representation of SU p3q must satisfy q " pλ 1`λ2 q mod 3, (4.8) if the gauge group is U p3qˆSU p2q. Note in particular that any colour singlets must have charge q P 3Z, as is the case for the SM leptons.
The Γ 6 quotient case Finally, we discuss the case with gauge group G " G SM {Γ 6 . Consider a field in an arbitrary representation of this gauge group, corresponding to the pλ 1 , λ 2 q representation of SU p3q, the isospin-j representation of SU p2q, and with U p1q charge q. The hypercharge constraint is that q " 2j mod 2 " pλ 1`λ2 q mod 3 (4.9) (see Appendix D). For example, for a field with j " 1{2 and pλ 1 , λ 2 q " p1, 0q, i.e. corresponding to the bifundamental representation of SU p3qˆSU p2q, this constraint reduces to q " 1 mod 6. (4.10) The only SM fermion transforming in the bifundamental representation of SU p3qˆSU p2q is the left-handed quark doublet Q, and sure enough the charge of Q is one.
Having established these constraints on the hypercharges of fermion fields for these four versions of the SM gauge group, we now turn to our main concern, which is to compute the bordism groups of BG for each of the four possible gauge groups G, which detect potential global anomalies theories with these gauge groups. We begin with the simplest case.

Ω Spin
For the simplest case where G " G SM " SU p3qˆSU p2qˆU p1q with a regular spin structure, we use the AHSS associated with the fibration (3.6) to compute the bordism groups Ω Spin dď5 pBG SM q. To begin, we have that B rSU p3qˆSU p2qˆU p1qs " BSU p3qˆBSU p2qˆBU p1q. (4.11) Together with the Künneth formula in cohomology, this means that the cohomology ring of BG SM is generated by the Chern classes associated with each factor of the gauge group, where x P H 2 pBG SM ; Zq indicates the first Chern class associated with the U p1q factor, c 1 2 P H 4 pBG SM ; Zq indicates the second Chern class of SU p2q, and c 2 P H 4 pBG SM ; Zq and c 3 P H 6 pBG SM ; Zq indicate the second and third Chern classes respectively of the SU p3q factor. We thus have the following low dimension cohomology groups with all cohomology groups in odd degrees vanishing. Because of this, and because these groups are all torsion-free, there is a (non-canonical) isomorphism yielding the homology groups that we need to populate the entries of the second page of the AHSS relevant for computing the bordism groups Ω Spin Since the action of the Steenrod square on the generators of H ‚ pBSU pnq; Z{2q, which are the universal Chern classes, is given by the formula [10] Sq 2 pc i q " pi´1q c i`1 the Steenrod square action on each of the generators of the cohomology ring (4.12) is then given by Sq 2 pxq " x 2 , Sq 2 pc 1 2 q " 0, Sq 2 pc 2 q " c 3 , where x 2 is a shorthand notation for x Y x, the cup product of cohomology classes. This follows from the third line of Eq. (3.16) and naturality of the Steenrod squares, as discussed at the end of §3. We see from Fig. 5 that there is only a single entry on the diagonal p`q " 5 which is thus relevant to the computation of Ω Spin Let us denote the generators of E 2 4,1 -pZ{2q 3 as r x 2 , r c 1 2 , and r c 2 , which are dual to the generators x 2 , c 1 2 , c 2 P H 4 pBG SM ; Z{2q by the Kronecker pairing (denoted ¨,¨ ) between homology and cohomology. Then we see that where Ą Sq 2 denotes the dual Steenrod square. Hence, the kernel of β is ker β -pZ{2q 2 , generated by r c 1 2 and r c 2 .
The differential labelled γ in Fig. 5 is the composition of the dual Steenrod square and the reduction mod 2: where the relevant Steenrod square is 20) where to deduce x 2 Þ Ñ 2x 3 we have used Cartan's formula (3.16) and the fact that Sq 1 pxq " 0 as H 3 is trivial. Again using the Kronecker pairing, we deduce that Ą Sq 2 kills r x, and sends r c 3 to r c 2 . Therefore im γ -Z{2, generated only by r c 2 . We can then take the homology with respect to the differentials β and γ to turn the page of the AHSS and deduce the p4, 1q element of the third page, Since the entries in every odd column vanish, there are no non-trivial differentials on the third page, and so we can turn to the fourth page with E 4 p,q " E 3 p,q for all pp, qq. On the fourth page the only differential relevant to computing Ω Spin 5 pBG SM q is d 4 : E 4 4,1 Ñ E 4 0,5 , which is a homomorphism from Z{2 to Z and is thus trivial. So the p4, 1q entry stabilises to E 4,1 8 -Z{2, and since this is the only non-zero element on the p`q " 5 diagonal it follows that Ω Spin where we can identify the potential global anomaly in this theory with the Witten anomaly associated to the SU p2q factor.
To see that this must be the case, consider a theory with gauge group G SM and a single fermion transforming as a doublet under SU p2q and a singlet under both SU p3q and hypercharge. Using the Dai-Freed prescription for the fermionic partition function one obtains an anomalous theory because exp 2πiη "´1 on S 4ˆS1 . This must therefore correspond to the non-trivial class in Ω Spin 5 pBG SM q. We can continue to compute the bordism groups of BG SM in lower degrees in a similar fashion. From Fig. 5  pBG SM q, we need the differential as well as the map d 2 2,1 : Z{2 Ñ Z{2. The dual Steenrod square is precisely the same as for the map β, which maps r x 2 Þ Ñ r x, and the other generators to zero, so we have that im α " Z{2. Then, we do not need to compute the map d 2 2,1 to deduce that its kernel must be Z{2, because we know that im α Ă ker d 2 2,1 . Hence, taking the homology, we deduce that E 8 2,1 " 0. All elements on the p`q " 3 diagonal thus stabilise to zero and we have that To compute Ω Spin 4 pBG SM q, we know from above that the map β into E 2 2,2 has image im β -Z{2, generated by the element r x P H 2 pBG SM ; Z{2q. The map out of E 2 2,2 is to zero and so its kernel is Z{2; turning to the next page, this element therefore stabilises at Z{2 Z{2 " 0. More care is required to deduce ker α, as follows. We have that r c 1 2 and r c 2 certainly map to zero, where note that the elements r x 2 , r c 1 2 , and r c 2 are here valued in integral homology (rather than in homology with coefficients in Z{2). Thus, while r x 2 P H 4 pBG SM ; Zq maps to the non-zero element r x P H 2 pBG SM ; Z{2q, the element 2 r x 2 P H 4 pBG SM ; Zq maps to zero in H 2 pBG SM ; Z{2q. Hence, the map α has a kernel ker α -Z 3 (which may look strange given its image is non-zero), and so we deduce E 8 4,0 -Z 3 . Given also that E 8 0,4 -Z, we compute

pBpG SM {Γ 2 qq
We now turn to compute the bordism groups for the variants of the SM involving quotients of G SM by discrete subgroups of its center, as listed in Eq. (1.1). Recall from §4.1 that pU p1qˆSU p2qˆSU p3qq{Γ 6 Z Z{2 epZ{3, ZˆZ{2q 0 epZ{3, epZ{3, Z 4 qq 0 Hence BpG SM {Γ 2 q " BU p2qˆBSU p3q using (3.8). This is useful, because the cohomology ring of the classifying space of the groups U pnq is well-known.
Using the usual fibration pt ÝÑ BpG SM {Γ 2 q ÝÑ BpG SM {Γ 2 q, the second page of the AHSS is given by E 2 p,q " H p`B U p2qˆBSU p3q; Ω Spin q pptq˘, as shown in figure 6. Recall  where c i , c 1 i are the ith Chern classes (which are cohomology classes in degree 2i) for SU p3q and U p2q, respectively. Thus, we have the integral cohomology groups (4.30) Again, because these are torsion-free and the cohomology groups all vanish in odd degrees, we deduce from these the integral homology groups, Thus far, this appears superficially identical to the case of no discrete quotient considered above, and indeed the second page of the AHSS is populated by the same groups; however, the action of the Steenrod squares is subtly different, meaning the action of the differentials (and, specifically, the maps α, β, and γ) is not necessarily the same as above. It turns out that an important difference shall be in the map γ. In particular, since the action of the Steenrod square on the generators c i of H ‚ pBU pnq; Z{2q -Z{2rc 1 , . . . , c n s is given by [34] we have that its action on the generators of the cohomology ring of BpU p2qˆSU p3qq is Sq 2 pc 1 1 q " c 12 1 , Sq 2 pc 1 2 q " c 1 1 Y c 1 2 , Sq 2 pc 2 q " c 3 , Sq 2 pc 3 q " 0.  where ρ denotes reduction modulo 2. Since Sq 2 : H 2 Ñ H 4 maps c 1 1 Þ Ñ c 12 1 , we see that both α, β map Ă c 12 1 Þ Ñ r c 1 1 and others to zero. Moreover, α maps 2 Ă c 12 1 to zero. So we have, using similar arguments as before, that ker α -Z 3 , im α -Z{2, ker β " pZ{2q 2 , im β -Z{2, (4.35) which is as it was in the previous case.
We now turn to the map γ. The relevant Steenrod square is here  and thus that this version of the SM has no global anomalies, no matter what the fermion content. One can compute the bordism groups in lower degrees using the same methods as in the previous example, and one finds no other differences in the results, which are again recorded in Table 1.
We thus arrive at a seemingly curious result; there are no global anomalies in this version of the SM, for arbitrary fermion content. The reader might wonder what has happened to the Witten anomaly, and the condition that there must be an even number of SU p2q doublets in the theory. We discuss the resolution to this puzzle (which also occurs in the case G " G SM {Γ 6 ) in §4.6. For now, it might be useful to remark on what goes wrong with the argument of the previous Section, in which we considered a theory with a single fermion in the spin-1 2 representation of SU p2q (and a singlet under both SU p3q and U p1q), and claimed exp 2πiη "´1 ‰ 1 on S 1ˆS4 . We cannot use such an argument when G " G SM {Γ 2 , because the hypercharge constraints presented in §4.1 mean there is no such representation of the gauge group, because any SU p2q doublet fermion must have odd (and thus non-zero) hypercharge. We must then take care to ensure that local anomalies associated with hypercharge cancel, before we turn to the global anomalies. We return to this issue in §4.6.

pBpG SM {Γ 3 qq
Our approach for tackling this variant of the SM is qualitatively very similar to that employed for the Z{2 quotient in the previous Subsection. Recall from §4.1 that the gauge group here may written as One may tackle this variant of the SM using the same methods employed for the Z{2 quotient in the previous Subsection. Thus, to avoid repetition, we relegate the calculations for this gauge group to Appendix C. The upshot is that we find corresponding to the Witten anomaly associated with the SU p2q factor in (4.39). The lower-degree bordism groups are tabulated in Table 1.
For this gauge group, an alternative fibration exists which we can also use to compute the bordism groups, based on the Puppe sequence. Reassuringly, using this other fibration yields the same bordism groups, and we include the details of both methods in Appendix C. We will need to employ such a Puppe-induced fibration shortly in §4.5 to compute the bordism groups of BpG SM {Γ 6 q.

pBpG SM {Γ 6 qq
The Z{6 quotient in the case G " G SM {Γ 6 is generated by the element ξ given by (4.2), and there is no straightforward way to write the group G SM {Γ 6 as a product, as we did in the previous two cases. This means a direct attempt to use the AHSS to compute the bordism groups of G SM {Γ 6 seems unlikely to work, given we do not know how the differentials on the second page act.
The second page of the AHSS associated with this fibration is given by While this may look like a rather unwieldy expression, note that the bordism groups Ω Spin q pBpU p2qˆSU p3qqq are precisely those that we have already computed in our study of global anomalies for the case G " G SM {Γ 2 , as recorded in the second line of Table 1. These groups only feature factors of Z and Z{2, and the homology groups of the Eilenberg-Maclane space KpZ{3, 2q valued in Z and Z{2 are [37] i 0 1 2 3 4 5 We can thence compute all the entries (4.43) in the second page of the AHSS. These are shown in Fig. 7.
Somewhat fortunately (for the sake of being able to perform the computation), all the entries on the p`q " 5 diagonal relevant for the computation of Ω Spin 5 pBG SM {Γ 6 q vanish already on the second page. This is just as well, because for this fibration we do not know any formulae for the action of the differentials (with which to turn to the next page) in terms of Steenrod squares (or indeed any other operation on (co)homology). 22 We thus conclude that Ω Spin 5 pBpG SM {Γ 6 qq " 0. Since all relevant homomorphisms are trivial, all entries E p,q with p`q ă 5 stabilise on the second page. We can then compute the remaining bordism groups with degree lower than 5 without ambiguities apart from Ω Spin 2 pBpG SM {Γ 6 qq and Ω Spin 4 pBpG SM {Γ 6 qq due to 22 Note that the similar-looking fibration Z{2 ÝÑ U p3qˆSU p2q ÝÑ G SM {Γ 6 does not yield such simplifications, and so cannot be used to compute the relevant bordism group because there are unknown differentials on the second page. This is roughly because the homology of KpZ{2, 2q is 'more complicated' than that of KpZ{3, 2q. (4.46) The notation epA, Bq denotes a group extension of A by B, that is, a group that fits into the following short exact sequence We tabulate our results in Table 1.
Note added: since this article appeared in preprint form, the Adams spectral sequence has been used to resolve the ambiguities we found (using the AHSS) in Eq. (4.46) [15]. It was therein found that Ω Spin 2 pBpG SM {Γ 6 qq " ZˆZ{2. Comparing with our result (4.46), this corresponds to the non-trivial extension 0 ÝÑ ZˆZ{2 ÝÑ ZˆZ{2 ÝÑ Z{3 ÝÑ 0, (4.49) where the first map is multiplication by 3 on the first factor and the identity on the second. In Ref. [15] it was also found that also corresponding to a non-trivial solution to the extension problem (4.46).

Interplay between global and local anomalies
It is interesting that there are no possible global anomalies in the cases with quotients by Z{2 and Z{6, whereas in the case of a quotient by Z{3 (or the case with no quotient at all) there is a Z{2 global anomaly which we have identified with the familiar Witten anomaly associated with the SU p2q factor.
This might at first appear puzzling. We know that cancellation of the Witten anomaly in an SU p2q gauge theory, and in the SM, requires n L´nR " 0 mod 2 if there are n L (n R ) left-handed (right-handed) fermions in SU p2q doublets. More generally, the Witten anomaly receives contributions from any fermions in SU p2q representations with isospin 2r`1{2, r P Z. Does the fact that we have computed that there are no such conditions for global anomaly cancellation in two variants of the SM mean that in these cases we can dispense with Witten's condition, and consider extensions of the SM with odd numbers of SU p2q doublets? The answer is no, due to a subtle interplay between global and local anomaly cancellation, which we now describe.
The key point is that taking discrete quotients of G SM changes the set of representations that fermions can carry, since every fermion must be in a bona fide representation of the group G. This leads to constraints on the possible hypercharges for fermions transforming as electroweak doublets. As we derived in §4.1, when we quotient G SM by Z{2 or Z{6, any field transforming in the pj, qq representation of the SU p2qˆU p1q factor must satisfy the isospin-charge relation q " 2j mod 2. Of course, one is free to perform an overall rescaling of all the U p1q charges in the theory, so the precise statement is that there must exist a normalisation of the U p1q gauge coupling such that the charge constraints (4.51) are possible. We assume such a normalisation for the U p1q charges in the following. 23 Now consider the cancellation of local anomalies. Suppose we have N j fermions transforming in the SU p2q representation with isospin j, and that these have charges denoted tq paq j u, where a " 1, . . . N j , and q paq j " 2j mod 2. We assume that all fermions have lefthanded chirality. The SU p2q 2ˆU p1q anomaly coefficient is then proportional to where the sum over j is over the different values of isospin, and T pjq denotes the Dynkin index (defined such that Tr`t a j t b j˘" 1 2 T pjqδ ab , where tt a j u denotes a basis for sup2q in the isospin´j representation), which is given by the formula This formula implies that T pjq is odd when j " 2r`1{2, r P Z, and is even otherwise.
When the anomaly condition (4.52) is reduced mod 2, only the contributions to (4.52) from isospins 2r`1{2 remain, since it is only these irreps for which both T pjq and the charges q paq j are necessarily odd. We thus obtain ÿ jP2Z`1{2 N j " 0 mod 2. (4.54) In other words, in the theories with gauge groups G SM {Γ 2 or G SM {Γ 6 , the total number of fermions transforming in isospin 2r`1{2 representations must be even, in order for the local SU p2q 2ˆU p1q anomaly to cancel -even though there is no global anomaly in either of these cases. This is equivalent to the condition, in the SU p2qˆU p1q case, that the usual Witten anomaly vanishes.

A generalisation of the SM
The Standard Model with gauge group G SM " SU p3qˆSU p2qˆU p1q is the starting point of a 2-parameter family of anomaly-free chiral gauge theories [38,39]. The gauge group for this family of generalised Standard Model theories is G GSM " SU pN qˆSppM qˆU p1q, N ą 2 and odd, M ě 1 (5.1) It was shown in Ref. [39] that theories in this family have the same phase structure as the Standard Model when one varies the relative strength between the strong force and as we wish.
the weak force. It is also not far-fetched to assume that this family of theories exhibits similar features in the infrared. This generalisation subjects the Standard Model to the framework of large-N expansion, which could potentially be used to analyse the dynamics of this family of chiral gauge theories perturbatively in a more controlled fashion.
The left-handed doublets of fermions that couple to the weak force in the Standard Model now become 2M -tuplets in the fundamental representation of SppM q. Since there are N`1 chiral fermions in the fundamental representation of SppM q, we need N to be odd to cancel the Z{2 global anomaly. In order to have sufficient number of chiral fermions to cancel the local anomalies, the right-handed fermions must proliferate, and we end up with M copies each of right-handed electrons E α , right-handed down quarks D α , right-handed up quarks U α , and right-handed neutrinos N α , with α " 1, . . . , M . There are also M copies of the Higgs field, H α . The matter content of this generalised theory and its representations under the gauge group G GSM is given in full in Table 2. The simplest case with M " 1 and N " 3 gives the Standard Model. The hypercharges given in Table 2 are chosen so that the theory is free of local anomalies, and the theory is moreover free of Witten anomalies associated with the SppM q factor. It is natural to ask whether this generalisation is really consistent for every pN, M q by considering our more general criterion for global anomalies, detected by Ω Spin 5 pBG GSM q. Fortunately, we do not need to repeat our calculation of the spin bordism group for this new gauge group as it is the same as the calculation in §4.2. To see this, first recall that the relevant entries on the second page of the AHSS are given by which are the same as those of SU p2q and SU p3q, respectively. Therefore, the relevant entries on the second page of the AHSS are still given by Fig. 5. Moreover, the action of the Steenrod square on the generators of lowest degrees of the cohohomology rings of BSppM q and BSU pN q are the same as in the Standard Model case, giving rise to the same relevant differentials in Fig. 5. The calculation given in §4.2 then goes through unaltered. We then have that Ω Spin implying that there is no additional global anomaly except the usual Witten anomaly associated with the SppM q factor of the gauge group (for any choice of M ) .

Global anomalies in BSM theories
In this Section, we show how to extend these methods to compute whether there are any potential global anomalies in BSM theories, by considering various popular examples. Firstly, we consider extensions of the SM by an arbitrary product of gauged U p1q symmetries (such as in theories featuring heavy Z 1 gauge bosons). We then turn to a number of grand unified theories, including the Pati-Salam model, trinification models, and a five-dimensional theory based on SOp18q.

Multiple Z 1 extensions of the SM
We consider a four-dimensional gauge theory with gauge group corresponding to an extension of the (usual) SM gauge group by arbitrary U p1q factors, with a priori arbitrary fermion content. The corresponding Z 1 bosons in such a theory have been posited to address many phenomenological questions -for a review, see e.g. Ref. [40]. We will compute whether there are potential global anomalies in such a BSM theory.
The cohomology ring for BG m is where x k is the first Chern class associated with the kth U p1q factor, and the remaining Chern classes are defined as in Eq. (4.12). In particular, we have the following lowdimensional cohomology groups with all cohomology groups in odd degrees vanishing, which of course coincides with the SM case when m " 1. Again, these groups are isomorphic to the corresponding groups in homology, with which we can deduce the entries E 2 p,q of the AHSS, which are shown in Fig. 8.
We task ourselves here with the computation of Ω Spin 5 pBG m q, which measures the potential global anomalies in the four-dimensional gauge theory we are interested in from the point of view of BSM. The relevant entries of the AHSS, lying on the p`q " 5 diagonal, are highlighted in Fig. 8. To turn to the third (and thence fourth) page, we thus need to compute the differentials here labelled α and β. This is again similar to the case of the SM considered above. The map β is the dual to the Steenrod square Sq 2 : H 2 pBG m ; Z{2q ÝÑ H 4 pBG m ; Z{2q so the kernel of β is spanned by r c 2 , r c 1 2 , and Č x i Y x j with i ă j. Hence ker β -pZ{2q  To calculate im α, where α " Ą Sq 2˝ρ , we first look at the corresponding Steenrod square Sq 2 : H 4 pBG m ; Z{2q ÝÑ H 6 pBG m ; Z{2q Thus the image of Ą Sq 2 , and also of α, is spanned by r c 2 and Ą x i x j , for i ă j. Thus im α -pZ{2q 1 2 mpm´1q`1 . Taking the quotient then yields On the E 4 page (see Fig. 8) the only relevant differential must be trivial as it is a homomorphism from Z{2 to Z, so the p4, 1q entry stabilises to E 8 4,1 -Z{2 and it follows that Ω Spin 5 pB pU p1q mˆS U p2qˆSU p3qqq -Z{2, where we can again identify the potential global anomaly in this theory with the Witten anomaly associated to the SU p2q factor. Thus we find that there are no potential new global anomalies associated with extending the usual SM gauge group by an arbitrary torus, and indeed by arbitrary fermion content coupled to such a gauge group. There have been a number of recent studies [41][42][43] attempting to classify the space of U p1q extensions of the SM that are free of local anomalies; here, we show that all such models are automatically free also of global anomalies, provided of course that there is no Witten anomaly associated with SU p2q. It is also straightforward to calculate the lower-degree bordism groups for this example, which we simply tabulate in the first line of Table 4. We find that the additional U p1q factors do indeed affect the bordism groups in lower degrees, in particular in degrees two and four. Table 4: Summary of results from our bordism computations of relevance to BSM physics. The first row corresponds to theories with multiple Z 1 bosons, the second row to a Pati-Salam model, and the last two rows to trinification models.

Pati-Salam models
Here we consider the simplest incarnation (for our purposes) of the Pati-Salam model, in which the SM gauge group is embedded in the larger group The cohomology ring for BpPSq is where c L{R 2 denote the second Chern classes of the SU p2q L{R factors, and c 1 i denotes the ith Chern class of SU p4q. A notable difference between this example and all those considered previously is that the second homology group is here vanishing. This only serves to simplify the computation of the AHSS, and so we choose to omit the details for brevity. The upshot is that we find Ω Spin 5 pBpPSqq -Z{2ˆZ{2. (6.10) We identify the two Z{2-valued global anomalies with the Witten anomalies associated with each SU p2q factor in the Pati-Salam group, a result that follows straightforwardly from Witten's original arguments. We quote the remaining results of our calculations for all bordism groups Ω Spin dď5 pBpPSqq in Table 4. We note in passing that there are variants on the Pati-Salam gauge group that involve various discrete factors, which complicate the computation of the bordism groups. For example, left-right symmetric models have been proposed in which G " PS¸Z{2, and there are also models featuring a quotient by a Z{2 subgroup. Unfortunately, neither of the bordism computations for these gauge groups succumb to attack using the simple fibrations considered in this paper.

Trinification models
In trinification models of grand unification [44], the underlying gauge group is either where the Z{3 quotient is the diagonal subgroup of the pZ{3q 3 centre symmetry. In both cases, the SM quarks are packaged into representations p3, 1, 3q and p3,3, 1q, with the leptons transforming in the p1, 3,3q. The model also contains multiple Higgs fields transforming in the p1, 3,3q representation (each of which contains three SM-like Higgs doublets), needed to break the gauge symmetry down to a SM subgroup; the first option in (6.11) is broken down to G SM {Γ 2 , while the second is broken to G SM {Γ 6 . Like Pati-Salam models, trinification models are attractive in part because all the gauge, Yukawa, and quartic couplings in the lagrangian can be run to arbitrarily high energies without hitting any Landau poles, thereby exhibiting 'total asymptotic freedom' [45].

No quotient
To find out whether there are potential global anomalies when the gauge group is SU p3q 3 , we compute Ω Spin d pBSU p3q 3 q. Since the method is very similar to that used in previous Sections, we will only quote the results here to avoid repetition. We find Since Ω Spin 5 pBSU p3q 3 q " 0, the trinification models based on this gauge group are free of any global anomalies, regardless of the fermion content. Now let us consider the option involving a permutation symmetry among the three SU p3q factors, i.e. where G " SU p3q 3 {pZ{3q. We have the fibration Z{3 Ñ SU p3q 3 Ñ G, which we can use the Puppe sequence to turn into the following fibration BSU p3q 3 ÝÑ BG ÝÑ B 2 pZ{3q -KpZ{3, 2q. (6.12) Using this fibration, we can now form the AHSS to find Ω Spin 5 pBGq. The second page, as we have seen so many times, is given by , Ω Spin q pBSU p3q 3 qw hich can be constructed using the results for Ω Spin pt pBSU p3q 3 qq, which were already calculated in this Subsection. It is displayed in Fig. 9. One can see immediately that all entries with p`q " 5 stabilise already at this page. We can again conclude that The other entries with p`q ă 5 also stabilise on this page because all relevant homomorphisms are trivial. The spin bordism groups of lower degrees can be calculated uniquely apart from Ω Spin 4 which involves non-splitting group extensions. It is given by The full results are given in Table 4. A grand unification model based on the gauge group SOp18q in five dimensions has been recently proposed by Reig et al. in Ref. [14] in an attempt to unify all gauge interactions as well as all fermion generations. A new mechanism was suggested for the symmetry breaking pattern SOp18q Ñ SOp10qˆSOp8q (which produces another grand unification theory, which is then ultimately broken to the SM), through an orbifold construction.

SOp18q family unification
To check whether this model might suffer from a global anomaly, we have to compute the sixth spin bordism group with the SOp18q-bundle structure preserved. This is in some ways a more challenging task than those we have considered so far, and could potentially give us an interesting and non-trivial anomaly cancellation condition due to the fact that the cell-structure of the bundle becomes more intricate as we go higher in dimensions. (One might say that in low dimensions 'accidents' are more likely to happen; it is largely due to the vanishing of Ω Spin 3 pptq and Ω Spin 5 pptq that Ω Spin 5 pBGq turns out to be especially simple for many groups -and we haven't yet found a G for which Ω Spin 3 pBGq doesn't vanish.) We note that the bordism groups Ω Spin d pBSOpnqq for n ě 8 with d ď 5 were computed in Ref. [10], as was Ω Spin 6 pBSOp3qq. Importantly for this discussion, it was there found that Ω Spin 5 pBSOp18qq " 0. (6.14) This means that every five-manifold in an SOp18q gauge theory is null-bordant, and thus that the Dai-Freed prescription can always be used to write down the candidate fermionic partition function. Here, we are interested in global anomalies in such a five dimensional SOp18q gauge theory, which are thus captured by Ω Spin 6 pBSOp18qq, which has not been computed before.
To compute Ω Spin 6 pBSOp18qq, we need input in the form of homology and cohomology groups of BSOp18q. Using results on the integral cohomology ring of BSOpnq from Ref. [46] as well as the universal coefficient theorem, together with the fact that the mod 2 cohomology ring of BSOpnq is generated by the Stiefel-Whitney classes w i viz.
H ‚ pBSOp18q; Z{2q " Z{2 rw 2 , w 3 , . . . , w 18 s , (6.15) we can tabulate all the relevant homology and cohomology groups, as shown in Table 5. The E 2 page for the Atiyah-Hirzebruch spectral sequence is given in Fig. 10. Since we use the fibration of the form pt Ñ BG Ñ BG, the differentials δ, η, θ are the duals of the Steenrod squares. The action of the Steenrod squares on the Stiefel-Whitney classes was worked out a long time ago by Wu [47], and is given by Using this, together with Cartan's formula (3.16), we can work out what the differentials δ, η, and θ are.
The dual map to δ is the Steenrod square Sq 2 from H 3 pBSOp18q; Z{2q -Z{2, generated by w 3 , to H 5 pBSOp18q; Z{2q -Z{2ˆZ{2 , generated by w 2 w 3 and w 5 . Its action on w 3 can be read off directly from (6.16) to be Sq 2 pw 3 q " w 2 w 3 . (6.17) Therefore the differential δ maps Č w 2 w 3 and Ă w 5 to Ă w 3 and 0, respectively. Therefore, we obtain The dual to η is given by where H 4 pBSOp18q; Z{2q -Z{2ˆZ{2 is generated by w 2 2 and w 4 , while H 6 pBSOp18q; Z{2q -pZ{2q 4 is generated by w 2 3 , w 3 2 , w 2 w 4 , and w 6 . Combining Wu's formula and Cartan's formula, we find Sq 2 pw 4 q " w 2 w 4`w6 , and Sq 2 pw 2 2 q " w 2 3 , (6.21) whence one can show that the action of η is given by Therefore, im η -Z{2ˆZ{2, and we get Similarly, one can easily show that the action of θ on the generators of H 7 pBSOp18q; Z{2q -pZ{2q 3 is given by We see that the image of θ is generated by one generator Ă w 5 , so im θ -Z{2 and The differential γ can be written as γ " Ą Sq 2˝ρ where ρ is the reduction mod 2 in homology. This Ą Sq 2 has the same action as η given by (6.22). Moreover, it was shown in [10] that the image of ρ : H 6 pBSOp18q; Zq Ñ H 6 pBSOp18q; Z{2q is generated by Ć w 2 w 4 , Ă w 6 , and Ă w 3 2 . Therefore, only one generator of H 6 pBSOp18q; Zq maps to 0 P H 4 pBSOp18q; Z{2q under γ, resulting in ker γ -Z{2. (6.26) We now have all the relevant entries on the E 3 page of the spectral sequence, as shown in Fig. 11. We can clearly see that all entries E p,q with p`q " 6 have now stabilised apart from E 3 2,4 , since there is an unknown differential from E 3 5,2 into it. Unfortunately, we have no means of evaluating this differential. All we can say is that E 2,4 stabilises to either 0 or Z{2. Moreover, the extension that we need to solve to get the spin bordism group does not split, resulting in an ambiguous result: Even though we cannot compute the group unambiguously, we can still infer some interesting consequences for anomaly cancellation in a five-dimensional SOp18q theory. Most obviously, Ω Spin 6 pBSOp18qq is certainly not zero, so there are non-trivial constraints on the fermion spectrum if the theory is to be free of Dai-Freed anomalies. While we cannot compute definitively what these constraints are, we can get a feel for what they might be by first noting that Eq. (6.27) implies that Ω Spin 6 pBSOp18qq is an abelian group of order 2, 4, or 8. Thus, given global anomalies are captured by the exponentiated η-invariant which is a homomorphism from Ω Spin 6 pBSOp18qq to U p1q, in the 'most anomalous scenario' there may be a chiral fermion that contributes a factor e 2πi{8 to the anomaly. Thus, one is guaranteed to obtain an anomaly-free theory if there are eight such fermions. Recalling that Ω Spin 5 pBSOp18qq " 0, which means that the Dai-Freed prescription for the partition function can be used for arbitrary five-dimensional spacetimes, Eq. (6.27) then tells us that the particular unified theory presented in Ref. [14] must be carefully checked for global anomalies before being taken seriously.
Note added: since this article appeared in preprint form, it was found in Ref. [15], by using the Adams spectral sequence, that the solution to the extension problem (6.27) is Ω Spin 6 pBSOp18qq -Z{2ˆZ{2, (6.28) corresponding to the trivial extension for the first line in (6.27). This group is order-2, so in fact the global anomalies in such a five-dimensional GUT will necessarily cancel if there are pairs of fermions in any given representation. For the SOp18q GUT that was proposed in Ref. [14], there is a single fermion transforming in the 256-dimensional representation, so there is potentially a global anomaly.

(B)SM theories with spin c structures
Part of the motivation for the bordism-based criterion for anomaly cancellation that we have used in this paper is the desire to define the SM (or our favourite BSM extension) on arbitrary four-manifolds, or at least within some suitable class of four-manifolds. Such a requirement can be motivated by locality, and is certainly a requirement in a quantum theory of gravity in which the geometry (and thus topology) of spacetime cannot be held fixed.
In order to define fermions, one needs to equip spacetime with a spin structure, or a variant thereof with which to stitch together locally-valued spinor fields into globallydefined ones. It is well known that not all orientable four-manifolds admit a spin structure (with CP 2 being a well-known example of an orientable four-manifold that is not spin). The obstruction to being spin is measured by the second Stiefel-Whitney class which takes values in H 2 pΣ, Z{2q. While H 2 pΣ, Z{2q " 0 for all orientable manifolds in dimension three or fewer, it does not vanish for all four manifolds. One might therefore ask whether the SM and related theories we have explored in this paper can be defined on all orientable four-manifolds, by not assuming the presence of a spin structure. We invite the reader to consult Appendix A, in which we provide more details regarding the definitions of spin structures and the like.
As we noted in §2, in the presence of a U p1q gauge symmetry it becomes possible to define spinors using only a spin c structure on spacetime. The transition functions on a spin c bundle over an oriented four-manifold Σ are valued in the group Spin c p4q, which can be defined by the short exact sequence where U p1q A denotes a gauged symmetry. Since all orientable four-manifolds admit a spin c structure (the obstruction here being in the third Stiefel-Whitney class), one can in principal try to define a four-dimensional gauge theory on all orientable four manifolds by using a spin c structure. These observations were first made back in 1977 [48], motivated by the authors' desire to define a theory of quantum gravity on all orientable spacetimes.
In order to define all fermions using a spin c structure, for a particular non-abelian gauge theory (such as one of the SMs), requires there exists a U p1q subgroup of the gauge symmetry, here denoted by U p1q A , such that all fermions in the theory transform in bona fide representations of the group (7.1). Using similar arguments to those given in §4.1, this results in constraints on the allowed U p1q A charges of fermions, which here depend on their spin. We begin our discussion by recapping what these 'spin-charge relations' are, which was recently discussed (in the context of defining similar theories on spin c manifolds) in Ref. [35].

Spin-charge relations
To derive the spin-charge relations, we require that the SM fermions transform in bona fide representations of both Spin c p4q and G, where G is one of the four SM gauge groups listed in Eq. (1.1). It is here helpful to write A Weyl fermion transforms in the p 1 2 , 0q or p0, 1 2 q representation of the SU p2q LˆS U p2q R factor. So, when considering Weyl fermions we may restrict our attention to a subgroup of Thus, by the same argument we used in §4.1, one deduces that there exists a normalisation of charges such that all Weyl fermion have odd charges under U p1q A , in order to define the theory using this spin c structure.
The question then is, is there any U p1q A subgroup of G in which all the SM fermions have odd charges? It turns out the answer is no. To see why, consider U p1q A to be generated by where Y is the generator of hypercharge, s the Cartan generator of (electroweak) SU p2q, are the Cartan generators of SU p3q (in a non-standard normalisation which is convenient for our purposes). Eq. (7.4) defines a general U p1q A subgroup of G.
We then need to decompose all the SM fermion fields into eigenstates of (7.4). To wit, consider the left-handed doublet of quarks, Q. This needs both an SU p2q index (which we denote by an upper Greek index α P t1, 2u) and an SU p3q index (which we denote by a lower Latin index i P t1, 2, 3u). In this notation, Q αi denotes 2ˆ3 " 6 Weyl fermions. We thus denote the SM fermion content by the fields tQ α i , L α , U i , D i , Eu, which number fifteen in total.
The charges of all the SM fields under the generator (7.4) are then There are no rational values for a, b, c, and d such that all the charges in this table are odd numbers. To see why, note firstly that the oddness of the charge of e requires that a " p2n`1q{2. But then there is no value of d such that both d 3 and u 3 have odd charge.
We hereby see the restrictiveness of the spin-charge relations: there is in fact no U p1q gauge symmetry in the SM which one can use to define the theory using a spin c structure. This fact was pointed out in Ref. [10]. Hence, given only the gauge symmetries and the fermion content of the SM, one cannot define it on all four-manifolds using a spin c structure. 24

Gauging B´L
One can instead define a theory on all orientable four-manifolds in which the SM gauge group is extended by an additional U p1q gauge symmetry for which the spin-charge relations are satisfied, such as gauging B´L, 25 where B is baryon number and L is lepton number.
Under U p1q B´L all the SM fermions have odd charges (either´1 or 3), and so this gauge symmetry can be used to define a spin c structure [10].
Of course, B´L is free of local ABJ-type anomalies. Here we consider global anomalies in SMˆU p1q theories defined on all spin c manifolds, such as gauged B´L, by computing the bordism groups Ω Spin c 5 pBGq, for the SM gauge groups listed in Eq. (1.1). These bordism groups can be computed using the AHSS associated to a fibration of the form F Ñ BG Ñ B. For example, given the 'trivial' fibration pt Ñ BG Ñ BG, the second page of the AHSS is now E 2 p,q " H p pB; Ω Spin c q pF qq, (7.6) where the bordism groups of spin c q-manifolds equipped with maps to a point are [49] q 0 1 2 3 4 5 6 7 8 9 10 Interestingly, these groups do not feature any torsion, and moreover they vanish in all odd degrees, at least up to Ω Spin c 9 pptq. It then follows immediately that for all odd d ď 9, (7.8) because non-zero entries in E 2 p,q can only appear when p`q is even (since H p pBG, Zq also vanishes in all odd degrees for these gauge groups). In particular, these groups vanish in degree d " 5, so there are no possibilities of global anomalies in any of these theories.
The case where G " G SM {Γ 6 is only slightly less straightforward. We may as before proceed via the Puppe sequence to deduce the fibration BpU p2qˆSU p3qq Ñ BpG SM {Γ 6 q Ñ KpZ{3, 2q, (7.9) and write down the corresponding AHSS, from which one immediately sees that Ω Spin c 5 pBG SM {Γ 6 q " 0, (7.10) so again such a theory is automatically free of global anomalies. These conclusions hold when the SM fermion content is extended arbitrarily.
symmetry G SMˆU p1q. Rather, the vector field A µ defines a spin c connection on Σ.
to define an analogue of the spinor, 27 and hence to define fermions. The idea here is very similar to defining spinors in the case that Σ p was orientable, except that now the transition functions of the tangent bundle are valued in Oppq, rather than SOppq, because they need not preserve orientation. Consequently, the structure group of the 'pinor' bundle is a double cover of Oppq, which is called a Pinppq group. But now there is not just one such double cover of Oppq, but two possible choices called Pin`and Pin´, as follows. One may choose a spatial reflection R to satisfy R 2 " 1 when acting on spinors, which defines the double cover Pin`, or choose R 2 "´1, which defines the double cover Pin´. A pin structure is then defined in a similar way to a spin structure; the Oppq-valued transition functions of the tangent bundle are lifted to (say) Pin`-valued functions, which must satisfy a consistency relation on triple overlaps. A non-orientable manifold that admits a (say) pin`structure is, not surprisingly, called a pin`manifold. Again, there are topological obstructions (involving Stiefel-Whitney classes) to defining such pin structures, which are different for pin`and pin´structures. Notably, every non-orientable 2-manifold and 3-manifold admits a pinś tructure, but not necessarily a pin`structure. 28 In both the orientable and non-orientable cases, one may in fact still define fermions using weaker structures on Σ p , provided there are additional gauge symmetries acting on the fermions. For example, a manifold that is not spin may nonetheless admit a spin c structure, which is defined analogously to a spin structure, but where the transition functions can be valued in the Spin c ppq group rather than Spinppq. The group Spin c ppq can be defined by the short exact sequence 0 Ñ U p1q Ñ Spin c ppq Ñ SOppq Ñ 0; in an intuitive sense, this "allows" the transition functions to vary by a (local) U p1q-valued phase, which can be used to "stitch together" transition functions where a spin structure might not be possible. If a fermion is acted upon by a U p1q gauge symmetry, then it is invariant under such local U p1q rephasings, and so will be well-defined using only the spin c structure. The obstruction to a manifold admitting a spin c structure now lies in its third Stiefel-Whitney class valued in Z (rather than Z{2). Importantly, all orientable manifolds in dimension p ď 4 are spin c . 29 Analogously defined pin c structures may be used to define fermions on non-orientable spacetimes with a U p1q gauge symmetry. 27 In the unorientable case, the fermion might better be called a 'pinor'. 28 For example, the manifold RP 2 admits only pin´structures. 29 Even 'weaker' structures have been used to define fermions on general spacetimes in the quantum gravity literature, using the idea of spin-G structures for various Lie groups G [50,51]. The use of spin-SU p2q structures, for an SU p2q gauge theory, has recently been used to derive a new kind of global anomaly [19].
B Computation of H 6 pKpZ{3, 2q, Zq In Ref. [10], a theorem from Ref. [52] was used to show that the homology groups H i pKpZ{3, 2q; Zq are given by where C is an abelian group of exponent less than or equal to 6, i.e., the degree of any element in C does not exceed 6. This means that, a priori, it has the form with h i ě 0. We will use the Serre spectral sequence to show that C must be of the form Recall that for a fibration F Ñ X Ñ B, the pp, qq entry on the second page of the Serre spectral sequence is given by [53] E 2 p,q " H p pB; H q pF qq (B. 3) The spectral sequence converges to H ‚ pXq, that is, the homology groups of X is determined from the last page of the spectral sequence by 30 H n pXq " where ‹ is a contractible space. The second page of the Serre spectral sequence is given in figure 12.
Since H i p‹q " 0 for i ą 0, any entry in the Serre spectral sequence apart from E 0,0 must stabilise to 0. In particular, the entry E 6,0 must stabilise to 0. Since the differential δ acts trivially on Z{2, Z{4, and Z{5, these factors would be present in E 8 6,0 unless h 2 " h 4 " h 5 " 0. We can also see that h 6 " 0 by a similar argument. Suppose that h 6 ‰ 0. Let δ 6 be a homomorphism from Z{6 to Z{3. There are three choices depending on where it sends the element 1. The first choice is δ 6 p1q " 0, which is the trivial homomorphism, in which case the kernel is Z{6. The second choice and third choice are sending 1 to 1 or 2, both of which result in the same kernel: ker δ 6 -Z{2. In subsequent pages, the homomorphisms from the p6, 0q entry go into either 0 or Z{3, and can never result in a trivial kernel. Therefore, E 8 6,0 ‰ 0, which is a contradiction. Hence h 6 " 0. This is enough for our purpose: we have determined that In this Appendix we give the details of the computation of the spin bordism groups of the SM quotient by Z{3. We present two methods, associated with two different fibrations.

Method 1
Firstly, we use the AHSS associated to the fibration pt Ñ U p3qˆSU p2q Ñ U p3qˆSU p2q, (C.1) for which the second page of the AHSS is given by E 2 p,q " H p pBpU p3qˆSU p2qq; Ω Spin where c i , c 1 i are the ith Chern classes for BU p3q and BSU p2q, respectively. From this, together with the Künneth formula in cohomology, we find that H 2 pBpG SM {Γ 3 qq is generated by c 1 , H 4 pBpG SM {Γ 3 qq by c 2 1 , c 2 , c 1 2 , and H 6 pBpG SM {Γ 3 qq by c 3 1 , c 1 c 1 2 , c 1 c 2 , c 3 , and again the absence of torsion means these cohomology groups are isomorphic to the corresponding groups in homology.
We again form the AHSS associated to the trivial fibration over a point. The entries on the second page of the AHSS are identical to those of the previous two cases, albeit with different action of the differentials, so we choose not to reproduce the diagram for a third time. Again, the difference to the previous cases shall enter in the action of the differential labelled γ.
We turn to the action of γ. The relevant Steenrod square is here  Since the discrete Z{3 quotient is here embedded 'orthogonally' to the SU p2q factor in G, we feel safe in suggesting that this Z{2 captures the Witten anomaly coming from the SU p2q factor. As for the previous example, the lower-degree bordism groups are unchanged (see Table 1).

Method 2
We provide here an alternative proof that Ω Spin 5 pBpG SM {Γ 3 qq " Z{2 using an alternative fibration, After we apply the Puppe sequence, this fibration turns into Using the results for the homology groups of KpZ{3, 2q up to degree 6 given in Appendix B, we can work out the E 2 page of the Atiyah-Hirzebruch spectral sequence, given in Figure  13. 31 Moreover, we can deduce that the differential d in the E 6 page must be trivial, since it is a homomorphism from a product of Z{m factors with m odd to Z{2. All the entries E p,q with p`q " 5 now stabilise, and we can read off the spin bordism group as

D Decomposing U pnq irreducible representations
The purpose of this Appendix is to decompose an irreducible representation of U pnq -SU pnqˆU p1q Z{n in terms of the U p1q charge and SU pnq irreducible representation using character theory, from which we extract the charge constraints presented in §4.1.
Let G be a group and V a d-dimensional representation of G. An element g P G is represented by a dˆd matrix R V pgq. The character of g in the representation V , denoted by χ V pgq, is defined by d d Figure 13: The E 2 and E 6 pages of the Atiyah-Hirzebruch spectral sequence for G " G SM {Γ 3 from the fibration (C.7).
(We use the normalised character where we have χ V peq " 1 for all finite irreducible representation V .) From this definition, it is easy to see that the character of g is a class function, that is, it only depends on the conjugacy class of g χ V pgq " χ V phgh´1q, for any h P G (D.2) We now specialise to the case G " U pnq. Since any unitary matrix can be diagonalised by a unitary matrix, any element g P U pnq is conjugate to a diagonal matrix of the forms g " diag pz 1 , z 2 , . . . , z n q , |z i | " 1.
Therefore, a U pnq character can be thought of as a function χ U pnq V : T n Ñ C, where T n is the maximal torus of U pnq.
A U pnq irreducible representation labelled by λ " pmq n`µ can be written uniquely in terms of the SU pnq irreducible representation V pλq and the U p1q charge qpλq as follows.
(D. 9) To see this, we first write g P U pnq in terms of a U p1q element e iθ and an elementg P SU pnq as g " e iθg . Then the coordinates z of T n is given in terms of θ and the coordinates y of T n´1 by z 1 " e iθ z 1 , z 2 " e iθ y 2 y´1 1 , . . . , z n´1 " e iθ y n´1 y´1 n´2 , z n " e iθ y´1 n´1 . (D. 10) In the representation pq, V q, g is represented by e iqθ R V pgq. This can be phrased in terms of characters as χ U pnq V pz 1 , . . . , z n q " e iqθ χ SU pnq V py 1 , . . . , y n´1 q , (D.11) By direct substitution of (D.10) into (D.6), it is easy to show that s λ pzq " e ipnm`|µ|qθ s µ`y1 , y 2 y´1 1 , . . . , y´1 n´1˘, (D. 12) whence our claim that pV, qq " pµ, nm`|µ|q follows.
Therefore, for an irreducible representation pµ, qq of SU pnqˆU p1q to be a bona fide irreducible representation of U pnq, we need q to be equal to the number of boxes in µ modulo n.
This result can be applied to a more complicated scenario. As an example, we consider the group G " G SM {Γ 6 which can be realised as G " pU p3qˆU p2qq {U p1q, where we identify the overall U p1q factor in U p3q with the one in U p2q. Our result (D.9) tells us that, for a representation pν, µ, qq of SU p3qˆSU p2qˆU p1q to be a bona fide representation of G, we must have q " |µ| mod 2, and q " |ν| mod 3. (D.13)