Global anomalies&bordism of non-supersymmetric strings

The three tachyon-free non-supersymmetric string theories in ten dimensions provide a handle on quantum gravity away from the supersymmetric lamppost. However, they have not been shown to be fully consistent; although local anomalies cancel due to versions of the Green-Schwarz mechanism, there could be global anomalies, not cancelled by the Green-Schwarz mechanism, that could become fatal pathologies. We compute the twisted string bordism groups that control these anomalies via the Adams spectral sequence, showing that they vanish completely in two out of three cases (Sugimoto and $SO(16)^2$) and showing a partial vanishing also in the third (Sagnotti 0'B model). We also compute lower-dimensional bordism groups of the non-supersymmetric string theories, which are of interest to the classification of branes in these theories via the Cobordism Conjecture. We propose a worldvolume content based on anomaly inflow for the $SO(16)^2$ NS5-brane, and discuss subtleties related to the torsion part of the Bianchi identity. As a byproduct of our techniques and analysis, we also reprove that the outer $\mathbb{Z}_2$ automorphism swapping the two $E_8$ factors in the supersymmetric heterotic string is also non-anomalous.


Introduction
It is widely known that there are five supersymmetric string theories in ten dimensions [1]. It is slightly less known that there are several other non-supersymmetric string theories in ten dimensions, many of which have tachyons; and there are just three known models in ten dimensions which are both non-supersymmetric in spacetime and tachyon free: the SO(16) × SO (16) string [2,3], the Sugimoto model [4], and the Sagnotti 0'B model [5,6]. These non-supersymmetric models and their compactifications have been the subject of a renovated interest in the recent literature (see e.g. ), presumably because they constitute a promising arena to study quantum gravity away from the supersymmetric lamppost.
In spite of this recent surge of work, we still know remarkably little about the three tachyon-free non-supersymmetric models in ten dimensions, particularly when compared with their supersymmetric counterparts. In particular, the spectrum of all three nonsupersymmetric models is chiral in ten dimensions, and so it is potentially anomalous. Local anomalies have long been known to cancel via non-supersymmetric versions of the Green-Schwarz mechanism [2][3][4][5][6]. However, to our knowledge, except for an inconclusive analysis in [36] these models have not been shown to be free of global anomalies, as was done early on in [37][38][39] for the supersymmetric chiral theories. 1 A global anomaly could lead to an inconsistency of the model, and having to discard it, or to new topological couplings that cancel it [40].
The purpose of this paper is twofold: On one hand, we compute the potential global anomaly of the three tachyon-free non-supersymmetric string theories in quantum gravity (albeit only a subclass of anomalies for the Sagnotti 0'B model), showing that it vanishes. We do this using cobordism theory, and computing the relevant twisted string bordism groups. This is a standard technique for proving anomaly cancellation results; see [40][41][42][43][44][45][46][47] for recent anomaly cancellation theorems in string and supergravity theories using this technique.
The second important result of our paper is precisely the calculation of these twisted string bordism groups (which have not appeared in the literature before), which are summarized in Table 1 in the conclusions. The physics use of these bordism groups is that they can be used to predict new, singular configurations (branes) of the corresponding non-supersymmetric string theories, by means of the Cobordism Conjecture [48] (see [49][50][51] for similar recent work in type II and supersymmetric heterotic string theories). Furthermore, the calculation of the bordism groups themselves by means of the Adams spectral sequence is interesting in its own right, and we expect that similar techniques can be used to compute string bordism groups of e.g. six-dimensional compactifications, and more generally, to study anomalies of any theory with a 2-group symmetry or a Green-Schwarz mechanism.
Along the way, we will encounter and comment on issues such as whether the heterotic Bianchi identity can be taken to take values on the free part of cohomology or the torsion piece must be included, or the connections between anomaly cancellation in eleven-dimensional backgrounds and anomaly inflow on non-supersymmetric NS5 branes on these theories. We also include a quick introduction to the Green-Schwarz mechanism in the modern formalism of anomaly theory, providing for the first time a candidate for the worldvolume degrees of freedom for the NS5 brane in the SO(16) × SO (16) string. We also study global anomalies in the Z 2 outer automorphism swapping the two factors of the SO(16) × SO (16) string, showing that anomalies vanish.
The upshot of our paper is: • The bordism group controlling anomalies of the Sugimoto string, Ω String-Sp (16) 11 , vanishes (theorem 3.48), and therefore the theory is anomaly-free 2 .
• The bordism group Ω String-SU (32)⟨c 3 ⟩ 11 , controlling the anomaly of the Sagnotti 0'B model, is isomorphic to 0 or Z 2 (theorem 3.63). We do not know whether the anomaly vanishes, although it does in all specific backgrounds we looked into.
• For the SO(16) × SO (16) heterotic string, (where the identity component of the global form of the gauge group is actually Spin(16) × Spin (16), since the massless spectrum contains both spinors and vectors 3 ), the bordism group Ω String-Spin(16) 2 11 controlling the anomaly vanishes (theorem 3.78), and therefore this theory is anomaly-free.
• There is also a Z 2 gauge symmetry swapping the two factors of Spin (16), whose anomaly we also studied. The bordism group controlling the anomaly has order 64 -but nevertheless (theorem 4.30), the anomaly vanishes.
As a consequence of our calculations, we also can cancel an anomaly in a supersymmetric string theory.
• When one takes into account the Z 2 symmetry of the E 8 × E 8 heterotic string swapping the two copies of E 8 , the anomaly vanishes (corollary 4.34).
The cancellation of this anomaly is not a new result: it is a special case of the more general work of [42]. Our argument rests on different physical assumptions and is a different mathematical result; for example, we do not assume the Stolz-Teichner conjecture. Thus we answer a question of [46], who showed the bordism group controlling this anomaly has order 64 but did not address the anomaly, and asked for a bordism-theoretic argument that the anomaly vanishes.
One key application of this Z 2 symmetry of the E 8 × E 8 heterotic string is constructing the CHL string [53], a nine-dimensional string theory obtained by compactifying the E 8 × E 8 heterotic string on a circle, where the monodromy around the circle is the Z 2 symmetry we discussed above. An anomaly in the Z 2 symmetry would have implied an inconsistency in the CHL string. We find that the anomaly vanishes, in agreement with the results in [42], which showed this from a worldsheet perspective; by contrast, we approach the question from a pure spacetime perspective.
One can make an analogous construction for the SO(16) × SO (16) heterotic string, compactifying it on a circle whose monodromy exchanges the two bundles. The result is a ninedimensional non-supersymmetric string theory whose gauge group is (perhaps a quotient of) Spin (16). This theory has only been studied very recently as far as we can tell, in [35,54]. Analogously to the CHL string, an anomaly in the Z 2 symmetry of the SO(16) × SO (16) heterotic string would lead to an inconsistency of this new theory, and our anomaly cancellation result implies a consistency check for this theory on backgrounds where the gauge group is Spin (16). It would be interesting to study this theory on more general backgrounds.
We have also identified a plethora of non-trivial bordism classes on these theories. It is a natural direction to explore the nature and physics of the bordism defects associated to these branes [49][50][51], a task we will not pursue in this paper. Furthermore, representatives of the bordism classes we encountered provide natural examples of interesting compactification manifolds for these non-supersymmetric strings to various dimensions; studying these, finding out whether moduli are stabilized (including SUSY-breaking stringy corrections to the potential) etc. is another important open direction to study. This paper is organized as follows: In Section 2 we provide a lightning review of modern methods to study anomalies and how these cancel via the Green-Schwarz mechanism, as well as a detailed description of how this happens for each of the three tachyon-free, nonsupersymmetric string theories. We believe this is the first time these important results are collected together in a single reference, and with a unified notation. Section 3 contains our main result -the calculation of bordism groups for these theories using the Adams spectral sequence -together with a discussion of the natural cohomology theory for the Bianchi identity to take values in. We also study in detail the relationship of higher-dimensional anomaly cancellation to the worldvolume theory of magnetic NS branes. Section 4 extends the anomaly calculation to the SO(16) × SO (16) string including the (gauged) automorphism swapping the two Z 2 factors. This multiplies the number of interesting bordism classes, but anomalies still cancel. Finally, Section 5 presents a table with our results, conclusions, and potential further directions, including a few comments on how these anomalies might be studied from a worldsheet point of view, in the line of [42,45].

Local anomalies and the Green-Schwarz mechanism
The word "anomaly" describes the breaking of a classical symmetry by quantum effects. In a Lagrangian theory, anomalies correspond to a lack of invariance of the path integral under a symmetry transformation. They can arise for both global and gauge symmetries in field theories. Anomalies in global symmetries only point to the fact that the symmetry cannot be gauged; they can lead to anomaly matching conditions that heavily constrain the RG-flow and strong coupling dynamics of the theory [55]. In contrast, anomalies in gauge theories point to true inconsistencies: a gauge symmetry is by definition a redundancy of the theory and as such can never be broken. In this paper, we will only consider anomalies in gauge symmetries.
The anomalies under consideration arise in field theories when they are coupled to gauge fields and dynamical gravity. They then correspond to a lack of invariance of the path integral under a gauge transformation/diffeomorphism (for a review, see [56]). When this transformation can be continuously connected to the identity, we speak of local anomalies. Their cancellation heavily constrains a theory; for example the gauge group of N = 1 supergravities in ten dimensions is constrained by anomaly cancellation to be either one of four gauge groups: (a quotient of) Spin(32), E 8 × E 8 , U (1) 496 or U (1) 248 × E 8 . The last two can be ruled out as low-energy EFTs of a consistent theory of quantum gravity by demanding the consistency of the worldvolume theory of brane probes in [57] (see also [11] for developments in the context of orientifold models) and using more general arguments in [58].
When the anomalous symmetry transformation cannot be continuously connected to the identity, then we speak of global anomalies (not to be confused with anomalies in global symmetries!). Global anomalies and their cancellation will be at the heart of this paper. It only makes sense to study them once local anomalies cancel. We therefore review local anomaly cancellation in the remainder of this section, before discussing global anomalies in the next ones.
The most direct way to study local anomalies is to compute certain one-loop Feynman diagrams involving external gauge bosons and/or gravitons, and chiral fermions in the internal legs; for ten-dimensional theories, the relevant diagram has 6 external legs [59].
There is, however, a much more concise way of studying such anomalies, through what is called an anomaly polynomial [56]. This is a certain formal polynomial in the gauge-invariant quantities trF m and trR m , which are certain contractions of Riemann and gauge field strength tensors that do not involve the metric. If the theory we wish to study lives in d spacetime dimensions, the anomaly polynomial is of degree (d+2). Although the anomaly polynomial is often discussed in the physics literature directly in terms of trF m and trR m , we find it more natural and convenient to write it down in terms of (the free part of) Chern and Pontryagin characteristic classes, more common in the mathematical literature. These can be written as linear combinations of trF m and trR m via Chern-Weil theory as follows. The i-th Chern class c r,i is associated to a complex vector bundle, in some representation r of the gauge group. Via Chern-Weil theory, they are represented in cohomology by the following characteristic polynomial of the field strength (here, t is just a dummy variable) 4 : 1) or, expanding the determinant, i c r,i t i = 1 + i tr r (F ) 2π t + tr r (F 2 ) − tr r (F ) 2 8π 2 t 2 + · · · (2. 2) The traces are over the gauge indices and as such the i-th Chern class c r,i is a 2i-form. The Pontryagin classes are characteristic classes associated to a real vector bundle, which we will always take to be the spacetime tangent bundle. One way to define them is in terms of the Chern classes of the complexification of the vector bundle. The total Pontryagin class is the sum of the Pontryagin classes and its first few terms are as follows: Similarly to the case of the Chern classes, the Pontryagin class p i is a 4i-form. The reason we prefer these characteristic classes over the trace notation trF m and trR m is that, as will be clear later, the anomaly polynomial is a sum of Atiyah-Singer indices, and these indices are written in terms of these classes in the mathematical literature. The precise relationship between local anomalies and the anomaly polynomial is as follows. The anomalous variation of the quantum effective action δ Λ Γ can be related to the (d + 2)-dimensional anomaly polynomial through what is called the Wess-Zumino descent procedure, which we briefly outline here. Since the characteristic classes are (locally) exact, the anomaly polynomial itself is also (locally) exact. We can therefore locally write it as P d+2 = dI d+1 (2.5) where I d+1 is called the (Lagrangian density of the) anomaly theory and locally satisfies the descent property δ Λ I d+1 = dI d . It is related to an anomalous variation of the effective action δ Λ Γ, which is local, by extending the spacetime manifold X d into a (d + 1)-dimensional manifold Y d+1 whose boundary is X d . One finds where the anomaly I d is only defined up to a closed form and α(Y d+1 ) is called the anomaly theory. The anomaly corresponding to a given gauge transformation is then computed as the integral of I d over the spacetime manifold X d . It follows that local anomalies vanish if and only if the anomaly polynomial vanishes. The anomaly polynomial can be written as a sum of contributions from all of the fields in the theory. Each contribution is given by an index density in (d + 2)-dimensions, whose integral over a compact manifold (with suitable structure, e.g. spin for fermions) gives the index of the corresponding Dirac operator via the Atiyah-Singer index theorem. We now list some of these contributions that will be relevant in what follows. The index density associated to a left-handed Weyl fermion in the representation r of a gauge group with field strength F is: where the term tr r e iF/2π is sometimes referred to as the Chern character. The notation [· · · ] d+2 means that one should select the (d + 2)-form part of the enclosed expression.Â(R) is called the A-roof polynomial, and it can be expanded as: Note that expression (2.7) can be easily applied to a fermion that is a singlet under the gauge group, in which case the Chern character reduces to 1. The contribution to the anomaly polynomial corresponding to a left-handed fermion singlet is therefore simply: The index density associated to a left-handed Weyl gravitino in (d + 2) dimensions is: Finally, a self-dual tensor gives a contribution: where L(R) is called the L-polynomial or Hirzebruch genus, and it can be expanded as follows: Armed with these index densities, we can now review anomalies in ten-dimensional N = 1 supergravities. With this supersymmetry, the only multiplets are the gravity and vector multiplets [59]. There are pure gravitational anomalies coming from the chiral gravitini in the gravity multiplet as well as Majorana-Weyl spinors in both gravity and vector multiplets. There can also be gauge anomalies involving the fermions in the vector multiplet, which transform in the adjoint representation of the gauge group (they are gaugini). The anomaly polynomial of these theories is computed using (2.7)-(2.11) summing over all chiral fields (dilatino, gravitino, and gaugini), and reduces to the following expressions for the Spin(32)/Z 2 and the E 8 × E 8 theories respectively: (2.14) Here, the Chern classes are expressed in the vector representations of Spin(32)/Z 2 and the SO(16) subgroups of each E 8 respectively, and the (1) and (2) indices differentiate between each of the two E 8 gauge groups. These expressions agree with the ones in [59] once (2.1)-(2.4) are used to put Chern and Pontryagin classes back into traces. The anomaly polynomial does not vanish, and at first sight this would mean that the theories are inconsistent. However, both anomaly polynomials factorize in the following schematic form: where the X 4 is a four-form and X 8 is an eight-form. This factorization property is the key to cancelling the anomaly, through what is known as the Green-Schwarz mechanism [59,60]. There is another field in these theories that can contribute to the aforementioned diagrams: the massless Kalb-Ramond B-field. Consistency of the 10-dimensional supergravity requires that the B-field not be invariant under gauge and gravitational interactions, and in fact it must satisfy an identity of the form where H 3 ≡ dB 2 − ω CS is the gauge-invariant curvature of the B-field built from the Chern-Simons forms of gauge and spin connections that appears in the (super)gravity couplings. X 4 = dω CS is a linear combination of characteristic classes of gauge and gravitational bundles.
There is more to the Bianchi identity at the global level, a subject which we will discuss in Section 3.1.
In this case, the anomaly corresponding to a factorized anomaly polynomial (2.15) can be cancelled by introducing the following term in the action: This term in the action is actually generated by string perturbation theory, as shown in [61] in the heterotic case, and it contributes a term −dB 2 ∧ X 8 to the Lagrangian density of the anomaly theory. Using the Bianchi identity and (2.5), we see that this adds a term −X 4 X 8 to the anomaly polynomial, exactly canceling the term in (2.15) so that the total local anomaly vanishes. This is the basic principle of the Green-Schwarz mechanism where the anomaly is cancelled by introducing an extra term in the action.
The main focus of this paper is however the three known 5 ten-dimensional non-tachyonic, non-supersymmetric string theories, which also feature a B-field and a Green-Schwarz mechanism to cancel local anomalies. The diagram in Figure 1 sums up how these non-supersymmetric theories are related to eachother and to the supersymmetric ones via gaugings of various worldsheet symmetries. We will now briefly describe the matter content of these theories as well as their anomalies.

• The Sugimoto model
The Sugimoto string [4] can be thought of as the non-supersymmetric sibling of the 5 It was recently proven that there are no other examples in the heterotic context [62].  supersymmetric type I Spin(32)/Z 2 string. The main departure for us is that the gauge algebra is sp(16) instead, and thus the Chern classes are taken in the fundamental representation of this group 6 . This distinction arises from the different kind of orientifold projection of type IIB, which introduces anti-D9 branes and an O9 plane with positive Ramond-Ramond charge and tension. The sign change in the reflection coefficients for unoriented strings scattering off the O9 is such that the Chan-Paton degeneracies reconstruct representations of the symplectic group Sp (16).
As in the type I case, the closed-string sector arranges into an N = 1 supergravity multiplet, while the chiral fermions from the open-string sector arrange into the antisymmetric rank-two representation of the gauge group, leading to the same anomaly polynomial formally. This representation is however reducible and contains a singlet; this is nothing but the Goldstino that accompanies the breaking of supersymmetry [63][64][65][66]. The low-energy interactions comply with the expected Volkov-Akulov structure of nonlinear supersymmetry [67][68][69], although there is no tunable parameter that recovers a linear realization. All in all, since the anomaly polynomial is formally identical to the type I case, it factorizes as follows: The type 0'B string [5,6] of Sagnotti is built from an orientifold projection of the tachyonic type 0B string, where the unique tachyon-free choice involves an O9 plane with zero tension. The resulting gauge group is U (32), Ramond-Ramond p-form potentials with p = 0 , 2 , 4 (the latter having a self-dual curvature) survive the projection and they get anomalous Bianchi identities for the gauge-invariant curvatures, where X 2 = c 32,1 , X 4 is formally identical to the one in Sugimoto and type I strings, and X 6 is a polynomial in p 1 , c 32,1 , c 32,2 and c 32,3 . The Bianchi identity for X 2 tells us that the low-energy gauge group reduces to SU (32), since c 32,1 is set to zero. All these RR fields with anomalous Bianchi identities play a crucial role in the cancellation of local anomalies via a more complicated Green-Schwarz mechanism involving a decomposition of the anomaly polynomial [6] of the form As we will see, setting X 6 to zero implies that c 32,3 is also trivial, and so we shall impose this condition when studying global anomalies of this theory.

• The heterotic model
The case of the SO(16) × SO(16) string [2,3] is slightly different. Along with its two supersymmetric counterparts, it is the unique ten-dimensional heterotic model that is devoid of tachyons. It is built from a projection of either of the two heterotic models, most directly the E 8 ×E 8 one under the projector built from a combination of spacetime fermion number and an E 8 lattice symmetry. As a result, it does not have any chiral fields that are uncharged under the gauge symmetry, and in particular it does not have a gravitino. Its anomaly polynomial was derived in [2,3] and factorizes as: where the (1) and (2) indices differentiate between each of the two SO(16) gauge groups. The Green-Schwarz mechanism is carried by the Kalb-Ramond field, which survives the projection as befits a heterotic model [70].
All in all, local anomalies vanish for all three non-supersymmetric string theories, by the Green-Schwarz mechanism (or a more complicated version of it). This was already known in the literature, but leaves open the possibility for the presence of global anomalies. Global anomalies are those that arise in gauge/diffeomorphism transformations that cannot be continuously connected to the identity. These anomalies are not detected by the anomaly polynomial. In the following section we detail how one can study these anomalies and we evaluate them for the case of the three non-supersymmetric tachyon-free string theories.

Global anomalies and bordism groups
In the previous Section, we have summarized prior results in the literature regarding anomaly cancellation of ten-dimensional non-supersymmetric string theories via the Green-Schwarz mechanism. Importantly, the Green-Schwarz mechanism only guarantees cancellation of local anomalies -it guarantees that the (super)gravity path integral is gauge invariant as long as we only consider gauge transformations infinitesimally close to the identity. More generally, one also need discuss global anomalies, namely anomalies in gauge transformations that cannot be continuously deformed to the identity. The archetypal example of such a global anomaly is Witten's SU (2) anomaly [71]. If one includes topology-changing transitions, one has even more general anomalies (dubbed Dai-Freed anomalies in [72]), involving a combination of gauge transformations and spacetime topology change. In this paper, we will take the point of view that such anomalies should cancel in a consistent quantum theory of gravity, where spacetime topology is supposed to fluctuate. The framework of anomaly theories introduced briefly in the previous section (2.5) can also be used to study global anomalies of Lagrangian theories such as the ones we are interested in. Given a d-dimensional quantum field theory, an anomaly on a manifold X d (possibly decorated with gauge field, spin structure, etc.) means that the partition function Z(X d ) is not invariant under gauge transformations (or diffeomorphisms, for the case of a gravitational anomaly). In a modern understanding (see [72][73][74][75][76] for detailed reviews, and also [77,78] for a discussion in the context of the 6d Green-Schwarz mechanism), the anomaly can be captured by an invertible (d + 1)-dimensional field theory α with the property that, when evaluated on a manifold with boundary Y d+1 with ∂Y d+1 = X d , the product is invariant under gauge transformations. The d-dimensional QFT arises as a boundary mode of the (d + 1)-dimensional invertible field theory α, and the anomaly is re-encoded in the fact that (3.1) is not the partition function of a d-dimensional quantum field theory -its value depends in general on Y d+1 and the particular way on which the fields on X d are extended to Y d+1 . In general, it may be very difficult to determine α. However, in weakly coupled Lagrangian theories, we have a prescription to associate an anomaly theory to each of the chiral degrees of freedom involved. For instance, the anomaly theory for a Weyl fermion in d-dimensions (d even) is given by the so-called eta invariant of a (d + 1)-dimensional Dirac operator with the same quantum numbers as the fermion we started with [74,79], If one has several fermions, the total anomaly theory is simply the product of these (so that the η invariants add up). There are other topological couplings that can also contribute to the anomaly theory, as we will see below.
Two different open manifolds Y d+1 and Y ′ d+1 , both having X d as a boundary, will yield values for the partition function (3.1) differing by a factor The manifold Y d+1 ∪Y ′ d+1 is just a general closed (d+1)-dimensional manifold. In an anomaly free-theory, the partition function in (3.1) should not depend on the choice of extension; therefore, in this picture, anomaly cancellation is simply the statement that the anomaly theory α(Ỹ d+1 ) be trivial when evaluated on a closed manifoldỸ d+1 . The particular case in whichỸ d+1 itself is a boundary,Ỹ d+1 = ∂Z d+2 , corresponds to local anomalies, which allows us to connect the discussion to the preceding Section. The η invariants introduced above, that give the anomalies for chiral fermions, can in this case be evaluated by means of the APS index theorem [80], where P d+2 is the anomaly polynomial of the previous subsection, and the "Index" is an integer. We thus recover the usual, perturbative, anomaly cancellation condition in terms of the anomaly theory. In theories where anomalies are cancelled via the Green-Schwarz mechanism, another ingredient is necessary. The ten-dimensional action has an extra Green-Schwarz term (2.17), which is the boundary mode of an 11d invertible field theory The total anomaly theory is therefore the sum of the fermion anomaly and α GS (Y 11 ). On a manifold which is itself a boundary,Ỹ 11 = ∂Z 12 , where in the last equality we used the constraint that we are restricting to twisted string manifolds satisfying the (anomalous) Bianchi identity dH = X 4 . Taking this last contribution into account, we see that the local anomaly coming from the GS term can cancel that of the fermions, provided that the anomaly polynomial factorizes as discussed in Section 2.
In the rest of this paper, we will assume that local anomalies cancel, and ask what is the value of the total anomaly theory, e 2πiαtot = e 2πiα fermions e 2πiα GS (3.7) when evaluated on 11-dimensional closed manifolds which are not boundaries. This task seems daunting at first, since, depending on the collection of background fields, there can be infinitely many such manifolds. Fortunately, one can prove 7 [74] that the partition function of the anomaly theory α tot (mod 1) is a bordism invariant, 8 This reduces the problem significantly: since α tot (mod 1) is a bordism invariant, one need only evaluate it on a single representative per bordism class. Furthermore, these classes form an abelian group, the bordism group (of manifolds suitably decorated with a twisted string structure and gauge bundle). Bordism groups have appeared prominently in the field theory and quantum gravity literature, and there are many techniques available for their computation (see [51] for a detailed introduction). Thus, to compute these anomalies, we will just compute the relevant bordism groups and evaluate the anomaly theory on generators. Notice that if it happens that the relevant bordism group Ω 11 is 0, there are no global anomalies to check! That this happens was in fact shown by Witten in [37,38] for the E 8 × E 8 string when one does not take into account the Z 2 symmetry switching the two E 8 gauge fields. 9 See [39] for an analysis of type I string.
More recently, [42,45] used the Stolz-Teichner conjecture to analyze global anomalies in supersymmetric, heterotic string theory even in stringy backgrounds, lacking a geometric description. In this paper we content ourselves with the target space treatment described above, which may miss anomalies of non-geometric backgrounds. In the following, we present the calculation and results for the three ten-dimensional non-supersymmetric string theories described in Section 2. But before that, we will describe and justify more carefully the precise structure that will be assumed in our bordism calculations.

Bianchi identities and twisted string structures
As described in the previous Subsection, the computation of global anomalies can be organized in terms of a bordism calculation and an anomaly theory, which is just a homomorphism from the bordism group to U (1). The precise bordism group to be used (i.e. the particular 7 The proof is a straightforward application of the APS index theorem (3.4), see [74]. 8 There is also a more theoretical and more general proof that the partition function of the anomaly theory is a bordism invariant, due to Freed-Hopkins-Teleman [81] and Freed-Hopkins [82]; they show that up to a deformation, which is irrelevant for anomaly calculations, the partition function of any reflection-positive invertible field theory is a bordism invariant. 9 If one does want to take this Z2 symmetry into account, for example to study the CHL string, the relevant Ω11 is nonzero [46,Theorem 2.62], and it was not known whether the global anomaly cancels. We will show that it does cancel in this paper, in Section 4. structure that our manifolds are required to have) depends on the theory we are interested in. For instance, all heterotic string theories under consideration include fermions, so we will consider only manifolds (and bordism between them) carrying a spin structure; the anomaly theory is related to the η invariant for a certain Dirac operator on this manifold. This means that the second Stiefel-Whitney class of the allowed manifolds where the anomaly theory is to be evaluated will vanish, w 2 = 0. (3.9) In heterotic string theories, also the Bianchi identity (2.16) needs to be taken into account. Equation (3.6) illustrates that cancellation of perturbative anomalies requires us to assume dH = X 4 even off-shell 10 . Therefore, we will restrict our bordism groups to consist of 11dimensional manifolds in which (2.16) is satisfied. In particular, we will set for any closed 4-manifold M 4 . The precise expression of X 4 in terms of characteristic classes depends on the particular theory under study. The particular case has been studied in the mathematical literature, and receives the name of a string structure. The X 4 's that appear in heterotic string theories are always of the form and we will refer to the data of a solution to this equation for chosen a and b as a twisted string structure. This notion appeared in the mathematical literature in [83,Definition 8.4]. The bordism groups related to the three non-supersymmetric string theories we are going to consider are Ω String−Sp (16) 11 , Ω String−SU (32)⟨c 3 ⟩ 11 , Ω String−Spin(16) 2 11 , (3.13) for the Sugimoto, Sagnotti, and SO(16) × SO(16) heterotic theories, respectively; these are the bordism groups of twisted string manifolds where the particular choice of twisted string structure is spelled out by the Green-Schwarz mechanisms for these theories as we discussed in Section 2.
Remark 3.14. Before presenting the results for the bordism groups, we must discuss an important subtlety, which affects the bordism calculation. Up to this point in this paper, we have been cavalier when writing down characteristic classes such as "p 1 " or "c 2 ", and defined these characteristic classes as closed differential forms (e.g. in (2.4)) by way of Chern-Weil theory. However, these differential forms have quantized periods, as is the case for data coming out of any quantum theory, and a proper treatment of the Green-Schwarz mechanism should take this into account. There are two ways to do this.
1. The simplest approach is to lift to Z-valued cohomology: the quantized periods are a reminder that the de Rham classes of the Chern-Weil forms of p 1 , c 2 , etc., lift canonically to classes in H 4 (BG; Z) for various Lie groups G, and on many manifolds M , these integer-cohomology lifts of these characteristic classes can be torsion! Thus it is natural to wonder whether the B-field should be an element of H 3 (-; Z) and the Bianchi identity (3.12) should take place in H 4 (-; Z). The definitions of string structure and twisted string structure in mathematics assume this lift has taken place.
2. Alternatively, one could lift to differential cohomologyȞ 4 (-; Z), which amounts to observing that it is not just the Z-cohomology lift which is natural, but also the data of the Chern-Weil form; differential cohomology is a toolbox for encoding both of these pieces of data. Indeed, for any compact Lie group G and class c ∈ H * (BG; Z), there is a canonical differential refinementč ∈Ȟ * (B ∇ G; Z) [84,85], where B ∇ G is the classifying stack of G-connections. 11 Thus we could instead ask: should we begin with a B-field iň H 3 (-; Z) and ask for the Bianchi identity to take place inȞ 4 (-; Z)? This combines the two other formalisms we considered, differential forms and integral cohomology.
The answer in the mathematics literature is often the second option, beginning with Freed [39, §3] and continuing in, for example, [46,[86][87][88][89][90][91][92][93][94][95][96][97][98][99][100][101][102][103]. In particular, [93,94] interpret the data entering into the Green-Schwarz mechanism as specifying a connection for a Lie 2-group built as an extension of the gauge group by BU (1), providing an appealing physical interpretation of the lift to differential cohomology. We are interested in classifying anomalies, and while there is an interesting differential refinement of the story of the bordism classification of anomalies due to Yamashita-Yonekura [104][105][106], the deformation classification of anomalies ultimately can proceed without differential-cohomological information, because it boils down to studying bordism groups. Because of this, we will work with characteristic classes in integral cohomology, noting here that the correct setup of the Green-Schwarz mechanism taking torsion and Chern-Weil forms into account uses differential cohomology, and that for our computations it makes no difference.
Note that cancellation of perturbative anomalies around (3.6) only requires the free part of X 4 ∈ H 4 (-; Z) to be trivial in a compact manifold, and poses no obvious restriction on torsion. Reference [38] studies a particular example suggesting that this should be the case, but does not attempt to make a general argument. To ascertain whether the torsion piece of X 4 must also be trivialized or not, consider the physical origin of the Bianchi identity, which is itself a two-dimensional version of the Green-Schwarz mechanism described above (see e.g. [57,107]). Consider a worldsheet wrapped on a 2-manifold Σ 2 of the ambient tendimensional spacetime manifold M 10 . The configuration should be invariant under spacetime diffeomorphisms, and gauge transformations, which are manifested as global symmetries of the worldsheet. However, in heterotic or type I theories, the worldvolume degrees of freedom are chiral, and anomalous under these transformations. The anomaly theory, which we denote α worldsheet , is encoded by a three-dimensional η invariant. Applying the APS index theorem (3.4), we obtain exp(2πiα worldsheet ) = exp 2πi X 4 , (3.15) where X 4 is a certain differential form built out of characteristic classes, and which is precisely the X 4 appearing above (indeed, (3.15) is usually taken to give the definition of X 4 ). As things stand, any configuration with an insertion of a fundamental string worldsheet on Σ 2 has a gravitational anomaly; however, the worldsheet also has an electric coupling to the B-field, exp 2πi whose anomaly theory is simply Now, the total worldsheet anomaly is The physical consistency condition is that the total anomaly is trivial exp 2πiα worldsheet total = 1, for all M 3 .

(3.19)
When M 3 is a boundary, anomaly cancellation is achieved, as above, by setting dH = X 4 , precisely the Bianchi identity described above. However, this is not all there is to (3.19).
Assuming that anomalies vanish when M 3 is a boundary, exp(2πiα worldsheet total (M 3 )) is actually only dependent on the integer homology class of M 3 . In fact, since it is a map that assigns a phase to each 3-cycle in the ambient 10-dimensional manifold M 10 , it can be regarded as an element of H 3 (M 10 ; U (1)), with U (1) coefficients. Using the long exact sequence in cohomology associated to the short exact sequence of groups Z → R → U (1), we obtain that [75] H 3 (M 10 ; R) → H 3 (M 10 ; U (1)) → H 4 (M 10 ; Z) → H 4 (M 10 ; R), (3.20) where the third map is taking the free part of the integer cohomology class. In general, exp(2πiα worldsheet total ) will have pieces both in the image of the first map and in its cokernel. An example where the anomaly theory has a non-trivial piece in the image of the first map of (3.20) can be obtained by compactifying heterotic string theory on a Bieberbach 3-manifold, a fixed-point free quotient of the torus T 3 . 12 Since T 3 is Riemann-flat, a quick analysis would suggest that the Bianchi identity is satisfied automatically with no gauge bundle or B-field turned on. However, trying to implement this manifold directly in the worldsheet results in a theory which is not level-matched. The problem is that exp(2πiα worldsheet total ) with no gauge bundle turned on is nontrivial for most Bieberbach manifolds, and so the anomaly theory is a nontrivial class in H 3 (M 10 ; R). Cancelling this anomaly forces either a B-field (discrete torsion) to be turned on, or a non-trivial flat gauge bundle to be present.
The rest of the anomaly theory is in the image of the second map in (3.20), and can therefore be represented by a certain torsion integer cohomology class in H 4 (M 10 ; Z), whose free part vanishes. We will now show that this is in fact the torsion part of X 4 −dH. Consider a torsion 3-cycle M 3 of order k, i.e. such that kM 3 is the boundary of some 4-manifold N 4 . Let us see how to compute the anomaly in this case. First, the anomaly theory α worldsheet total is a linear combination of η invariants, which in this particular case can be re-expressed as linear combinations of gravitational and gauge Chern-Simons numbers as discussed above. The Chern-Simons invariant is additive on disconnected sums, and so, we have Next, we can use the fact that kM 3 = ∂N 4 , to write (after exponentiation) This expression is not obviously independent of the choice of N 4 , but when (X 4 − dH) is pure torsion, it actually is. The reason is that the quantity N 4 (X 4 − dH) may be rewritten as a linking pairing in homology [108]. If we Poincaré dualize (X 4 − dH) tor to a torsion 6-cycle M 6 , the linking pairing between M 6 and M 3 is constructed by choosing a boundary N 4 for kM 3 and computing N 4 (X 4 − dH) modulo k. Importantly, the result does not depend on the choice of N 4 (see [108] for a review and proof of these facts).
In short, the full analog of the Bianchi identity is (3.19). Unpacking this condition, we recover that: • There is the condition on any 3-manifold M 3 that where CS X 4 3 is a (local) Chern-Simons form obeying dCS X 4 3 = X 4 . This will force discrete B-fields to be turned on in certain situations, such as on Bieberbach manifolds (these were referred to as "worldsheet discrete theta angles" in [78]). 12 We thank Cumrun Vafa for pointing out this example to one of us.
• As a consequence of the previous point, when M 3 is a boundary, we get that the Bianchi identity dH = X 4 must hold over the integers.
The general analysis we just carried out is somewhat abstract; in the next Subsection, we will verify its correctness by explicitly checking, in a variety of backgrounds, that anomalies in ten dimensions only cancel if the torsional part of the Bianchi identity holds.
Finally, we comment on another possible way in which the anomaly calculation could have been set, avoiding the calculation of string bordism groups altogether, as in [78]. Anomalies are always studied with respect to a choice of background fields. The approach we have followed here takes the metric g, the gauge field A, and the 2-form field B as background fields, and imposes the Bianchi identity as a restriction on the allowed backgrounds. However, in a quantum theory of gravity, there are no global symmetries, and therefore, there are no background fields either. This is manifested in the fact that all three of g, A, B are actually dynamical fields that we are supposed to path-integrate over. Treating these as backgrounds is justified if there is some sort of weak coupling limit in which the fields become frozen. This is automatically the case at low energies in any ten-dimensional string theory, since the couplings of all of g, A, B are dimensionful and become irrelevant in the deep IR. It is not the case e.g. in six dimensions, where antisymmetric tensor fields are often strongly coupled and cannot be treated perturbatively. In such cases, the only approach available is to explicitly perform the path integral over the tensor fields, compute their effective action, and verify that the resulting path integral indeed cancels against the contributions of other chiral fields. There is no meaningful analog of the notion of having a string structure, since no weak coupling notion is available. The anomaly theory (as a function of the metric and background gauge fields) can then studied on general spin manifolds (and not just string manifolds), and anomalies cancel in a standard way, because the B-field (which is integrated over) couples to background 4 and 8-forms X 4 and X 8 , and has a mixed anomaly captured by the anomaly polynomial X 4 X 8 , just what is needed to cancel the anomaly of the fermions. From a perturbative string worldsheet point of view, we feel it is more natural to keep B as a background field; furthermore, the techniques we use in this paper can be extended to compute lower-dimensional string bordism groups, which control solitonic objects in these non-supersymmetric theories via the Cobordism Conjecture [48].

Evidence for torsional Bianchi identities
In the previous Subsection, we gave an argument that the Bianchi identity holds at the level of torsion, too. The argument relies heavily on string perturbation theory, and one may worry e.g. that it does not capture strongly coupled situations. In this Section, we provide independent evidence, which does not rely on the worldsheet at all, that the Bianchi identity holds at the level of integer cohomology. We do so by computing Dai-Freed anomalies of supersymmetric and non-supersymmetric string theories on simple eleven-dimensional lens spaces. Lens spaces are quotients of spheres by Z p groups; they are the simplest examples of manifolds whose cohomology is purely torsional (except in bottom and top degrees, as usual).
In particular, their first Pontryagin classes are torsion; the upshot of the calculation in this Section is that spacetime anomalies on lens spaces seem to vanish if and only if the Bianchi identities are satisfied at the level of integral cohomology, including torsional classes. Now we turn to the details of evaluating anomalies on lens spaces; we refer the reader to [40,51] for more on lens spaces and the corresponding expressions for eta invariants. (Eleven-dimensional) lens spaces are defined to be quotients of the form where the Z p action acts as scalar multiplication by e 2πi/p on the six complex coordinates C 6 and where we embed the covering S 11 as the unit sphere. An important property of these lens spaces is that the Green-Schwarz term, will automatically vanish on lens spaces, since H 3 (S 11 /Z n ; Z) = 0. As a result, the calculation of the full anomalies of string theories on lens spaces reduces to determining the anomalies of the chiral fields. We will now evaluate the anomaly theory of the Type I and the Heterotic string theories (supersymmetric and non-supersymmetric) on certain eleven-dimensional lens spaces.

Type I and HO heterotic theories
As the Green-Schwarz (GS) contribution to the anomaly theory vanishes on lens spaces, the remaining fermion anomaly theory of the type I and HO heterotic theory is given by The Rarita-Schwinger eta invariant η RS arises from the anomaly theory of a ten-dimensional gravitino according to α gravitino = η RS − 2η D [40]. In order to evaluate this anomaly theory on L 11 p , one can derive the branching rules for the adjoint representation of Spin(32) in terms of the charge-q irreducible Z p representations L q . This branching depends on how Z p is included in the gauge group. We choose a family of inclusions of the form according to which the (complexified) vector representation of Spin(2N ) splits as The parameter k denotes an inclusion that places the U (1) fundamental representation L in k diagonal blocks, in pairs L ≡ L ⊕ L −1 , and the rest in the trivial representation L 0 . Then, the vector representation of Spin(2N ) further splits into In order to find the branching rules for other representations, it is convenient to use Chern characters. Letting x ≡ c 1 (L), the Chern character of N (and N * ) decomposes into Then we can build the characters for adjoint, symmetric and antisymmetric SU (N ) representations, from which we can reconstruct the characters for Spin(2N ) representations of interest, such as the adjoint (antisymmetric) and spinorial. The resulting branching rules involve the representations L q ≡ L q ⊕ L −q . In particular, the adjoint of Spin(2N ) branches according to which gives the corresponding eta invariant. Using the expressions the anomaly simplifies to In order to compare the cases in which α = 0 mod 1 to the Bianchi identity, let us recall that the total Pontryagin class of L 2k−1 Thus p 1 (L 11 p ) = 6y, and one can show that the canonical choice of p 1 2 afforded by the spin structure is p 1 2 = 3y. On the other hand, according to the branching rule (3.29), the total Chern class of the associated vector bundle is c = (1 − y) 2k , and thus c 2 = −2k y. Therefore, vanishes if and only if k = 3 mod p. Plugging in k = 3 + mp with m integer, the anomaly does vanish (mod 1), and it does not vanish otherwise.

E 8 × E 8 theory
The calculation for the E 8 × E 8 theory is almost the same as in the preceding case. The anomaly theory has the same form of (3.26), the only difference being the branching of the adjoint representation adj = (248, 1) ⊕ (1, 248). We employ the same construction as before, embedding Z p into the Spin (16) subgroup of E 8 . The general construction is thus specified by a pair (k 1 , k 2 ) pertaining to the two E 8 factors. One then has to compute the branching for the 120 and the spinorial 128 of Spin(16) which compose the adjoint representation of E 8 . The former has been presented in the preceding section, now with N = 8, while the latter can be constructed computing Chern characters of antisymmetric representations of SU (8) whose direct sum gives the branching of the spinorial representation: The Chern character for the various antisymmetric representations can be found by expanding the graded Chern character for the exterior algebra Λ For instance for N = 8 and k = 1, summing the even or odd rank characters leads to which means that the spinorial representations branch according to 128 The anomaly for this particular choice (k 1 , k 2 ) = (1, 0) then simplifies to which vanishes (mod 1) for p = 2. Let us now look at the Bianchi identity. The Chern class of the adjoint E 8 × E 8 associated bundle is with y a generator of H 4 (L 11 p ; Z), and thus c 2 = −60y. For E 8 × E 8 we have to divide c 2 2 by 30 in the Bianchi identity, thus getting This class only vanishes if p = 2, which is the same value for which the anomaly vanishes! Similarly, for (k 1 , k 2 ) = (1, 1) the Bianchi class is y, which never vanishes (except for the trivial case p = 1), and accordingly the anomaly never vanishes either. One can carry on with more complicated embeddings computing the spinorial branching of 128: for (k 1 , k 2 ) = (2, 1) the Bianchi class vanishes, and indeed the anomaly turns out to always vanish mod 1. At first glance, the case (k 1 , k 2 ) = (2, 0) appears to present an exception, since the Bianchi class is y ̸ = 0 but the anomaly vanishes for p = 5. However, in order to find the relationship between torsional Bianchi identities and anomalies, for given torsion the anomaly should vanish for all allowed backgrounds, and the (1, 1) embedding has the same Bianchi class but nonvanishing anomaly for p = 5.
The general expression for any (k 1 , k 2 ) for the E 8 ×E 8 theory is more involved due to how the spinorial representations branch, but the procedure to compute the anomaly is systematic.

Non-supersymmetric theories
Let us now address the non-supersymmetric cases. An immediate consequence of the above result for the supersymmetric heterotic theories is that the anomaly on lens spaces satisfying the torsional Bianchi identity also vanishes for the non-SUSY heterotic theory, since its chiral matter content is in the virtual difference of the corresponding representations [70]. This fact will turn out to be useful when discussing fivebrane anomaly inflow in section 3.4.2.
For the Sagnotti model, the anomaly theory can be written as [5,6] since it contains a four-form RR field with self-dual curvature, similarly to type IIB. Following the same procedure as before, now with the simpler inclusion Z p → U (1) → SU (32), one can evaluate the fermionic anomalies; for the self-dual field, in the second line of eq. (3.44) we have used anomaly cancellation in type IIB supergravity to recast its anomaly theory in terms of fermionic contributions, along the lines of [40]. Thus we obtain The Chern class of the associated fundamental bundle is now c = (1 − y) k , so that c 2 = −k y and the Bianchi class vanishes for k = 6 mod p. Notice that c 3 = 0 as well for these bundles, since the total Chern class only contains powers of y ∈ H 4 (L 11 p , Z). Substituting k = 6 + mp for integer m, the anomaly vanishes as expected, but not otherwise.
The calculation for the Sugimoto model is essentially identical: the anomaly theory is simply α Sugimoto = − α 0 ′ B , since the antisymmetric fermion has now positive chirality and the gravitino and dilatino contribute the opposite of the self-dual tensor. The inclusion we (16), under which the 32 representation branches according to Since the resulting Bianchi class is also the same, one obtains the same result: the anomalies cancel on lens backgrounds which satisfy the Bianchi identity at the torsional level.

Vanishing bordism classes
We now turn to the main results of this paper -the calculation of string bordism groups with twisted string structures corresponding to the non-supersymmetric strings, by means of homotopy theory. These sections cover in some detail the mathematical aspects of the calculation; a table summarizing the results can be found in the Conclusions.

Sp(16)
At this point we make our first bordism computation: that every closed, spin 11-manifold M with a principal Sp(16)-bundle P satisfying the Green-Schwarz identity 1 2 p 1 (M )+c 2 (P ) = 0 is the boundary of a compact spin 12-manifold on which the Sp(16)-bundle and Green-Schwarz data extend. This implies that the anomalies we study in this paper vanish for the Sugimoto string.
To make these computations, we use the Adams and Atiyah-Hirzebruch spectral sequences. By now these are standard tools in the mathematical physics literature, so we point the reader to [51,109] for background and many example computations written for mathematical physicists. The computations in this paper are a little more elaborate: twisted string bordism rather than twisted spin bordism. There are fewer such calculations in the literature, but we found the references [46,[110][111][112][113] helpful.
On to business. The data of a G-gauge field and a B-field satisfying a Bianchi identity is expressed mathematically as a principal bundle for a Lie 2-group extension of G by BU (1). = 0, the lower-dimensional bordism groups are barely more work.
Theorem 3.48. The low-dimensional String-Sp(16) bordism groups are: (16) be the vector bundle associated to the defining representation; it is rank 64 as a real vector bundle. Then by an argument analogous to [51, §10.4], there is an isomorphism Ω (16). The Thom spectrum X V of V → X is a homotopy-theoretic object whose homotopy groups can be expressed as certain kinds of bordism groups by the Pontrjagin-Thom theorem; the upshot is that string bordism groups of X V are isomorphic to (X, V )-twisted string bordism groups of a point. See [51, §10.4] for more information and references.
If tmf denotes the spectrum of connective topological modular forms, then it follows that the map Ω String * (X) → tmf * (X) is an isomorphism in degrees 15 and below whenever X is a space or connective spectrum [111, Theorem 2.1] (the latter condition includes all Thom spectra we study in this paper). Therefore for the rest of the proof we focus on tmf * ((BSp(16)) V −64 ). These are finitely generated abelian groups, so we may work one prime at a time (see [51, §10.2]).
As input, we will need the following calculation of Borel. For large primes p (i.e. p ≥ 5), we want to show that tmf * ((BSp (16)) V −64 ) lacks ptorsion in degrees 11 and below. This follows because when p ≥ 5, the homotopy groups of the p-localization tmf (p) are free and concentrated in even degrees [116, §13.1], and the Z (p) cohomology of BSp(16) (hence also of (BSp(16)) V −64 , by the Thom isomorphism) is always free and concentrated in even degrees as a consequence of proposition 3.49 and the universal coefficient theorem, so the Atiyah-Hirzebruch spectral sequence computing p-localized tmf * ((BSp(16)) V −64 ) collapses with only free summands on the E ∞ -page, preventing ptorsion in tmf * ((BSp (16)) V −64 ) in the range we care about.
For p = 3, the 3-localized Atiyah-Hirzebruch spectral sequence does not immediately collapse, so we use the Adams spectral sequence (and we will see that this Adams spectral sequence does immediately collapse). The Adams spectral sequence takes the form Let us explain this notation. We pick a prime p; then A is the p-primary Steenrod algebra, the Z p -algebra of all natural transformations H * (-; Z p ) → H * +t (-; Z p ) that commute with the suspension isomorphism. The mod p cohomology of any space or spectrum X is thus naturally a Z-graded A-module, so we may apply Ext A , the derived functor of Hom A . This gives us two gradings: the original Z-grading on cohomology is the one labeled t, and the grading arising from the derived functors is the one labeled s. On the right-hand side of (3.50), π s * denotes stable homotopy groups, and (-) ∧ p denotes p-completion. We will not need to worry in too much detail about p-completion: we will only ever p-complete finitely generated abelian groups A, for which the p-completion carries the same information as the free summands and the p-power torsion summands of A. Thus we will typically be implicit about p-completion -in particular, Z p always denotes the cyclic group of order p, not the p-adic integers.
We are interested in tmf -homology (or really string bordism), rather than stable homotopy, which means replacing X with tmf ∧ X in (3.50); then the Adams spectral sequence converges to tmf t−s (X) ∧ p . By work of Henriques and Hill (see [110,116]), building on work of Behrens [117] and unpublished work of Hopkins-Mahowald, there is a change-of-rings theorem for the 3-primary Adams spectral sequence for tmf simplifying (3.50) to Here A tmf is the graded Z 3 -algebra with |β| = 1 and |P 1 | = 4. For the Adams E 2 -page, A tmf acts on H * (X; Z 3 ) by sending β to the Bockstein for 0 → Z 3 → Z 9 → Z 3 → 0 and P 1 to the first mod 3 Steenrod power. See [110][111][112][113] for more information and some example computations with this variant of the Adams spectral sequence. As input, we need to know how β and P 1 act on H * ((BSp(16)) V −32 ; Z 3 ). This is determined in [113,Corollary 2.37] from the input data of the action of the images of β and P 1 on the mod 3 Steenrod algebra on H * (BSp(16); Z 3 ). As the cohomology of BSp(16) is concentrated in even degrees, β must act trivially, and thus likewise for the Thom spectrum (BSp(16)) V −64 . Shay [118] computes the action of P 1 on mod 3 Chern classes; 13 the formula implies that in H * (BSp (16) (16) Definition 3.53. If M is a Z-graded module over a Z-graded algebra A, we will let Σ k M denote the same underlying A-module with the grading shifted up by k, i.e. if x ∈ M is homogeneous of degree m, then x ∈ Σ k M has degree m + k. We will write Σ for Σ 1 .
The notation Σ k is inspired by the suspension of a topological space, which has the effect of increasing the degrees of elements in cohomology by 1.
Then, there is an A tmf -module isomorphism where P is concentrated in degrees 12 and above (so we can ignore it). We draw the decomposition (3.55) in fig. 3, left. We need to compute Ext A tmf (N 3 , Z 3 ). To do so, we use the fact that the short exact sequence of A tmf -modules (which we draw in fig. 2 induces a long exact sequence on Ext groups; traditionally one draws the Ext of the first and third terms of a short exact sequence in the same Adams  We will draw the long exact sequence in Ext corresponding to (3.56) in fig. 2. To do so, we need Ext A tmf (Z 3 ), which is due to Henriques-Hill [110,116], and Ext A tmf (Cν), which is computed in topological degree 14 and below in [113, Figure 2]. Our notation for names of Ext classes follows [113, §3]; Ext A tmf (Σ 8 Z 3 ) is a free Ext A tmf (Z 3 )-module on a single generator, so call that generator z. 14 Most boundary maps are nonzero for "degree reasons," meaning that their domain or codomain is the zero group. For t − s ≤ 14, there are two exceptions: ∂(z) could be ±αy or 0, and ∂(αz) could be ±βx or 0. Since the boundary maps commute with the Ext A tmf (Z 3 )-action and α(αy) = βx, 15 these two boundary maps are either both zero or both 14 There are two classes which generate Ext A tmf (Σ 8 Z3) as an Ext A tmf (Z3)-module, and one is −1 times the other. For the purposes of this paper, it does not matter which one we call z and which one we call −z. 15 The equation α(αy) = βx is stated in [113, Remark 3.21], but not proven there. One way to prove it is to compare with the equivalent α-action αy → βx in Ext A tmf (N1) in the long exact sequence in (ibid., Figure  5): because ∂(αy) = ±βw and αβw ̸ = 0, and because α(∂(-)) = ∂(α · -), α(αy) ̸ = 0, hence must be ±βx, and we can choose the generator x so that we obtain βx and not −βx. The calculation of ∂(αx) in (ibid., Lemma 3.24) does not use any information about α(αy).
nonzero. To see that they are both nonzero, we use that Ext 0,8 is not the image of an Ext class for N 3 , so ∂(z) ̸ = 0. With this Ext in hand, we can draw the E 2 -page of the Adams spectral sequence in fig. 3, right. The spectral sequence collapses at E 2 in the range we study for degree reasons. The straight lines denote actions by h 0 ∈ Ext 1,1 , which lift to multiplication by 3, so we see there is no 3-torsion in degrees 11 and below. Lastly, for p = 2, we use the Adams spectral sequence again; the outline of the proof is quite similar to the p = 3 case, but the details are different. Specifically, we will once again use the Adams spectral sequence and a standard change-of-rings theorem to simplify the calculation of the E 2 -page, but the algebra of cohomology operations is different.
Let A(2) be the subalgebra of the mod 2 Steenrod algebra generated by Sq 1 , Sq 2 , and Sq 4 . There is an isomorphism H * (tmf ; [120,121], which by a standard argument simplifies the E 2 -page of the 2-primary Adams spectral sequence to The next thing to do is to determine how A(2) acts on H * ((BSp (16) where P contains no elements in degrees 11 or below, and hence will be irrelevant to our calculations. We draw (3.58) in fig. 4 (16) fig. 4, right. Looking at the E 2 -page, most differentials are ruled out by degree considerations or the fact that they must commute with the action of h 0 or h 1 Proof. Classes x ∈ E 0,• 2 of an Adams spectral sequence for G-bordism correspond naturally to (a subset of) Z 2 -valued characteristic classes c x for manifolds with G-structure, and x survives to the E ∞ -page if and only if there is a closed manifold M with G-structure such that M c x = 1; see [41, §8.4].
For the Adams spectral sequence for string-Sp(16) bordism at p = 2, the two classes corresponding to a basis of E 0, 8 2 are the mod 2 reductions of c 2 2 and c 4 . To finish this lemma, we will find closed string-Sp(16) 8-manifolds on which these classes do not vanish.
• The quaternionic projective plane HP 2 has a tautological principal Sp(1)-bundle P := S 11 → HP 2 ; let P ∨ → HP 2 be the same space with the quaternion-conjugate Sp(1)action, and let Q → HP 2 be the principal Sp (16) The map f * induces a pullback map on mod 2 cohomology and on Adams spectral sequences; the map on mod 2 cohomology is the quotient by all elements of degree greater than 4, so the effect on Adams spectral sequences is to kill all summands in Ext except for the red summands. As H * ((HP 1 ) f * V −64 ; Z 2 ) consists of two Z 2 summands in degrees 0 and 4, joined by a Sq 4 , the Adams spectral calculating its 2-completed tmf -homology is worked out by is an isomorphism. Thus this differential persists to the Sp(16) Adams spectral sequence.

U (32)
Now we discuss the Sagnotti string, whose gauge group is U (32). The Green-Schwarz mechanism for this theory involves three classes in degrees 2, 4, and 6 canceling c 1 , c 2 , and c 3 of the gauge bundle, respectively.
We may impose the degree-2, 4, and 6 conditions on BU (32) in any order. Starting with c 1 , we obtain BSU (32); then, let BSU (32)⟨c 3 ⟩ denote the fiber of the map (3.62) A map X → BSU (32)⟨c 3 ⟩ is equivalent data to a rank-32 complex vector bundle V → X with SU -structure and a trivialization of c 3 (V ). There is a tautological such vector bundle V t → BSU (32)⟨c 3 ⟩, which is the pullback of the tautological bundle over BSU (32). Finally, the degree-4 condition for a U (32)-bundle V over a manifold M asks for a trivialization of 1 2 p 1 (M )+c 2 (V ). Thus, we ask for a (BSU (32)⟨c 3 ⟩, V t )-twisted string structure on M , i.e. a map f : M → BSU (32)⟨c 3 ⟩ and a string structure on T M ⊕ f * V t ; the Whitney sum formula for 1 2 p 1 unwinds this into the usual Green-Schwarz condition. We will be a little casual with the notation and call a (BSU (32)⟨c 3 ⟩, V t )-structure a String-SU (32)⟨c 3 ⟩structure, even though we do not construct a Lie 2-group String-SU (32)⟨c 3 ⟩ realizing this twisted string structure (and indeed, there is no guarantee one exists). is isomorphic to either 0 or Z 2 .
The ambiguity is in a differential we were not able to resolve. Unfortunately, this means we were not able to use bordism-theoretic methods alone to calculate the anomaly of the Sagnotti string. Our proof also yields partial information on lower-dimensional bordism groups; there is ambiguity due to Adams spectral sequence differentials, some of which we suspect are nonzero.
As for the Sugimoto string, we work one prime at a time. Proof. It suffices to work with cohomology valued in the ring Z[1/6] of rational numbers whose denominators in lowest terms are of the form 2 m 3 n , as tensoring with Z[1/6] preserves all p-power torsion for p ≥ 5. Cartan [123] and Serre [124] computed H * (K(Z, n); Z[1/6]); their formulas imply that when p ≥ 5, H k (K(Z, 5); Z[1/6]) is torsion-free for k ≤ 12, and vanishes apart from Since this map commutes with differentials, this means the fate of all classes in E 0,• 2 is determined by their preimages in the spectral sequence for K(Z, 5) → * → K(Z, 6). For example, we know thanks to Serre [125, §10] that in that spectral sequence, E transgresses to the mod 2 reduction of the tautological class F of K(Z, 6). Therefore in the spectral sequence for BSU (32)⟨c 2 ⟩, E transgresses to the pullback of F , which is c 3 . The Leibniz rule then tells us d 6 (xE) = xc 3 for x ∈ H * (BSU (32); Z[1/6]); since this cohomology ring is polynomial, xc 3 ̸ = 0 as long as x ̸ = 0, so these differentials never vanish. Therefore the nonzero part of the E ∞ -page, at least in total degree 12 and below, is a quotient of E * ,0 2 = H * (BSU (32); Z[1/6]). Since H * (BSU (32); Z[1/6]) is free and concentrated in even degrees. This implies the E ∞ -page is also free and concentrated in even degrees in total degree 12 and below, which implies the lemma statement. Proof. We want to compute Ω String * ((BSU (32)⟨c 3 ⟩) Vt−64 ), and as noted above, we may replace Ω String * with tmf for the degrees k in the corollary statement. Because of lemma 3.65 and the fact that tmf (p) has homotopy groups concentrated in even degrees and lacks p-torsion, the Atiyah-Hirzebruch spectral sequence computing tmf * ((BSU (32)⟨c 3 ⟩) Vt−64 ) ∧ p collapses with no p-torsion in the range 11 and below.
As usual, p = 2 and p = 3 are harder.  Proof. This is a standard argument with the Serre spectral sequence for the fibration on the left in (3.64), so we sketch the details. For the mod 2 cohomology, we need as input H * (BSU (32); [115, §29]: these are the mod 2 reductions of the Chern classes. We also need H * (K(Z, 5); Z 2 ), which was computed by Serre [125, §10]. This is a polynomial ring on infinitely many generators; the six in degrees below 13 are E ∈ H 5 , the mod 2 reduction of the tautological class; G := Sq 2 (E); H := Sq 1 (H); I := Sq 4 (E); K := Sq 4 (G); and L := Sq 5 (G).
In the Serre spectral sequence, the class E transgresses to c 3 , and the proof is the same as in the proof of lemma 3.65. Similarly, we divine the fate of the other classes on the line p = 0: • In the Serre spectral sequence for the rightmost fibration in (3.64), G transgresses via d 8 to Sq 2 (F ) by the Kudo transgression theorem [126], so in the leftmost fibration, d r (G) = 0 for r ≤ 7, and d 8 (G) = Sq 2 (c 3 ) = 0. Thus G is a permanent cycle.
• In a similar way, H transgressing to Sq 3 (F ) via d 9 pulls back to imply d 9 (H) = Sq 3 (c 3 ) = 0, so H is also a permanent cycle. Likewise, E 2 is a permanent cycle, because in the fibration over K(Z, 6), it supports the transgressing d 11 (E 2 ) = Sq 5 (F ), and Sq 5 (c 3 ) = 0, and similarly K and L are permanent cycles.
The Leibniz rule then cleans up the rest of the spectral sequence in degrees 12 and below. This gives us the ring structure. For the Steenrod squares, we use the information of the A-action on H * (BSU (32); Z 2 ) coming from the Wu formula, together with the A-actions we gave when describing H * (K(Z, 5); Z 2 ) above. There is ambiguity in the Steenrod squares in H * (BSU (32)⟨c 3 ⟩; Z 2 ) coming from the loss of information passing to the associated graded on the E ∞ -page, which is the source of λ 1 , λ 2 , and λ 3 in the theorem statement. However, by pulling back to the analogous fibration over BSU (2), where the fiber bundle admits a section (as c 3 of an SU (2)-bundle is canonically trivial), so we can use the Künneth formula to compute Steenrod squares. Pulling back to SU (2) loses all information about c i for i > 2, so this leaves ambiguity in c 4 and c 6 as described in the theorem statement, but resolves the ambiguity involving c 2 . Some ambiguity can be erased by redefining generators, which is how we disambiguate Sq 4 (H) = L, but this still leaves the choices listed in the theorem statement.
We obtain P 1 (c i ) from Sugawara's calculations [119, §5] of Shay's formula [118]. With the fate of these classes known, the Leibniz rule cleans up the rest of the spectral sequence in total degrees 12 and below to obtain the theorem statement.
Now, just as in the proof of theorem 3.48, we run the Adams spectral sequences at p = 3 and p = 2. The twist by V t twists the action of P 1 at p = 3, and the action of Sq 4 at p = 2, in an analogous way. Reusing names of A tmf -modules from §3.3.1, we conclude that there is an A tmf -module isomorphism where P is concentrated in degrees 12 and above, so will not affect us. We draw (3.68) in fig. 5, left. We calculated Ext A tmf (N 3 ) in fig. 2; using this, we discover that, like for the Sugimoto string, in degrees 11 and below, the E 2 -page consists only of h 0 -towers in degrees 0, 4, and 8, so there can be no 3-torsion. See fig. 5, right, for a picture of this Adams spectral sequence. Last, p = 2. The ambiguity in the Steenrod actions is not severe enough to get in the way of the existence of an isomorphism of A(2)-modules where P is concentrated in degrees 12 and above, so will be irrelevant for us, and: • N 1 is isomorphic to A(2) ⊗ A(1) Q in degrees 6 and below (i.e. the quotients of these two A(2)-modules by their submodules of elements in degrees 7 and above are isomorphic); and • N 2 is isomorphic to A(2) ⊗ A(1) Q in degrees 3 and below.
Here Q is the "question mark," the A(1)-module which has a Z 2 -vector space basis {x 0 , x 1 , x 3 } with |x i | = i, and with Sq 1 (x 0 ) = x 1 , Sq 2 (x 1 ) = x 3 (all other A(1)-actions are trivial for degree reasons). The module Σ 11 N 2 is generated by U c 2 G. We draw the decomposition (3.69) in fig. 6 ). For N 1 and N 2 , we use that an isomorphism of A(2)-modules in degrees k and below implies the existence of an isomorphism of Ext groups in topological degrees k − 1 and below, so it is good enough to know Ext A(2) (A(2) ⊗ A(1) Q); then the change-of-rings theorem (see, e.g., [109, §4.5]) implies and Adams-Priddy [127, Table 3.11] compute Ext A(1) (Q). Putting all this together, we can draw the E 2 -page in fig. 6, right. In the range relevant to us, the E 2 -page is generated as an Ext A(2) (Z 2 )-module by the following ten summands.
2. Coming from Ext(Σ 7 N 1 ): b 1 ∈ Ext 7,0 and b 2 ∈ Ext 2,12 . These are subject to various relations: notably, if x is any one of these generators, h 2 x = 0. Once we take into account the fact that differentials commute with h 0 and h 1 , we still need to determine d 2 (b 1 ), d r (c), d r (d), d 2 (a 5 ), d 2 (b 2 ), and d r (e).  The values of these differentials are the same as for the corresponding classes in the Adams spectral sequence for the Sugimoto string, and the proofs are the same as we gave for them in §3.3.1.

Coming from Ext
Ultimately we need to address d 2 : E 0,11 2 → E 2, 13 2 . If this differential vanishes, there is also potential for d 6 : E 0,11 6 → E 6,16 6 to be nonzero. The fate of these two differentials determines whether Ω String-SU (32)⟨c 3 ⟩ 11 is nonzero, so it is unfortunate that the techniques we applied were unable to resolve them.
We are able to obtain some partial information, though. Here w 1 ∈ Ext 4,12 A(2) (Z 2 ) is the class whose image in Ext A(1) (Z 2 ) is the Bott periodicity class. Thus there are λ 1 , λ 2 ∈ Z 2 such that d 6 (e) = λ 1 w 1 h 2 1 a 1 + λ 2 h 4 0 b 2 . (3.74) induced by the inclusion A(1) → A(2) of algebras. It is possible to compute the right-hand Ext groups using the decomposition (3.69) and the techniques in [109]; one learns that e and w 1 h 2 1 a 1 both remain nonzero after (3.74), so it suffices to compute d 6 (e) in the ko-homology Adams spectral sequence. There, though, the submodule M a 1 of the Ext groups generated by a 1 splits off: because V t −32 is spin, there is a Thom isomorphism ko * ((BSU (32)⟨c 3 ⟩) Vt−32 ) ∼ = ko * (BSU (32)⟨c 3 ⟩), so ko * (pt) splits off; as this splitting lifts to the level of spectra, it also splits M a 1 off of the Adams spectral sequence, so all differentials into M a 1 from any other summand vanish. Thus d 6 (e) cannot be h 2 1 w 1 a 1 in the ko-homology Adams spectral sequence, so the same is true in the tmf -homology Adams spectral sequence.
Determining whether λ 4 = 0 appears to be difficult. This would be a good problem to address because if λ 4 ̸ = 0, so that d 2 (e) ̸ = 0, then the bordism group controlling the anomaly of the Sagnotti string would vanish, and the anomaly would cancel, at least on the class of backgrounds we studied.
Because the class e potentially causing a nonzero bordism group is in Adams filtration 0, the corresponding bordism invariant is the integral of a modulo 2 characteristic class, explicitly (3.77) The class G ∈ H 7 (BSU (32)⟨c 3 ⟩; Z 2 ) is a little mysterious, so we go into some more detail; it is an example of a secondary characteristic class in the sense of Peterson-Stein [128].
Recall that by a trivialization of a cohomology class z ∈ H k (X; A), where A is an abelian group, we mean a null-homotopy of a map f z : X → K(A, k) whose homotopy class represents z. There is a space of such trivializations, and a standard result in obstruction theory implies that its set of path components is a torsor over H k−1 (X; A). In other words, given two trivializations of f z , their difference is well-defined as an element of H k−1 (X; A).

Spin(16) × Spin(16)
Next we discuss the symmetry type of the non-supersymmetric heterotic string with gauge Lie algebra so(16) ⊕ so (16). Although usually called SO(16) 2 , there are fields transforming in spinor representations in the massless spectrum of the theory which means that we should instead consider Spin(16) 2 . There is a further subtlety: according to [52], the gauge group G is the quotient of Spin(16) × Spin(16) by the diagonal Z 2 subgroup ⟨(k, k)⟩, where k ∈ Spin(16) is either central element not equal to ±1. 16 As the computation of H * (BG) is complicated, we will make a simplifying assumption: only working with the double cover Spin(16) × Spin(16), as we mentioned above. Thus our anomaly cancellation results are only partial information: if we found an anomaly for Spin(16) 2 , it would imply the existence of an anomaly for the actual gauge group G. However, we found that anomalies cancel for Spin(16) 2 , which is only partial information: there could be an anomaly of the theory which vanishes when restricted to gauge fields induced from a Spin(16) 2 gauge field. It would be interesting to address the more general question of the anomaly for G. 17 Let String-Spin(16) 2 be the Lie 2-group which is the string cover of Spin × Spin(16) × Spin (16) corresponding to the degree-4 cohomology class 1 2 1 , where c (i) refers to the cohomology class c coming from the i th factor of BSpin or BSpin(n). 18 Quotienting String-Spin(16) 2 by the Spin(16) 2 factor produces a map to Spin; composing with Spin → O we obtain a tangential structure as usual. 16 Strictly speaking, the analysis of [52] does not take into account the full string spectrum. Therefore, a priori the correct gauge group G may differ from this particular quotient of Spin(16) 2 .
17 Like for any double cover, for any odd prime p, the quotient BSpin(16) × BSpin(16) → BG is a p-primary equivalence, so the lack of p-primary torsion we establish for Spin(16) × Spin(16) remains valid for G. 18 Elsewhere in the paper we have referred to 1 2 p L 1 and 1 2 p R 1 as Chern classes, and indeed they are Chern classes of the representations that play a role in the Green-Schwarz mechanism for this string theory. However, the bordism computation we perform in this section only depends on the characteristic class, not the representation (this is the thesis of [113]), so to emphasize this independence, we use the more intrinsic name 1 2 p1, as this class is one-half of the first Pontrjagin class of the vector representation of Spin(n).
The colors in the theorem statement will be explained below; they correspond to different summands in (an approximation to) MT (String-Spin (16)  → Ω String-Spin 2 k induced by i is an isomorphism. Therefore for the rest of this proof, we can work only with String-Spin 2 bordism without affecting the results.
Concretely, a string-Spin 2 structure on a vector bundle E → X is data of a spin structure on E and two virtual spin vector bundles V L , V R → X and a trivialization of 1 2 p 1 (E) − 1 2 p 1 (V L ) − 1 2 p 1 (V R ). Since 1 2 p 1 is additive in direct sums [46, Lemma 1.6], this is equivalent to a trivialization of 1 2 p 1 (E − V L − V R ), meaning that a string-Spin 2 structure is equivalent to the data of V L and V R and a string structure on W : The data (E, V L , V R ) and (E, W, V R ) are equivalent, as V L = E − W − V R , and the spin structure on V L can be recovered from the spin structures on E, W , and V R by the two-outof-three property (the string structure on W includes data of a spin structure). Therefore the data of a string-Spin 2 structure on E → X is equivalent to the following data: • a spin structure on E, • a virtual string vector bundle W → X, and • a virtual spin vector bundle V R → X.
Taking bordism groups, we learn Ω String-Spin 2 * ∼ = −→ Ω Spin * (BSpin × BString). (3.80) For any spaces A and B, the stable splitting Σ ∞ + (A) ≃ Σ ∞ A ∨ S and its analogue for B together imply a stable splitting implying that for any generalized homology theory h, Here h(X) denotes "reduced h-homology" of a space X, meaning the quotient h(X)/i * (h(pt)) induced by a choice of basepoint i : pt → X. Thus for example Ω Spin * (X) denotes reduced spin bordism, etc.
Apply (3.81b) for h = Ω Spin * , A = BSpin, and B = BString: The colors in (3.81c) indicating the pieces of this direct-sum decomposition correspond to the colors in the theorem statement displaying which pieces of the bordism groups come from which summands in (3.81c). The final step is to determine the four summands in (3.81c).
• Ω Spin * (BString) is not in the literature as far as we know, but in the range we need is easy to calculate: because BString is 7-connected, Ω Spin k (BString) vanishes for k < 8; for 8 ≤ k ≤ 11, use the Atiyah-Hirzebruch spectral sequence. Work of Stong [132] and Giambalvo [133] implies that in degrees 11 and below, for A = Z and Z 2 , H * (BString; A) consists of a single summand isomorphic to A in degree 8, and the remaining groups vanish. This suffices to collapse the Atiyah-Hirzebruch spectral sequence into the blue groups in the theorem statement.
Remark 3.82 (Analogy with E 8 × E 8 ). The two-step simplification of String-Spin(16) 2 (first replace Spin (16) with Spin, then recast as spin bordism of a space) is directly analogous to Witten's [134, §4] simplification of the symmetry type of the E 8 × E 8 heterotic string: first, there is a 15-connected map BE 8 → K(Z, 4), so in dimensions relevant to string theory we may replace the former with the latter; then Witten recast the data of the two maps to K(Z, 4) and the twisted string structure given by the Green-Schwarz procedure as a spin structure and a single map to K(Z, 4).
Remark 3.83 (Analogy with Spin(32) and detecting a non-supersymmetric 0-brane). The same two-step procedure also works for the Spin(32)/Z 2 heterotic string when one restricts to Spin(32)-bundles, showing that the relevant twisted string bordism groups coincide with Ω Spin * (BString), which vanishes in dimension 11. As with Spin(16) × Spin (16), this is only partial information towards a complete anomaly cancellation result.
However, the partial information provided by these bordism groups is already useful: combined with the Cobordism Conjecture [48], it detects Kaidi-Ohmori-Tachikawa-Yonekura's non-supersymmetric 0-brane [50]. To see this, consider Ω Spin 8 (BString) ∼ = Z 3 : two of the Z summands come from Ω Spin * (pt), and as such are generated by HP 2 and the Bott manifold; the third Z summand is represented by S 8 with the map to BString given by the generator of [S 8 , BString] = π 8 (BString) ∼ = Z. Tracing through the simplification from twisted string bordism of BSpin (32) to the spin bordism of BString, we see that this S 8 has the Spin(32)-bundle arising from the generator of π 8 (Spin(32)) ∼ = Z, which is detected by p 2 .
The Cobordism Conjecture predicts that associated to this bordism class (or rather its image in the corresponding bordism group for Spin(32)/Z 2 ), there is a 0-brane in Spin(32)/Z 2 heterotic string theory whose link is S 8 with this Spin(32)-bundle and twisted string structure. This is precisely the 0-brane discovered by Kaidi-Ohmori-Tachikawa-Yonekura [50]. Those authors also discuss a 6-brane in the Spin(32)/Z 2 heterotic string, but its description uses π 1 (Spin(32)/Z 2 ) ∼ = Z 2 , so it is invisible to the Spin(32) computation we made here.

Physical intuition from fivebrane anomaly inflow
As we have just seen, the relevant bordism groups vanish, and therefore there are no Dai-Freed anomalies (except possibly for the Sagnotti string). It is instructive to study the vanishing of anomalies more explicitly in particular examples, to better understand the physics at play. Let us recall from Section 3 the structure of the anomaly theory for ten dimensional theories that feature a Green-Schwarz mechanism: The boundary mode of this eleven dimensional field theory gives exactly the contribution of the Green-Schwarz term to the classical action: In this section, we consider simple backgrounds of the factorized form Y 11 = S 3 × M 8 for the anomaly theory (3.84). We will also take one unit of three-form H flux threading the sphere, so that the Green-Schwarz term gives a nontrivial contribution, and M 8 a spin manifold equipped with a gauge bundle E such that p 1 (M 8 )+c 2 (E) 2 is trivial in integer cohomology. Unlike more general backgrounds, these factorized ones allows for an intuitive understanding of how anomalies are cancelled, via inflow.
On these backgrounds, the eta invariant contribution to the anomaly theory (coming from the fermions) vanishes on account of the factorization property (3.86) where A is odd-dimensional. The eta invariant of fermions on S 3 H vanishes modulo 1, as it is the same as the eta invariant on a three-sphere, which is the boundary of R 4 . As a result, the anomaly theory simplifies to the Green-Schwarz term If we can now show that this quantity is always an integer, Dai-Freed anomalies will vanish on all such factorized backgrounds. This result does not hinge on the the precise bordism groups computed in the preceding section.
In order to prove that eq. (3.87) is always an integer, we can connect it with the anomaly inflow mechanism on a fivebrane. Specifically, S 3 H is a non-trivial bordism class, and one possible boundary for it in string theory is a fivebrane. The fivebrane is a codimension four object, and it is characterized precisely by the fact that the angular S 3 in the transverse space is threaded by one unit of H-flux. These fivebranes are precisely D5-branes in the orientifold models and NS5-branes in the heterotic model. We will now show that X 8 coincides with the anomaly polynomial of such a fivebrane, up to terms which vanish when the Bianchi identity holds. In the presence of fivebranes coupling to B 6 , the dual of B 2 , the classical gauge variation of the effective action is compensated by the quantum anomaly of the chiral worldvolume degrees of freedom. For this inflow mechanism to work, the anomaly polynomial of a single fivebrane has to be I 8 = X 8 (up to terms that vanish on a twisted String manifold). To see this, notice that the bulk action receives additional worldvolume contributions of the form where W denotes the worldvolume of the fivebrane(s) and B 6 the dual of B 2 . The bulk action S bulk , which describes the ten-dimensional effective (super)gravity theory, is accompanied by the Green-Schwarz term S GS of eq. (2.17) to cancel bulk anomalies. The brane is instead described by the worldvolume DBI action S wv accompanied by the magnetic coupling to B 6 , which is the relevant coupling in the following argument. The equation of motion for B 6 and the corresponding dual Bianchi identity are where H 3 = dB 2 and the δ is a distribution-valued four-form that describes the embedding of W in spacetime. Correspondingly, the Bianchi identity for the gauge invariant field strength H ≡ H 3 = dB 2 − ω CS , which ordinarily reads dH = X 4 , also receives a new localized contribution. Because of this Bianchi identity, there is a new classical contribution to the gauge variations. Using descent, 6 , one finds which cancels by inflow provided that the worldvolume theory of the D5-brane has an anomaly polynomial I 8 = µ X 8 [135,136]. With our choice of units, the elementary charge µ = n 5 ∈ Z counts the number of fivebranes. As described in Section 2, the anomaly polynomial is a sum of indices, given by the APS index theorem for each one of the anomalous degrees of freedom propagating on the fivebrane. As such, we know that I 8 is an integer and we can conclude from the previous discussion that X 8 must also be an integer. This anomaly inflow argument thus allows one to show that the anomaly (3.87) always vanishes.
For the orientifold models, this mechanism can be implemented explicitly, since these theories have D5-branes whose worldvolume degrees of freedom are known. We describe this in detail in section 3.4.1.
On the heterotic side, although the SO(16) × SO(16) theory is known to have NS5 branes, their worldvolume degrees of freedom are not known and so we have to resort to other arguments to prove that the anomaly (3.87) vanishes. The proof can be found at the end of Section 3.4.2. There is, however, a more physical way of understanding why anomalies cancel in the heterotic case: one can show that the anomaly polynomial of SO(16) × SO (16) can be directly related to that the supersymmetric heterotic theories. Therefore, one use this connection to show that anomalies cancel for SO(16) × SO (16) by showing that they cancel in the supersymmetric cases. This is done explicitly in Section 3.4.2.
Finally, since we have proven that anomalies vanish in the heterotic case, we can reverse the anomaly inflow argument above to speculate about the worldvolume degrees of freedom of the NS5 brane. Indeed, we identify what kind of degrees of freedom give rise to the correct anomaly polynomial so as to have X 8 = I 8 . We do so away from strong coupling effects, in the puffed-up instanton limit of the NS5 brane. This is detailed in section 3.4.3.

Sp(16) and U (32)
Let us begin with the orientifold models. This cancellation of anomalies by inflow was first constructed for the case of Spin(32)/Z 2 in [135,136]. The chiral fermions on the worldvolume of the D5-brane consist of one vector multiplet of Spin(32)/Z 2 and two gauge singlets, such that, where (X 8 ) Spin(32)/Z 2 and (X 4 ) Spin(32)/Z 2 can be read off from (2.13). This shows how one recovers I 8 = X 8 up to a term that vanishes on a twisted string manifold.
One may wonder where the extra term in (3.91) comes from, even if we know it to vanish on a twisted string manifold. This can be understood as follows; from the perspective of the ten-dimensional supergravity action, a D5-brane amounts to introducing a delta function localized on the brane. The D5-brane gives a localized contribution to the 10d action of the form B 2 ∧Y 4 ∧δ 4 where Y 4 is some 4-form which in this case is reduces to Y 4 = − 1 24 p 1 , expanding the A-roof genus in the Chern-Simons effective worldvolume action. Indeed, using the Bianchi identity for the H 3 flux, we see that this term contributes to the anomaly polynomial as: Therefore, the appearance of the extra term in (3.91) can be traced down to not properly taking into account the delta-function source that corresponds to the localized D5-brane. The same mechanism happens in the two non-supersymmetric orientifold models, as was found by [137] along the lines of [136,138]. The worldvolume degrees of freedom on D5-branes can be extracted from one-loop open-string amplitudes [139], and the chiral fermions arrange in the virtual representation of SO(N ) × Sp(16) (for the Sugimoto model 19 ) or U (N ) × U (32) (for the Sagnotti model).
In order to compare the anomaly polynomial I 8 (without worldvolume gauge field) with the bulk X 8 , one needs to decompose characteristic classes of the bulk tangent bundle in terms of the worldvolume tangent bundle T W and normal bundle N . In detail, (3.94) 19 In this case, a single brane corresponds to N = 2.
When the normal bundle of the worldvolume is trivial, one obtains and therefore the inflow mechanism implies that X 8 integrates to an integer on any spin 8-manifold with X 4 = 0. When the normal bundle is non-trivial, there are additional contributions to the above expression, proportional to the Euler class of N . However, the full brane action also contains another term [135,136] proportional to B 2 rather than B 6 , which induces another classical variation to be canceled by inflow. As a result, the anomaly polynomial of the fivebrane worldvolume theory is not quite the above I 8 , but has an additional contribution that cancels the normal bundle terms [136]. In more detail, adding a coupling of the type W B 2 Y 4 to the fivebrane worldvolume action contributed a new classical variation to the effective action, which arises by descent from ∆I 8 = − (X 4 + n 5 χ(N ))Y 4 . Therefore, the full anomaly polynomial of the fivebrane worldvolume ought to be I 8 = n 5 X 8 − ∆I 8 , again up to terms that vanish on twisted String backgrounds. This additional coupling can be shown to cancel the normal bundle terms in the anomaly [136] (see also [140] for a discussion in the context of M-theory).

SO(16) × SO(16)
For the heterotic model, no such result is available, since the worldvolume degrees of freedom of NS5-branes are not understood without supersymmetry or dualities at one's disposal. However, one can nonetheless express X 8 as an index of six-dimensional chiral fields; since index are manifestly integers, this will be enough to establish that anomalies cancel. In order to do so, let us observe that the formal difference of representations of the chiral fermions of the non-supersymmetric heterotic model can be rewritten as 20 The matter fields in the first line after the equal correspond precisely to the decomposition of the adjoint of e 8 ⊕ e 8 into representations of the so 16 ⊕ so 16 subalgebra; they are the field content that would arise after giving a vev to an adjoint e 8 ⊕ e 8 field. Similarly, the fields in the second line are (with reversed chirality) those fields that would arise after adjoint Higgsing from the so(32) algebra to its so 16 ⊕ so 16 subalgebra. What we are seeing here is that, at a formal level (as far as the chiral spectrum is concerned), the SO(16) 2 is equivalent to one copy of the E 8 × E 8 string stacked on top of a copy of the Spin(32)/Z 2 string, with opposite chirality, and Higgsed to a common subgroup with algebra so 16 ⊕ so 16 . Therefore, we can write, at the level of anomaly polynomials, the equality (3.97) where we have merely restricted to SO(16) 2 bundles inside of the two groups above. Since each of the supersymmetric string theories are anomaly-free by themselves, the formal linear combination will also be. This argument, which can be carried out at the level of eta invariants etc. and not just anomaly polynomials, is yet another proof of the fact that the SO(16) 2 theory is anomaly free 21 , without relying explicitly on bordism calculations. Furthermore, in particular, this holds for the Green-Schwarz terms, which are It is unclear whether this connection between the non-supersymmetric SO(16) × SO (16) theory and the supersymmetric theories persists beyond a formal equality at the level of (super)gravity, or whether on the contrary it has a deeper meaning. Some previous work [141,142] (see also [143]) identified connections between supersymmetric and non-supersymmetric strings via interpolating models, which are nine-dimensional compactifications recovering either supersymmetric or non-supersymmetric strings in different decompactification limits.
In particular, an interpolating model was constructed between the SO(16) × SO(16) theory and the supersymmetric Spin(32)/Z 2 theory, matching the worldsheet CFT descriptions and solitons in between the two 22 . The cancellation of anomalies by fivebrane inflow for the SO(16) × SO(16) theory thus follows from that of the two supersymmetric heterotic theories. The anomaly inflow in the case of Spin(32)/Z 2 was discussed above (3.91). The case of E 8 × E 8 is slightly more involved and we will discuss it now. The anomaly inflow of the NS5-brane was famously discussed in [148] where the limit in which an instanton in E 8 × E 8 becomes point-like was matched to the world volume theory of the NS5 brane at strong coupling. For our purposes, we can ignore strong coupling dynamics and focus on matching a 6-dimensional anomaly theory of chiral fermions to (X 8 ) E 8 ×E 8 which can be read off from (2.14). This guarantees that (X 8 ) E 8 ×E 8 is an integer and that local anomalies cancel in 10d. We now detail how this can be done.
One can show that (X 8 ) E 8 ×E 8 can be decomposed as follows: where I SD +2 I Dirac is the index of a self-dual form field and 2 fermion singlets in 8 dimensions.
The index of a self-dual form field in 8 dimensions can be shown to be an integer over 8 [42]. Indeed, it can be written in terms of the signature of the 8-manifold as follows [102]: On the other hand, the index of chiral fermions is always an integer. In order to simplify the first term in (3.99), we can rewrite the Chern classes in an embedded SU (2) subgroup of each E 8 , which are known to be integer-valued. Therefore, on a twisted string manifold, X 8 reduces to: 2,2 are the 2nd Chern classes in the fundamental of the SU(2) subgroup of the i-th E 8 . The Bianchi identity (X 4 ) E 8 ×E 8 = 0 gives us Plugging this into (3.101), we see that the condition for anomalies to vanish comes down to showing that the following quantity is an integer: As it happens, it was shown in [102] that the last two terms give (28 times) an integer. Indeed, one can show that: Now, to show that the first two terms of (3.103) are an integer, one can note that on a twisted string manifold, p 1 2 is a characteristic vector of H 4 (X; R). This, in particular, means that: Therefore we have shown that on a twisted string manifold, X 8 is always an integer; and so there can never be an anomaly. Given that the X 8 of SO(16) × SO(16) is a linear combination of those of E 8 × E 8 and Spin(32)/Z 2 , we can infer that (X 8 ) SO(16) 2 is an integer and so that all anomalies vanish for this non-supersymmetric theory. Nevertheless, for completeness, let us detail explicitly how (X 8 ) SO(16) 2 can be proven to be an integer. One can write (X 8 ) SO(16) 2 as follows: 16,2 + 3p 1 ) + I

(1)
Dirac is the contribution of a fermion that transforms in the 16 of SO(16) i , which is known to be integer-valued. Therefore, on a twisted string manifold, the cancellation of anomalies comes down to showing that the following quantity is an integer: Given that one can put the 2nd Chern classes in the SU(2) subgroup of SO(16) as c (i) 16,2 → 2c (i) 2,2 ; the proof goes exactly as in the E 8 × E 8 case.
One can sometimes read-off the chiral field content of a theory from the anomaly polynomial. For instance, for the E 8 × E 8 case, the anomaly polynomial (3.99) suggests that the chiral field content of the NS5 brane is a self-dual form field and 2 fermion singlets. There are no chiral fields charged under the gauge group, since all the gauge-dependent parts of (3.99) are in the factorized piece. As it happens, this exactly the chiral field content of a 6d (1, 0) tensor multiplet, which is precisely the worldvolume field content of the NS5 brane in E 8 × E 8 string theory. This answer is essentially determined by anomalies together with supersymmetry. In the non-supersymmetric case of the SO(16) 2 string, reading off the chiral field content from (3.106) in the same way suggests that the chiral field content of the SO(16) 2 NS5 brane is: • Four fermion singlets, • A fermion transforming in the (16, 1) ⊕ (1, 16) of SO(16), • A self-dual 2-form field.
There are some subtleties in assessing whether or not these are truly the chiral degrees of freedom propagating on this non-supersymmetric brane. First of all, there is no supersymmetry to constrain the worldvolume theory of the NS5 brane which can therefore carry any kind of chiral degrees of freedom. As we have seen from (3.104) and (3.100), indices can sometimes be exchanged for one another and yet give the same integer. This means that the X 8 does not completely fix the worldvolume content of the NS5 brane, and that any chiral field content with the same anomaly as the one proposed above remains a possibility. Another reason why we cannot be sure that (3.106) correctly describes the degrees of freedom propagating on the NS5 brane is that we cannot be sure that an NS5 brane (understood as a small instanton where the full spacetime gauge group symmetry gets restored) exists to begin with. Unlike in the supersymmetric case, in general we expect that the size modulus of the instanton, being non-supersymmetric, receives a potential due to quantum effects that may lead to the small instanton limit being obstructed. The study of the strong coupling effects near the small instanton limit is beyond the validity of effective field theory, and thus beyond the scope of this paper (although it may be amenable to a version of the constructions in [50]), but we point out that, if the limit does exist and the small instanton transition does survive, the transition point would be a natural place to look for a non-supersymmetric interacting CFT, a cousin of the E 8 SCFT. It would be interesting to explore this further. On the other hand, studying the anomaly inflow on the worldvolume of the puffed-up NS5 brane instanton is accessible within the effective field theory (see fig.7). We do this explicitly in the next section. Figure 7. A sketch of a fivebrane puffing up into an instanton which can be described within the effective field theory.

Anomaly inflow on puffed-up fivebrane instantons
The above result shows that there are no Dai-Freed anomalies on factorized backgrounds of the form S 3 H × M 8 , since X 8 integrates to an integer. Anomaly inflow on fivebranes dictates that X 8 be the anomaly polynomial associated to the worldvolume theory on a single fivebrane, possibly up to terms that vanish when the Bianchi identity is satisfied. As explained above, without direct access to the relevant degrees of freedom on the fivebrane worldvolume, studying the anomaly inflow on the point-like NS5 brane is impossible. Luckily, one can still examine the anomaly inflow on the worldvolume of puffed-up fivebrane instantons, which can be described in the low-energy approximation. Puffing-up the fivebrane corresponds to delocalizing it along its transverse dimensions, making it look like a four dimensional gauge instanton.

(3.111)
The next step is to evaluate X 8 on this background. This amounts to decomposing characteristic classes according to the branching rules, and one finds 2,2 ) 2 should be replaced by zero, since it is proportional to the square of the worldvolume current δ(W ) of the fivebrane. All in all, when the dust settles one arrives at where the second factor corresponds to X 4 for the unbroken piece of the gauge group. The inflow therefore works when X 8 and P 8 are equal on manifolds where the Bianchi identity holds. A similar argument works for more general choices of instanton bundles. For the orientifold models, one expects the small limit of the "fat" fivebrane instantons to yield the worldvolume degrees of freedom of D5-branes. This is a nice crosscheck that we detail now. For the Sugimoto model (the calculation is identical in the Sagnotti model), the anomaly polynomial P 8 associated to the fermion zero modes of the instanton is where the only charged contribution comes from the first term on the right-hand side. In the small limit, the SU (2) Chern classes vanish and the remaining ones are enhanced to Sp (16) classes, ending up with Thus reproducing the anomaly polynomial of a D5-brane worldvolume up to terms that vanish on the allowed backgrounds.
4 Anomalies and bordism for the swap Z 2 action

Overview and the bordism computation
The E 8 × E 8 heterotic string theory has a Z 2 symmetry given by swapping the two copies of E 8 , so it is possible to expand the gauge group of the theory to (E 8 × E 8 ) ⋊ Z 2 . To our knowledge, this fact first appears in [52, §I] (see also [149, §2.1.1]). The question of anomaly cancellation for this string theory is completely different in the absence versus in the presence of this extra Z 2 : without it, the anomaly is known to vanish, as Witten [134, §4] showed it is characterized by a bordism invariant Ω Spin 11 (BE 8 ) → C × , and Stong [150] showed Ω Spin 11 (BE 8 ) ∼ = 0. But with the Z 2 swapping symmetry turned on, the relevant bordism group has order 64 [46, Theorem 2.62] courtesy of a harder computation; even though we cannot determine this group exactly, we will show that the anomaly vanishes, in accordance with the results in [106] obtained from a worldsheet perspective.
In this section, we discuss a closely analogous story for the Spin(16) × Spin(16) nonsupersymmetric heterotic string. The gauge group Spin(16) × Z 2 Spin(16) (where the diagonal Z 2 we quotient by corresponds to either of the subgroups in each Spin(16) whose quotient is not SO(16)) admits a Z 2 automorphism switching the two Spin(16) factors, enlarging the gauge group of this theory to (Spin(16) × Z 2 Spin(16)) ⋊ Z 2 ; see [52,§III].
In this paper, we chose to work with Spin(16) × Spin (16), which simplifies the bordism computations at the expense of applying to only some backgrounds. The Z 2 symmetry enlarges the gauge group to G 16,16 := (Spin(16)×Spin(16))⋊Z 2 . The Green-Schwarz mechanism is analogous: if x ∈ H * (BSpin(16); A) for some coefficient group A, let x L and x R denote the copies of x in H * (B(Spin(16) × Spin(16)); A) coming from the first, resp. second copies of Spin(16) via the Künneth formula. Then the class 1 2 p L 1 + 1 2 p R 1 , which was the characteristic class of the Green-Schwarz mechanism in the absence of the Z 2 symmetry, descends through the Serre spectral sequence for the fibration B(Spin(16) × Spin (16) to define a class in H * (BG 16,16 ; Z), and the Green-Schwarz mechanism asks, on a spin manifold M with a principal G 16,16 -bundle P → M , for a trivialization of   where either i = 1, j = 4, and k = 4, or i = 2, j = 6, and k = 5, and A is an abelian group of order 64 isomorphic to one of The fact that Ω G 16, 16 11 ̸ = 0 implies that the Spin(16) × Spin(16) heterotic theory with its Z 2 swapping symmetry could have an anomaly; we will nevertheless be able to cancel it later in this Section.
Proof. The proof is nearly identical to the analogous calculation for the E 8 × E 8 heterotic string, which is done in [46, §2.2, §2.3]; therefore we will be succinct and direct the reader there for the details. Let V → BSpin(16) × BSpin(16) be the direct sum of the tautological vector bundles on the two factors. The Z 2 swapping action on BSpin(16) × BSpin(16) lifts to make V into a Z 2 -equivariant vector bundle, so V descends to a vector bundle we will also call V over BG 16,16 . Since the action of Z 2 is compatible with the spin structures on the two tautological bundles, V → BG 16,16 is spin, so w 1 (V ) = 0 and w 2 (V ) = 0; and essentially by definition, 1 2 p 1 (V ) = 1 2 p L 1 + 1 2 p R 1 . Therefore just as for the other theories we studied, there is an isomorphism Ω (4.5) In the degrees relevant to us, H * (BSpin (16); Z 3 ) is generated by the Pontrjagin classes p 1 and p 2 with no relations in degrees 11 and below, so we obtain the following additive basis for H * (BG 16,16 ; Z 3 ) in degrees 11 and below: 1, Using this, we determine the A tmf -module structure on H * ((BG 16,6 ) V −32 ; Z 3 ) using [113,Corollary 2.37]: if U denotes the Thom class, β(U ) = 0 and P 1 (U ) = −U (p L 1 + p R 1 ) (as 1 2 x = −x in a Z 3 -vector space). Using this and the Cartan formula, we find an A tmf -module isomorphism H * ((BG 16,16 (4.6) where N 3 is as in definition 3.54 and P is concentrated in degrees 12 and above, and will be irrelevant for us. We draw (4.6) in fig. 8 [151] of a theorem of Nakaoka [152,Theorem 3.3] gives us the following additive basis for these cohomology groups in degrees 13 and below: • classes of the form c L + c R , where c ranges over a basis of H * (BSpin(16); Z 2 ) in degrees 13 and below; • the classes w L 4 w R 4 , w L 6 w R 6 , w L 4 w R k + w L k w R 4 for k = 6, 7, 8, and (w L 4 ) 2 w R 4 + w L 4 (w 2 4 ) R ; and • finally, we have classes of the form x m , w L 4 w R 4 x m , and w L 6 w R 6 x m , where x is the generator of H 1 (BZ 2 ; Z 2 ), pulled back by the quotient G 16,16 → Z 2 by the normal Spin(16) × Spin(16) subgroup.   Figure 9. The A(2)-module structure on H * (B((Spin(16) × Spin (16)) ⋊ Z 2 ) V −32 ; Z 2 ) in low degrees.
The figure includes all classes in degrees 13 and below. Here α :  This means that, for our string-theoretic applications, it does not matter whether we use BG 16,16 or BG ′ 16,16 .
Proof. We want to show that the map MT G 16,16 → MT G ′ 16,16 of bordism spectra is an isomorphism on π k for k ≤ 14. By the Whitehead theorem we may equivalently use H k (-; Z), and by the Thom isomorphism, it suffices to show the map BG 16,16 → BG ′ 16,16 is an isomorphism on Z-cohomology in degrees 14 and below. The cohomology rings of these spaces can be computed in two steps: first the Serre spectral sequence for the fibration B(Spin(16) × Spin(16)) → BG 16,16 → BZ 2 , then the Serre spectral sequence for the fibration B 2 U (1) → BG 16,16 → BG 16,16 ; and analogously for BG ′ 16,16 with Spin in place of Spin (16). For each of these two steps, the map Spin(16) → Spin induces a map of Serre spectral sequences. and because H * (BSpin; Z) → H * (BSpin (16); Z) is an isomorphism in degrees 15 and below, we learn that at each of the two steps, the two spectral sequences are isomorphic in degrees 14 and below, which implies the map BG 16,16 → BG ′ 16,16 induces an isomorphism on cohomology in degrees 14 and below. such that the pullback map on cohomology corresponding to the projection MT G ′ 16,16 → Q is a map given by the inclusion of an A(2)-module L → H * ((BG 16,16 ) V −32 ; Z 2 ), followed by applying A ⊗ A(2) -; the quotient of L by all classes in degrees 14 and above is isomorphic to To see the statement on cohomology, one can look at the edge morphism in the Serre spectral sequence for B 2 U (1) → BG ′ 16,16 → BG ′ 16,16 .
As we already know spin bordism groups in the dimensions we need, we focus on computing π * (Q) ∧ 2 . Because the cohomology of Q is of the form A ⊗ A(2) L, the change-of-rings theorem simplifies the Adams spectral sequence for Q to the form (4.14) we will then add on the summands coming from Ω Spin * to obtain the groups in the theorem statement. The first thing we need is Ext A(2) of M 2 , M 3 , M 4 , M 5 , M 7 , M 8 , and M 9 .
5. Finally, for M 7 , M 8 , and M 9 , we only need to know their Ext groups in degrees 12 and below. For i = 7, 8, 9, there is a surjective map M i → Σ 12 Z 2 whose kernel is concentrated in degrees 14 and above, so (e.g. using the long exact sequence in Ext associated to a short exact sequence of A(2)-modules [109, §4.6]) for t − s ≤ 12, Ext of each of these modules is isomorphic to Ext(Σ 12 Z 2 ), which was computed by May (unpublished) and Shimada-Iwai [156, §8].
The next step is to evaluate the differentials. Unlike the other Adams spectral sequences we considered in this paper, there are several differentials to address, even after using that differentials commute with the action of h 0 , h 1 , and h 2 : • d 2 on a 1 , a 2 , a 3 , c, d, e, f , and g, • d 3 on a 1 , a 2 , and a 3 , and  Proof. If ξ ′ denotes the tangential structure identical to G 16,16 except with K(Z, 4) in place of BSpin (16), then the class 1 2 p 1 , interpreted as a map BSpin(16) → K(Z, 4), induces a map of tangential structures from G 16,16 -structure to ξ ′ -structure, hence also a map of Thom spectra, hence a map of Adams spectral sequences. The E 2 -page for Ω ξ ′ * is computed in [46, Figure 3] in the range t − s ≤ 12, and looks very similar to our E 2 -page in fig. 10; using the comparison map between these two spectral sequences, we conclude the differentials in the lemma statement.
The comparison map would also tell us d 2 (c), except that the fate of this differential in ξ ′ -bordism is not known.
Lastly, we address the class d ∈ E 0,11 2 . Since d has topological degree 11, its fate affects the size of Ω G 16, 16 11 , hence the possible anomaly theories for the Spin(16) × Spin(16) theory.
Lemma 4.20. Y 11 has a spin structure.
Proof. To prove this, we will stably split the tangent bundle of Y 11 . This is a Recall that, since the normal bundle to S k → R k+1 is trivialized by the unit outward normal vector field ⃗ v, there is an isomorphism ϕ : T S k ⊕R ∼ = R k+1 ; since ⃗ v is O(k+1)-invariant, ϕ promotes to an isomorphism of O(k + 1)-equivariant vector bundles, where O(k + 1) acts trivially on the normal bundle and via the defining representation on R k+1 .
Applying this thrice, we have an isomorphism of vector bundles The Z 2 -action on S 4 × S 4 × S 3 we used to define in Y 11 in definition 4.18 extends to a linear action on R 5 × R 5 × R 4 , upgrading (4.21) to an isomorphism of Z 2 -equivariant vector bundles. In a little more detail: • Z 2 acts on T (S 4 × S 4 × S 3 ) as the derivative of the involution (4.19).
• Z 2 acts on R 5 ⊕ R 5 ⊕ R 4 as the Z 2 -representation described by the same formula (4.19).
• Z 2 acts on the normal R 3 by inverting and swapping the first two coordinates, and inverting the third: (x, y, z) → (−y, −x, −z). 23 The isomorphism (4.21) of Z 2 -equivariant vector bundles descends through the quotient by Z 2 to an isomorphism of vector bundles on Y 11 ; trivial bundles made equivariant by a Z 2representation descend to vector bundles associated to that representation and the principal In particular, if σ π → Y 11 denotes the line bundle associated to π and the sign representation σ of Z 2 on R and R denotes the trivial representation, then the Z 2 -representation (x, y) → (−y, −x) on R 2 is isomorphic to σ ⊕ R. Using this, we obtain an isomorphism of vector bundles Therefore we have an isomorphism of virtual vector bundles For any vector bundle V , V ⊕4 is spin, as can be verified with the Whitney sum formula, and the existence of a spin structure is an invariant of the virtual equivalence class of a vector bundle, so we can conclude.
P → Y 11 is nonzero. To do this, first factor f P into the following diagram of three fibrations: Here X is the S 4 × S 4 -bundle over BZ 2 defined analogously to Y 11 but using S ∞ = EZ 2 instead of S 3 . The map j : S 4 × S 4 → BSpin(16) × BSpin (16) is the map classifying L and L * . The diagram (4.26) induces maps between the Serre spectral sequences of the three fibrations; using it, one can compute the pullback of w L 4 w R 4 x 3 to Y 11 and see that it is nonzero, as promised. The last step in this calculation is to address extensions. The argument is nearly identical to [46, Lemma 2.59 and Proposition 2.60], though one now has the extra classes h k 0 a 1 for k ≥ 0, h 1 a 1 , and h 2 1 a 1 in degrees 8, 9, and 10 respectively which were not present in the E 8 × E 8 spectral sequence. Fortunately, this new ambiguity is fully resolved by applying the "2η = 0 trick" to classes of the form h 1 x in standard ways, for example as in [157,Corollary F.16(2)], [158, (5.47)  Proof. It suffices to find a bordism invariant ψ : Ω G 16, 16 8 → Z m for some m such that the value of ψ on (S 8 , P ) is linearly independent from the values on the other generators, and also not a multiple of any other element of Z m . For X 8 and RP 7 × S 1 nb , this will be vacuously true, because the G 16,16 -bordism classes of these manifolds are torsion, so we focus on the two HP 2 s and the Bott manifold described in [46, §2.2.1 (8)].
Let m = 2 and ψ be given by the two Z-valued invariants p 2 (M ) and (p 2 (V L ) + p 2 (V R )). The latter is a priori an invariant of manifolds with a Spin(16) × Spin(16)-bundle, but it survives the Serre spectral sequence to define an invariant of G 16,16 -bundles and therefore of G 16,16 -manifolds.
For the G 16,16 -structure on the Bott manifold and both G 16,16 -structures on HP 2 specified in [46, §2.2.1(8)], p 2 (M ) ̸ = 0. However, the former because T S 8 is stably trivial and the latter somewhat tautologically from the definition of V . Therefore ψ(S 8 , P ) is linearly independent from ψ evaluated on the other bordism classes we have considered. Finally, we know that the bordism class of (S 8 , P ) is not a multiple of some other class because (4.29b) is ±1, and if the class of (S 1 , P ) were a multiple, the values of all Z-valued bordism invariants on it would be divisible by some natural number greater than 1.
Thus we have found generators for all classes except for c and h 1 c, which may or may not be trivial, depending on the value of an Adams differential.

Cancelling the anomaly
Now that we have the generators of Ω G 16, 16 11 in hand, we proceed to calculate the partition function of the anomaly theory on these generators and show that it is trivial. We are able to do this without knowing the isomorphism type of Ω G 16, 16 11 , similarly to Freed-Hopkins' approach in [41]. Proof. Recall that α ∼ = α f ⊗ α X 8 , where α f is the anomaly of the fermionic fields and α X 8 is the anomaly coming from the term − B 2 ∧ X 8 (2.17) that the Green-Schwarz mechanism adds to the action, as we discussed in §2.
We will calculate α on a generating set for Ω G 16, 16 11 . Based on [51] (16) gauge bundles are trivial: the Z 2 symmetry switches two copies of the trivial bundle. This implies X 8 = 0, so α X 8 is trivial. For α f , we must calculate the η-invariants of the spinor bundles associated to the gauge bundles. We will first dimensionally reduce our theory on B, to obtain a 2d effective theory, and study the corresponding anomaly, which is the dimensional reduction of α f , on RP 3 . As the defining property of a Bott manifold is that the Dirac index is 1, and the gauge bundle is switched off in our example, the 2d spectrum is identical to the ten-dimensional one, so showing that the anomaly on RP 3 is trivial will imply α f (B × RP 3 ) = 1.
We need to know the gauge bundle on RP 3 , 24 but because the gauge bundle is trivial on B, we can describe the G 16,16 -bundle on RP 3 as induced from the Z 2 -bundle S 3 → RP 3 . Thus we should see how the G 16,16 -representations describing the fermions branch when we restrict to Z 2 .
• The 10d fermions in the (128, 1) ⊕ (1, 128) give a total of 128 2d fermions transforming as singlets of the swap, and another 128 transforming in the sign representation.
• For the 10d fermions in the (16,16), the swap is implemented via a matrix with sixteen blocks each having eigenvalues ±1, again giving 128 fermions in each of the trivial and sign representations of the swap Z 2 . Since the 10d fermions have opposite chirality to those in the previous point, the resulting 2d fermions also come in opposite chirality.
With these matter assignments, we obtain a total of 128 Z 2 charged fermions of each chirality, which collectively are anomaly-free (and therefore, gravitational anomalies cancel). Therefore, there is no anomaly under the swap on any background, such as RP 3 : the η-invariants all cancel out. Thus α f (B ×RP 3 ) vanishes and the overall anomaly α f ⊗α X 8 vanishes on B ×RP 3 . For Y 11 , which is an (S 4 × S 4 )-bundle over RP 3 , we perform a twisted compactification on S 4 × S 4 and study the anomaly of the resulting 2d theory on RP 3 . Because Y 11 is not a product, we must take a little more care with this procedure, but it is not so difficult to show that the assignment from a string 3-manifold N with principal Z/2-bundle P → N to the manifold κ(N ) := (S 4 × S 4 ) × Z 2 N, (4.32) where the two copies of S 4 are given the same Z 2 -action and Spin(16)-bundles as we used in the construction of Y 11 , produces a G 16,16 -manifold for all N and is compatible with bordism, allowing κ to define a functor of bordism categories and therefore a twisted compactification as promised.
The covering S 4 has Spin(16) 2 bundles characterized by a second Chern class c SO(16),i 2 = (−1) i (b 1 + b 2 ), (4.33) 24 In general keeping track of tangential structures on dimensional reductions can be complicated (see, e.g., [159, §9], but because B has a string structure and the tangential structure of the theory is a twisted string structure, we do not need to worry about this detail. where b 1 , b 2 are the volume forms of both S 4 factors. Now, rather than explicitly computing the dimensional reductions of α f and α X 8 on RP 3 , we take advantage of the fact that α is a deformation invariant, so we may deform our 2d theory into something where the value of the anomaly on RP 3 is more obviously trivial. 25 Specifically, we can take a limit in moduli space where the instantons become singular and pointlike, turning into a non-supersymmetric version of the heterotic NS5-brane; as explained in Section 3.4.2, the resulting theory becomes symmetric between the two Spin(16) factors, implying that, just like in the supersymmetric heterotic string theories, small instantons of both gauge factors are identified. After deforming in this way the gauge bundle on both Spin(16) factors, we are left with a single pointlike NS5 and a single anti-NS5 in each sphere, which annihilate, leading to a trivial and therefore anomaly-free configuration for the compactified theory, and implying that α(Y 11 ) = 1.
As a bonus, we can answer a question of [46], giving a bordism-theoretic argument for the analogous anomaly cancellation question for the E 8 × E 8 heterotic string. This anomaly cancellation result was first established by Tachikawa-Yamashita in [42] by a different argument.
Corollary 4.34. The anomaly field theory α for the E 8 × E 8 heterotic string theory taking into account the Z 2 swap symmetry is trivial.
Proof. The argument for Y 11 also works in the supersymmetric E 8 × E 8 theory, since the instantons may also be embedded in E 8 . In the supersymmetric case, the pointlike limit of the instanton is the ordinary, supersymmetric heterotic NS5-brane, as illustrated in [148], so the E 2 8 anomaly vanishes on Y 11 . For B × RP 3 , dimensional reduction leads to 248 singlet and 248 fermions (from the E 8 adjoints) charged under the sign representation. Since the relevant anomaly is controlled by and 248 is a multiple of 8, we conclude there is no swap anomaly either. Finally, we already know that gravitational anomalies cancel in B × RP 3 , since if we forget about the swap this is just an ordinary string background.
In summary, we have shown that anomalies vanish under both generators of the swap bordism group, both for Spin(16) 2 and E 8 × E 8 . The supersymmetric case is covered by the worldsheet analysis in [42], which takes into account twists including the swap we just discussed. Thus, we recover a special case of the general anomaly cancellation result there. On the other hand, our approach covers the non-supersymmetric SO(16) 2 case (for the case of geometric target spaces only).

Conclusions
Our world is non-supersymmetric, and that fact alone means that non-supersymmetric corners of the string landscape warrant much more attention than they have received so far, both as a source of interesting backgrounds that might connect more directly to our universe, as well as a new trove of data to check and refine Swampland constraints. In this paper we have moved a bit in this direction by computing the bordism groups and anomalies associated to twisted string structures in the three known non-supersymmetric, tachyon free string models in ten dimensions. The results we obtained are summarized in Table 1 for the Sugimoto and Spin(16) 2 groups; for the more complicated Sagnotti 0'B model, we were just able to show that there is a potential Z 2 anomaly. Table 1. Twisted string bordism groups computed in this paper for the Spin(16) 2 theory with and without including the swap (second and third columns), for the Sugimoto string (fourth column), and for the Sagnotti string (fifth column). In the second column, i, j, k are unknown integers, and A is an abelian group of order 64 (see Section 4 for details). In the fifth column, there are ambiguities due to undetermined differentials in the Adams spectral sequence; see §3.3.2 for details. In some cases, the bordism group vanishes in degree 11, which automatically implies the corresponding theory has no anomalies; we also show the anomaly can be trivialized for the Z 2 outer automorphism of the Spin(16) 2 string, even though the bordism group is nonzero. The results in this table can be further used to classify bordism classes and predict new solitonic objects in these non-supersymmetric string theories following [48,51].
From the results of the table, it is clear that both Spin(16) 2 and Sugimoto models are free of global anomalies. One might have expected this from the fact that they have a consistent worldsheet description. However, there can be non-perturbative consistency conditions that are not automatically satisfied by the existence of a consistent worldsheet at one-loop, see for instance [160], where a K-theory tadpole is not detected by the closed string sector. It would be very interesting to determine in full generality whether existence of a consistent worldsheet is sufficient to guarantee consistence of the target spacetime. Although we have not settled the question of consistency in the Sagnotti string, we expect that it is also free of anomalies; for instance, upon circle compactification, it can be related to a "hybrid" type I' setup involving an O8 + plane and an O8 − , both of which are individually consistent [137].
Perhaps the more interesting result of our work is Table 1 itself, listing the bordism groups of the Spin(16) 2 and Sugimoto theories. An obvious follow-up to this paper is to use the Cobordism Conjecture [48] together with the groups in Table (1) to predict new, non-supersymmetric objects in the non-supersymmetric string theories, similarly to what has been done in type II in [51]. While it is natural to expect these new branes to be nonsupersymmetric, it may be worthwhile to pursue this direction in more detail.
One subtlety that must be kept in mind, when considering our results, is that we did not necessarily use the correct global form of the gauge group in our calculations. With the exception of the SO(16) 2 ⋊ Z 2 , we focused on simply connected versions of all the groups, which immensely simplified the calculations. Since any bundle before taking a quotient is still an allowed bundle after taking the quotient, our results show that a very large class of allowed bundles in the Sugimoto and SO(16) 2 theories are anomaly free 26 , but particularly in the Sugimoto case there may be more bundles to check if the gauge group is actually Sp(16)/Z 2 . In the type I theory, we know the group is Spin(32)/Z 2 and not just SO(32) due to the presence of K-theory solitons transforming in a spinorial representation [161]. In the Sugimoto theory, the relevant K-theory is symplectic, and there do not seem to be any such solitonic particles [4], suggesting that the group might actually be Sp(16)/Z 2 . It would be interesting to elucidate this point and figure out whether there really are any global anomalies beyond those studied here.
Another result of our paper is a series of arguments and checks that in any heterotic string theory, the Bianchi identity must hold at the level of integer coefficients. Furthermore, satisfying the Bianchi identity even at the level of integer coefficients is not enough to guarantee consistency of the string background; there is also a consistency condition (tadpole) that is detected by H 3 (M ; R). The general consistency condition is of course that the anomaly of probe strings vanishes; more generally, it is natural to expect that all consistency conditions (tadpoles) of any quantum gravity background come from consistency of probe branes in said background.
Another limitation of our study is that, by following a (super)gravity approach, we must restrict to studying anomalies on smooth backgrounds. String theories, both with and without spacetime supersymmetry, make sense on much larger classes of backgrounds that do not admit a geometric description, such as orbifolds, and which are only analyzed from a worldsheet perspective. These cases are not covered by our analysis. Using modular invariance one can show that the Green-Schwarz mechanism always cancels local anomalies in any consistent worldsheet background [70,162], with or without spacetime supersymmetry. The question of whether global anomalies also cancel in these non-geometric backgrounds was addressed in [42,45], where all global anomalies are shown to cancel for all gauge groups and dimensions in the ordinary supersymmetric heterotic string theories. This remarkable result rests on the validity of the Segal-Stolz-Teichner conjecture [163], which connects deformation classes of worldsheet theories (or, more generally, two-dimensional (0, 1) supersymmetric QFTs) to the spectrum of (connective) topological modular forms (TMFs) [116,164] (see also [165]). The physical interpretation of this more refined generalized cohomology theory is related to "going up and down RG flows" [166], and it includes the familiar string bordism deformations of the target space manifold of a sigma model as well as more exotic, "non-geometric" deformations.
To construct an ordinary, spacetime-supersymmetric heterotic model, all that one needs is a (0, 1) SQFT. Such a QFT always has a notion of a right-moving worldsheet fermion number F w R , which is gauged by the usual GSO projection to construct a modular-invariant partition function. The original Segal-Stolz-Teichner conjecture applies precisely to (0, 1) SQFT's. If we wanted to make such an argument for a spacetime non-supersymmetric string theory (tachyonic or not), we face the obstacle that the GSO projection is different, and it involves additional worldsheet symmetries. For instance, the SO(16) 2 theory has a "diagonal" modular invariant partition function, which requires a notion of a left-moving worldsheet fermion number in addition to the (0, 1) SQFT structure. Thus, valid SO(16) 2 worldsheet theories are equipped with an additional left-moving Z 2 symmetry, or equivalently, they are equipped with both a spin structure and a Z 2 symmetry. To repeat the argument of [42,45], one must work with Z 2 -equivariant TMF; it would be very interesting to do so.
When we started this project we were actually quite surprised that we could not find a comment on global anomalies of non-supersymmetric tachyon-free strings 27 anywhere in the literature. After all, these constructions are all 25+ years old, and they have a quite distinguished role in the string landscape. In a sense, they look more like our universe than the more familiar, supersymmetric theories! Maybe the reason for this neglect is simply lack of workforce; the last 25 years have brought so much progress on so many areas that the community just had to focus on the most novel or promising ones, and simply left many important questions unanswered. The physics of non-supersymmetric string theories was a victim to this rapid progress. Despite this, recent research in this direction has yielded e.g. metastable vacua [145,167], novel end-of-the-world defects [168][169][170][171][172][173][174], and checks of Swampland constraints [12,20,175]. The results that we have presented in this paper are yet another step in this direction. We believe (and hope to have convinced at least some readers) that non-supersymmetric string theories constitute a very interesting arena where there seems to be an abundance of low-hanging fruit that is likely to yield novel lessons both in the Landscape and the Swampland. particularly grateful to Yuji Tachikawa for explanations on equivariant topological modular forms. MM gratefully acknowledges the support and hospitality of KITP during the "Bootstrapping Quantum Gravity" Program, supported in part by the National Science Foundation under Grant No. NSF PHY-1748958, as well as the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452, in the context of the program "Traversing the Particle Physics Peaks". The travel of MM to ACP was supported by a grant of the Simons Foundation. During the early stages of this work, MM was supported by a grant from the Simons Foundation (602883,CV) and by the NSF grant PHY-2013858, by the Maria Zambrano Scholarship (UAM) CA3/RSUE/2021-00493, and currently by the Atraccion del Talento Fellowship 2022-T1/TIC-23956 from Comunidad de Madrid.