Anomaly interplay in $U(2)$ gauge theories

We discuss anomaly cancellation in $U(2)$ gauge theories in four dimensions. For a $U(2)$ gauge theory defined with a spin structure, the vanishing of the bordism group $\Omega_5^{\text{Spin}}(BU(2))$ implies that there can be no global anomalies, in contrast to the related case of an $SU(2)$ gauge theory. We show explicitly that the familiar $SU(2)$ global anomaly is replaced by a local anomaly when $SU(2)$ is embedded in $U(2)$. There must be an even number of fermions with isospin $2r+1/2$, for $r\in \mathbb{Z}_{\geq 0}$, for this local anomaly to cancel. The case of a $U(2)$ theory defined without a choice of spin structure but rather using a spin-$U(2)$ structure, which is possible when all fermions (bosons) have half-integer (integer) isospin and odd (even) $U(1)$ charge, is more subtle. We find that the recently-discovered `new $SU(2)$ global anomaly' is also equivalent, though only at the level of the partition function, to a perturbative anomaly in the $U(2)$ theory, which is this time a combination of a mixed gauge anomaly with a gauge-gravity anomaly. This perturbative anomaly can be cancelled only if there is an even number of fermions with isospin $4r+3/2$, for $r\in \mathbb{Z}_{\geq 0}$, recovering the condition for cancelling the new $SU(2)$ anomaly. These statements highlight an interplay between perturbative and global anomalies in closely related theories.


Introduction
An SU (2) chiral gauge theory in four dimensions suffers from a non-perturbative global anomaly when there is an odd number of fermion multiplets in isospin 2r +1/2 representations, for r ∈ Z ≥0 [1]. Such a theory is anomalous because the (Euclidean) partition function changes sign under an SU (2) gauge transformation that corresponds to the non-trivial element in π 4 (SU (2)) = Z/2. Equivalently, the anomaly can be seen from a constant gauge transformation by the central element −1 ∈ SU (2), in the background of a single instanton, as we review in §2.
One might be forgiven for guessing that a U (2) chiral gauge theory suffers from a similar global anomaly, given that π 4 (U (2)) = Z/2 also, and given that U (2) is locally equivalent to SU (2) × U (1) which has a global anomaly associated with the SU (2) factor. It turns out that this is not the case. A quick way of reaching this conclusion is to recall that global anomalies are detected by the exponentiated ηinvariant [2,3], 1 which becomes a bordism invariant when perturbative anomalies vanish. Because the spin-bordism group Ω Spin 5 (BU (2)) = 0 (1.1) (which can be straightforwardly adapted from calculations in [5,6]), the exponentiated η-invariant for a 4d U (2) gauge theory in which perturbative anomalies cancel must be trivial, so there can be no global anomalies. In contrast Ω Spin 5 (BSU (2)) = Z/2, which allows for a possible global anomaly in the SU (2) theory. In this paper, our first goal is to explain why there is no global anomaly in a U (2) gauge theory, defined with a choice of spin structure. This is the subject of §3. The argument is simple enough to summarise in this Introduction. Recall firstly that U (2) may be written as where the Z/2 quotient is generated by the central element (−1, e iπ ) ∈ SU (2)×U (1). As for the SU (2) case, one could make a constant gauge transformation by the element (−1, 1) ∈ SU (2) × U (1) in the background of a single instanton, and might thus be tempted to reach the same conclusion that there can be a global anomaly. However, this gauge transformation is equivalently described by the element (1, e iπ ) ∈ SU (2) × U (1). Thus, the anomalous transformation is in fact a local U (1) transformation, and we can compute the variation of the fermionic partition function using the appropriate counterterms in the effective action. The noninvariance of the path integral measure (when there is an odd number of multiplets with isospin 2r + 1/2) arises because there is a mixed SU (2) 2 × U (1) perturbative anomaly. We show explicitly that the perturbative SU (2) 2 × U (1) anomaly can vanish only if there is an even number of multiplets with isospin 2r + 1/2, by reducing the anomaly cancellation condition modulo 2. Note that this is only true when the global structure of the gauge group is strictly U (2). The argument does not follow for the (locally isomorphic) gauge group SU (2) × U (1), even though the formula for the perturbative anomaly is the same, because not every representation of SU (2) × U (1) corresponds to a representation of U (2). Having realised that the apparently global SU (2) anomaly is manifest in U (2) rather as a local anomaly, we may conclude from (1.1) that there can be no other new global anomalies in a U (2) theory (defined with a spin structure).
Understanding the absence of global anomalies in a U (2) gauge theory, but nonetheless the necessity of the condition on isospin 2r + 1/2 multiplets, is of some phenomenological interest, because U (2) could be the gauge group for the electroweak theory [7]. For example, anomaly cancellation in such a theory provides constraints on the electroweak quantum numbers of field content in the context of going beyond the Standard Model.
We then turn to the more subtle case of a U (2) gauge theory defined without a spin or spin c structure, and perform a similar analysis relating to the 'new SU (2) (global) anomaly' that afflicts an SU (2) gauge theory that is similarly defined without a spin structure [8]. Recall that fields in such a theory are instead defined using a spin-SU (2) structure, which requires that all fermions (bosons) have half-integer (integer) isospin. The SU (2) theory is anomalous if there is an odd number of fermion multiplets with isospin 4r + 3/2, for r ∈ Z ≥0 . The partition function for such a theory, defined on certain manifolds that are not spin (in particular, on CP 2 ), changes sign under the combined action of a diffeomorphism ϕ and an SU (2) gauge transformation W . This is the new SU (2) anomaly, which we shall recap in §2.
The second goal of this paper is to understand what happens to the new SU (2) anomaly in the analogous situation in which the gauge group is enlarged from SU (2) to U (2). If the field content is such that all fermions (bosons) have half-integer (integer) isospins and odd (even) U (1) charges, then the U (2) gauge theory can be defined without a spin structure, using this time a spin-U (2) structure to parallel transport fields. Again, one might expect that a global anomaly should afflict such a theory, corresponding to the new SU (2) anomaly; and again, this turns out not to be the case, as we show in §4.
The new SU (2) anomaly enjoys a similar but subtly different fate to the old one. This time, because of the crucial role played by the diffeomorphism ϕ in deriving the new SU (2) anomaly, we find that the anomalous combination of ϕ and W cannot be replaced by a local U (2) gauge transformation, as was the case for the 'old' SU (2) anomaly. However, the anomalous combined action of ϕ and W has the same effect on the fermionic partition function as a local U (2) gauge transformation with determinant −1. This gives rise to a local anomaly, that is a combination of the perturbative SU (2) 2 × U (1) anomaly with the mixed gauge-gravity anomaly. By considering this particular combination of perturbative anomalies reduced modulo 4, we find that the U (2) gauge theory defined using a spin-U (2) structure can only be anomaly-free when there is an even number of fermion multiplets with isospin 4r + 3/2.
It is important to stress that, in the U (2) theory, this condition on isospin 4r+3/2 multiplets must be satisfied simply for perturbative anomalies to cancel; thus, unlike the new SU (2) anomaly, this condition persists even if we choose to restrict our attention to spin manifolds.
In §2 we review the pair of global anomalies in SU (2) gauge theory. In §3 we discuss the U (2) theory defined using a spin structure, before turning to the case without spin structure in §4. Finally, in §5 we interpret our results in terms of cobordism invariants. We thence explain why there are no other global anomalies in the U (2) theory defined using a spin-U (2) structure.

Review of the SU (2) global anomalies
The old anomaly We first review the global anomaly that occurs for an SU (2) gauge theory defined on a four-manifold M (which we take to be Euclidean) using a spin structure [1].
Consider a single fermion transforming in the isospin-j representation, coupled to a background SU (2) gauge field with curvature F . Let n + (n − ) denote the number of fermion modes with positive (negative) chirality (i.e. eigenvalue under γ 5 ). The Atiyah-Singer index theorem tells us that where p 1 (F ) ∈ Z is the first Pontryagin number (or instanton number), and is the Dynkin index defined via Tr(t a j t b j ) = 1 2 T (j)δ ab . Here {t a j } denotes a basis for the isospin-j representation of su (2). Because n + − n − is congruent to n + + n − ≡ N j modulo 2, the total number of fermion zero modes satisfies If N j is odd, then the partition function will change sign under the action of (−1) F , where F is the fermion number. But since (−1) F is equivalent to applying a gauge transformation by the central element −1 ∈ SU (2), this implies that SU (2) is anomalous in such a scenario.
Only fermions with isospin j = 2r + 1/2 can contribute to this anomaly, and only in backgrounds with odd instanton number, because it is only for these values of j that the Dynkin index (2.2) is odd. Thus, the anomaly vanishes if and only if the following holds Condition 1: There is an even number of fermions transforming in representations with isospin 2r + 1/2, for r ∈ Z ≥0 . (

The new anomaly
Suppose now that there is no spin structure available, and that fermions are instead defined using a weaker spin-SU (2) structure. 2 The transition functions for a spin-SU (2) bundle are valued in the group where the Z/2 quotient is generated by the central element −1 of SU (2) paired with the element (−1) F ∈ Spin(4). All fields must transform in representations of this group, which requires that all fermions have half-integer isospin, and all bosons have integer isospin. Such a theory can be defined on all orientable four-manifolds, including those that are not spin such as CP 2 . 3 In the simpler case that we discussed above, we saw how the usual SU (2) anomaly could be seen from the action of (−1) F on the path integral measure, since (−1) F is equivalent to an SU (2) gauge transformation by −1 ∈ SU (2). The new SU (2) anomaly is more subtle, and cannot be seen from a pure gauge transformation. Rather, the new SU (2) anomaly is the non-invariance of the path integral under a transformationφ which is a combined diffeomorphism ϕ of M (for certain non-spin manifolds M ) with an SU (2) gauge transformation W .
To see this anomaly one may take M to be CP 2 , and ϕ : z i → z * i to act by complex conjugation on the homogeneous complex coordinates {z i } of CP 2 . A spin-SU (2) connection A may be defined by embedding a spin c connection a in su(2), viz. A = σ 3 a, where σ 3 is the diagonal Pauli matrix. The spin c connection a obeys the following quantisation condition for any closed oriented 2-manifold S ⊂ M , where w 2 (T M ) is the second Stiefel-Whitney class, which is such that 2a defines a properly-normalised U (1) gauge field. In particular, choose a spin c connection a such that for some CP 1 ⊂ CP 2 . Such a spin c connection reverses sign under the diffeomorphism ϕ. The spin-SU (2) connection A, however, is invariant under the combined action of ϕ with any SU (2) gauge transformation W which also flips its sign, such as W = ( 0 −1 1 0 ). An anomaly in the transformationφ has to arise from the path integral over the fermion zero modes. On CP 2 the number of zero modes N j equals the index of the Dirac operator J j (they are not only congruent modulo 2 as before). 4 For a single fermion multiplet in the isospin-j representation coupled to the background spin-SU (2) connection A defined above, the Atiyah-Singer index theorem implies the index is [8] The zero modes come in pairs with eigenvalues +1 and −1 underφ. Hence, the fermionic partition function Z[A] transforms under the action ofφ by The index J j is even for all half-integer values of j, but is congruent to 2 mod 4 only when j = 4r + 3/2 for r ∈ Z ≥0 . For all other half-integer values of j, the index J j is divisible by 4. Hence, the partition function is invariant underφ, and the theory is non-anomalous, if and only if the following condition holds: 3 U (2) gauge theory with a spin structure We now turn to U (2) gauge theory. We begin with the simpler case of a U (2) gauge theory defined with a spin structure, for which the vanishing of the bordism group (1.1) implies there are no global anomalies. We will here give a physical explanation of this fact, previously noted in Refs. [5,6], which demonstrates the subtle interplay between local and global anomalies in U (2). The representation theory of U (2) plays a crucial role in the arguments used in this paper. Recall that an irreducible representation of U (2) ∼ = (SU (2) × U (1))/Z/2 is labelled an irreducible representation of SU (2), itself labelled by an isospin j, together with a U (1) charge q, subject to a restriction relating q and j. Namely, q and j must satisfy the following 'isospin-charge relation' 5 q ≡ 2j (mod 2), (3.1) in convenient units where both gauge couplings are set to one. Consider a theory with a single fermion with isospin j and charge q (satisfying (3.1)), coupled to a background U (2) gauge field with curvature F and defined on S 4 . Recall that the usual SU (2) anomaly occurs when the fermionic partition function changes sign under the gauge transformation by −1 ∈ SU (2). Embedding SU (2) ⊂ U (2), this global SU (2) transformation is equivalent to a U (1) gauge transformation by e iπ , which is a local gauge transformation.
The variation of the partition function Z[A] under a potentially anomalous U (1) gauge transformation can be computed using the appropriate counterterms in the effective action (see e.g. [12]). For a U (1) transformation by angle θ, we have that where the gravitational piece is proportional to the integral of Tr R∧R which vanishes for S 4 . Setting θ = π and the instanton number p 1 (F ) = 1, this reduces to We see that the path integral is invariant under this transformation if and only if qT (j) is even.
Recall that the Dynkin index T (j) is only odd for isospins j ∈ 2Z ≥0 + 1/2. The isospin-charge relation (3.1) means that q is also odd for these representations. Hence, there is necessarily an anomaly if there is an odd number of fermions in multiplets with isospin 2r + 1/2; in other words, precisely when condition (2.4) is violated. Thus, we find that the SU (2) global anomaly manifests itself rather as a perturbative anomaly when SU (2) is embedded in U (2). There are no global anomalies in the U (2) theory.
Indeed, one can directly derive that condition (2.4) must hold for a U (2) gauge theory by considering the equations for perturbative anomaly cancellation. Suppose that we have N j fermions transforming in isospin-j representations of U (2), with charges {q j,α }, where α = 1, . . . N j . We assume without loss of generality that all fermions have left-handed chirality. The SU (2) 2 × U (1) perturbative anomaly is proportional to The fact that T (j) is odd only for j ∈ 2Z ≥0 + 1/2, together with the isospin-charge relation, means that reducing mod 2 immediately yields j∈2Z+1/2 1 ≡ 0 (mod 2), (3.5) and hence that condition (2.4) must be satisfied to avoid a perturbative SU (2) 2 ×U (1) anomaly.

U (2) gauge theory without a spin structure
We now turn to the case where a spin structure is not available. Instead, we can use a spin-U (2) structure to parallel transport fields, provided that all fields transform in representations of the group The Z/2 quotient is generated by the product of the element (−1) F ∈ Spin(4) with the central element −1 ∈ U (2). Recalling also the effects of the Z/2 quotient within U (2), we have the following constraints on the allowed representations: where (q, j) label the U (2) representations as before.
In the analogous SU (2) theory, the new SU (2) anomaly is associated with a transformationφ that is a combined diffeomorphism ϕ plus gauge transformation W , as we reviewed in §2. Recall thatφ acts on the partition function as Let us first analyse the behaviour of the U (2) theory under this same transformation.
To that end, again take M to be CP 2 , and as in §2 defineφ to be the combination of the complex conjugation diffeomorphism ϕ : z i → z * i with the U (2) gauge transformation W = ( 0 −1 1 0 ). Moreover, we define a spin-U (2) connection A = σ 3 a, where a is the spin c connection satisfying Eqs. (2.6, 2.7), which is invariant underφ.
The diffeomorphism ϕ (on its own) is such that ϕ 2 = −1 when acting on fermions. More specifically, ϕ can be thought of as a certain spatial rotation through an angle π, corresponding (in certain coordinates) to the following transformation on a 2component Weyl fermion ψ a : where the index labels Lorentz SU (2) indices of the spin-1/2 fermion. Because the matrix appearing in (4.4) is not proportional to the identity, this diffeomorphism cannot therefore be subsumed by the U (1) phase degree of freedom in U (2). Thus, as in the SU (2) case, the transformationφ is necessarily not equivalent to a pure U (2) gauge transformation. Sinceφ is inequivalent to a local gauge transformation, in contrast to the situation in §3, we might suspect that this new SU (2) global anomaly will stick around in the U (2) theory. However, what we can do instead is construct a local U (2) gauge transformation whose action on the fermionic partition function Z[A] is identical to (4.3). Consequently, cancellation of perturbative anomalies shall guarantee that the suspected global anomaly in fact vanishes. To wit, consider a gauge transformation bỹ i.e. by a pure U (1) phase. Note that detW = 1 for θ / ∈ πZ, so that there is no corresponding gauge transformation in SU (2) by design. Let us now compute the transformation of Z[A] underW (θ), for a single fermion multiplet with isospin-j and charge q coupled to the spin-U (2) connection A. This time the gravitational contribution will be non-vanishing because CP 2 has non-zero signature. Taking into account the contributions from both the mixed gauge anomaly and the gauge-gravity anomaly, the shift in the partition function, for now on a general four-manifold M with metric g, is where in which the trace is only over the SU (2) gauge indices (we here choose to keep Lorentz indices explicit for clarity), and where Q is the generator of the U (1) factor in U (2), and the trace sums over all 2j + 1 components of the isospin-j representation. Recall thatF µν = 1 2 µνστ F στ and R µνστ = 1 2 µναβ R στ αβ , where R µνστ are the components of the Riemann tensor. We can relate both these integrals to characteristic classes of bundles over M , taking care with the various normalisation factors. Noting that τ a = σ a /2 are the generators of the SU (2) factor of U (2), the choice A = σ 3 a implies that F a µν = 2δ a3 f µν , where f = da is the curvature of the spin c connection a. We can thus reduce (4.7) to an integral over the spin c connection, The normalisation (2.7) of the spin c connection determines its first Pontryagin class in terms of the signature σ of M , viz. (4.10) Since σ = 1 for CP 2 we have that, when M = CP 2 , For the gravitational contribution, we use the fact that (4.12) and that Tr(Q) = (2j + 1)q to deduce that S gravity = + iθ 8 (2j + 1)q (4.13) The partition function therefore shifts by (4.14) Using the expression (2.2) for the Dynkin index, we find that the factor in square brackes is nothing but −iθJ j q, where J j is the same index from (2.8) that detected the new SU (2) anomaly. Therefore, setting θ = π/2 gives Recalling that all fermions in this theory have half-integral isospin j and odd charge q, and that T (j) ≡ 2 (mod 4) only when j ∈ 4Z + 3/2, we see that there is a perturbative U (2) anomaly when there is an odd number of fermion multiplets with isospin j ∈ 4Z ≥0 + 3/2; in other words, precisely when condition (2.10) is violated. Another way to see that the U (2) gauge transformation byW (π/2) has the same action on the path integral as the actionφ of the diffeomorphism ϕ plus SU (2) gauge transformation W is to consider the compositionφ(π/2) ≡φ ·W (π/2) of these two transformations. In other words, consider the combined action on Z[A] of the diffeomorphism ϕ plus a U (2) gauge transformation byW (π/2) · W = iW . The argument proceeds almost exactly as the argument for the new SU (2) anomaly, as summarised in §2; the only difference is that now the fermion zero modes transform in pairs underφ(π/2) with eigenvalues +i and −i (rather than +1 and −1) whose product is now +1 (rather than −1 as before). Thus, since there is an even number of zero modes, the action ofφ ·W (π/2) is always non-anomalous, and so each ofφ andW (π/2) must contribute the same mod 2 anomaly.
As we saw in §3 for the old SU (2) anomaly, we can again deduce the necessity of condition (2.10) directly from the equations for perturbative anomaly cancellation. This time, however, we also need to use the cancellation of the gauge-gravity anomaly, Reducing this equation modulo 4, and using the properties of J j noted above, we immediately obtain j=4r+3/2 1 ≡ 0 (mod 2), (4.18) recovering the condition (2.10) that, in the SU (2) case, is required to cancel the new SU (2) anomaly.
We have now seen how both conditions (2.4) and (2.10), for the cancellation of the old and new SU (2) anomalies, do not correspond to global anomalies when SU (2) is embedded as a subgroup of U (2). The arguments used for the two anomalies were, however, qualitatively different. In the case of the old SU (2) anomaly, for a theory defined using a spin structure, the global transformation in SU (2) corresponds to a local transformation in U (2), for which there is an associated perturbative anomaly if there are an odd number of multiplets with isospin j ∈ 2Z ≥0 + 1/2. For the new SU (2) anomaly, however, the mixed diffeomorphism plus gauge transformation is not equivalent to a local transformation in U (2). It nonetheless transpires to be equivalent to a local transformation in U (2) at the level of its action on the fermionic partition function. In this sense, the condition (2.10) emerges somewhat coincidentally in the U (2) theory, and should be thought of as 'trivialising' the new SU (2) global anomaly; for the old SU (2) anomaly, the correct interpretation is rather that there is no global anomaly.
As a result, the condition (2.10) enjoys a different 'status' in the SU (2) theory versus the U (2) theory. It is important to recall that the new SU (2) anomaly is no barrier to the consistency of an SU (2) gauge theory when formulated only on spin manifolds. In contrast, the constraint (4.18) on the U (2) theory is required by U (2) gauge invariance, and so its violation, like the violation of the original Witten anomaly, would render the U (2) theory inconsistent (even on spin manifolds).

Cobordism and the absence of U (2) global anomalies
Finally, we discuss the connection between our results and cobordism invariants in five dimensions. Such considerations will also enable us to conclude that there are no further anomalies in the U (2) gauge theories we have considered, defined either with or without a spin structure.

Case I: with a spin structure
For an SU (2) gauge theory defined on a four-manifold M equipped with spin structure, the original SU (2) anomaly is detected by the bordism group Ω Spin 5 (BSU (2)) = Z/2.
There is a corresponding cobordism invariant, namely the η-invariant, which reduces in this case to a 5d mod 2 index because the fermions are in real representations. Let I 1/2 denote this 5d mod 2 index for a single fermion with isospin-1/2. For anomalous fermion content, I 1/2 is non-vanishing on the mapping torus M × S 1 [1,8].
When SU (2) is embedded in U (2), a fermion with isospin-1/2 is necessarily in a non-trivial representation of U (1) by (3.1), and thus in a complex representation.
Hence, the η-invariant no longer reduces to a mod 2 index in this case. But this does not matter in the end, because one may calculate the bordism group directly to find that [5,6] Ω Spin 5 (BU (2)) = 0. (5.2) Hence, in the case that perturbative anomalies vanish and the η-invariant becomes a cobordism invariant, there are no cobordism invariants and thus the η-invariant must be trivial -even if we do not know how to calculate it directly. We therefore deduce that there are no global anomalies in this theory. This is consistent with our explicit calculation in §2, which realised the potentially anomalous global SU (2) gauge transformation to be equivalent to a local U (2) gauge transformation.

Case II: without a spin structure
Recall that for the SU (2) gauge theory defined without spin structure the corresponding bordism group is [13][14][15] Ω A possible basis is given by I 1/2 and I 3/2 , the 5d mod 2 indices associated with a single fermion with isospin-1/2 or 3/2 respectively [8]. The former corresponds to the old SU (2) anomaly, and the latter corresponds to the new one. Now consider the case of a U (2) gauge theory formulated without a spin structure, but rather using a spin-U (2) structure, as was the subject of §4. In Appendix A we calculate using the Adams spectral sequence that What is the interpretation of this 5d mod 2 cobordism invariant? And does it signify a possible new global anomaly that we have so far missed? Fermions in either the isospin-1/2 or 3/2 representations must have odd and thus non-vanishing charge under U (1). Thus, it is not clear how to relate the η-invariant for this theory to a mod 2 index such as I 1/2 or I 3/2 . Fortunately, we may follow Ref. [8] in identifying a mod 2 cobordism invariant dual to the generator of (5.4) to be where Y is a closed 5-manifold, and w 2,3 (T Y ) are Stiefel-Whitney classes. The crucial point is that J(Y ) is a mod 2 cobordism invariant of 5-manifolds with no further structure defined. Hence, J(Y ) is automatically a cobordism invariant of 5manifolds with spin-U (2) structure, albeit one that can only be detected on non-spin 5 manifolds. For example, and thus the Dold manifold (CP 2 × S 1 )/Z/2 6 is a suitable generator for the bordism group (5.4). Because J(Y ) vanishes trivially on spin manifolds, it does not appear in either (5.1) or (5.2). In Ref. [8], the cobordism invariant J(Y ) was identified, for any five-manifold with spin-SU (2) structure, with the mod 2 index I 3/2 , and thus with the new SU (2) anomaly, since the Dold manifold corresponds precisely to the action of the diffeomorphism plus gauge transformationφ on CP 2 . Since the action ofφ on the corresponding U (2) theory is equivalent, at the level of the partition function, to a local U (2) transformation as described in §4, the potential global anomaly corresponding to this cobordism invariant necessarily vanishes by perturbative anomaly cancellation. Since there are no other independent cobordism invariants, we conclude that there are no other possible global anomalies in the U (2) gauge theory defined using a spin-U (2) structure.

Acknowledgments
We thank Ben Gripaios and David Tong for discussions, and Pietro Benetti Genolini for reading the manuscript. JD is supported by the STFC consolidated grant ST/P000681/1. NL is supported by the DPST Scholarship from the Thai Government.

A Spin-U (2) bordism
In this Appendix we calculate the bordism group Ω Spin×U (2) Z/2 5 (pt), using the Adams spectral sequence. For a guide to using the Adams sequence to compute bordism groups, we recommend Ref. [16].
When there is no odd-torsion involved, the bordism group Ω G t−s (pt) can be evaluated via the Adams spectral sequence where A is the Steenrod algebra and M T G is the Madsen-Tillmann spectrum defined in terms of the Thom spectrum by M T G = Thom(BG, −V ), with V a stable bundle of virtual dimension 0 pulled back from the tautological stable bundle over BO by BG → BO. In our case, M T G can be written as with X G a Thom spectrum to be determined. For t−s < 8, this simplifies the Adams spectral sequence above to Ext s,t A 1 (H • (X G ), Z/2) ⇒ Ω G t−s (pt), (A.3) by the Anderson-Brown-Peterson theorem. Here A 1 denotes the subalgebra of A generated by the Steenrod operations Sq 1 and Sq 2 . To make the presentation clearer, we will write U n and SO n for U (n) and SO(n) in the rest of this Appendix.
Calculation of X G We will now show that the Thom spectrum X G when G = (Spin × U 2 )/Z/2 is given by X G = Σ −5 M SO 3 ∧ M U 1 . We follow the calculation of related examples in Refs. [6,15], whose method was based on Ref. [13]. The fibration Z/2 −→ G −→ SO × SO 3 × U 1 gives rise to the following fibration sequence of classifying spaces where w 2 ∈ H 2 (BSO), w 2 ∈ H 2 (BSO 3 ), and w 2 ∈ H 2 (BU 1 ) are the second Stiefel-Whitney classes for BSO, BSO 3 , and BU 1 , respectively. The fibration sequence (A.4) arises as a Puppe sequence, so the composite map is null-homotopic. Moreover, since these classes are valued modulo 2, this is equivalent to saying that the map w 2 • f is homotopy equivalent to w 2 • f + w 2 • f . Therefore, the following diagram which sends three bundles into a stable SO-bundle of virtual dimension 0. Using the Whitney product formula, the second Stiefel-Whitney class of the virtual bundle V + V 3 + V 2 − 5 is given by where we obtain the last equality using the pullback square (A.5). Therefore, the stable SO-bundle V + V 3 + V 2 − 5 can be lifted to a stable spin bundle, denoted by W , establishing the existence of a homotopy pullback (A.6). Therefore, the map −V : BG → BSO is homotopy equivalent to the map −W +V 3 +V 2 −5 from BSpin×BSO 3 ×BU 2 into BSO, giving rise to the identification of the Thom spectrum M T G = Thom(BG; −V ) with Finally, the rule for extracting the bordism groups can be roughly summarised as follows: an h 0 -tower containing m dots gives a factor of Z/2 m , and an infinite h 0 -tower gives a factor of Z. With this rule, the bordism groups of degree lower than six can be read off from the chart in Fig. 2 to be Ω G 0 = Z, Ω G 1 = 0, Ω G 2 = Z, Ω G 3 = 0, Ω G 4 = Z 3 , (A. 13) and, crucially for us, (A.14)