Symmetry TFTs from String Theory

We determine the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+1$$\end{document}d+1 dimensional topological field theory, which encodes the higher-form symmetries and their ’t Hooft anomalies for d-dimensional QFTs obtained by compactifying M-theory on a non-compact space X. The resulting theory, which we call the Symmetry TFT, or SymTFT for short, is derived by reducing the topological sector of 11d supergravity on the boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial X$$\end{document}∂X of the space X. Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in cohomology of the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial X$$\end{document}∂X, which in turn gives rise to the background fields for discrete (in particular higher-form) symmetries. We apply this framework to 7d super-Yang Mills, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X= \mathbb {C}^2/\Gamma _{ADE}$$\end{document}X=C2/ΓADE, as well as the Sasaki–Einstein links of Calabi–Yau three-fold cones that give rise to 5d superconformal field theories. This M-theory analysis is complemented with a IIB 5-brane web approach, where we derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply to both Lagrangian and non-Lagrangian theories, and allow for many generalisations.

Furthermore, theories with identical local dynamics can have different symmetries in this generalised sense [14,15], and different symmetry structures for a given set of local dynamics can sometimes be related by gauging [16], which in the presence of mixed 't Hooft anomalies can relate theories with more conventional symmetry structures to theories with less familiar ones [3,7].
A very useful way of organising these structures, that as we will see seems to arise naturally in string theory, is in terms of the following construction (which we learned from Dan Freed [17], here we are only sketching an outline of a more detailed construction): if the original theory T is formulated on d dimensional spacetimes M d , we introduce a generically non-invertible topological (d + 1) dimensional quantum field theory (which in this paper we will call the symmetry theory, or symmetry TFT (SymTFT) when we want to emphasize that it is topological 1 ) with the property that it admits a non-topological theoryT as the theory of edge modes on manifolds with boundary (a relative theory, in the framework of [18]), and also a gapped interface ρ to the anomaly theory A of the theory T: The anomaly theory is a well understood object (see [19,20] for reviews): it is an invertible theory that gives us a way of defining the phase of the partition function of T by evaluating the partition function of A on a d + 1 manifold with boundary (see [21] for the original discussion in the case of anomalies of fermions). The theory T, attached to its anomaly theory A, arises when we collide ρ withT.
We will argue in this paper that the picture that arises in string theory is the complementary one, in which we focus on the symmetry theory by sending ρ to infinity. More concretely, in this 1 In this paper we will be considering cases with d ∈ 2Z + 1, so there are no local anomalies. Figure 1: The cone C(L) over the link L, and the deformation, shown on the right, to a long cylinder where the singularity is at the far end.
paper we will consider singular string configurations, where we have a set of local degrees of freedom (often strongly coupled) living at the singular point of some non-compact cone X. We identify these local degrees of freedom withT. The choice of the actual symmetries of T (which in our picture above would be associated with a choice of ρ), has been previously argued to live "at the boundary of X" [22][23][24], a behaviour that is also familiar in the context of holography [25]. The goal of this paper is to sharpen this picture by giving a direct derivation of the symmetry theory from the string construction: we will see that we can obtain in a natural way a non-invertible topological theory encoding both the choices of symmetries for T and their anomalies.
Our methods do not require knowledge of a holographic dual, or of a weakly coupled description of the QFT. We find our results particularly illuminating in the case that the local degrees of freedomT are those of a strongly coupled CFT without a Lagrangian description (generically we know little about such theories, so any additional information is useful), but we do not require conformality of T either.
The main idea is as follows. In string theory we can construct the d-dimensional theories T by introducing defects or singularities extending along d-dimensional submanifolds of the 11dimensional spacetime. For concreteness we will focus in this paper on M-theory on singular spaces with a single isolated singularity. The non-trivial local dynamics arise from massless M2 branes wrapping vanishing cycles at the singularity. Close to the singular point the geometry will look like a real cone over some manifold L with dim(L) = 10 − d. We can deform the cone into an infinitely long cigar, with the singularity at the tip, and L as the base of the cylinder along the cigar, see figure 1. The information that we are after is topological, so it is reasonable to expect that we can still obtain it from this deformed background (our results will support this expectation). If we now dimensionally reduce the M-theory action on L we will obtain a theory on the remaining d + 1 dimensions, which look like M d × R ≥0 . We claim that the topological sector -i.e. couplings that are metric independent -arising from this reduction on L is precisely the symmetry theory forT.
In our specific context of M-theory we will obtain this topological sector by "reducing" the Chern-Simons sector of M-theory on L, additionally including the effect of flux-noncommutativity [26][27][28].
As we will see, flux-noncommutativity leads to choices of higher form symmetries which appear in a way familiar from holography. For example, in the AdS 5 × S 5 case studied in [25,29,30]  ABJM in [31] and the non-conformal dual to confinement in Klebanov-Strassler in [32].) We will obtain analogous couplings from compactification. The topological reduction that we consider also generates in a natural way the anomalies that are expected in cases where the answer from field theory is known.
We expect the general idea that we are putting forward to be much more general and applicable in a wide range of setups. 2 In order to illustrate this point, in section 5 we will apply the same topological reduction prescription for the same 5d SCFTs we analyse from the M-theory viewpoint, but now in terms of their realization from (p, q) 5-branes [33]. We will show how 1-form symmetries for 5d SCFTs are encoded in the brane-web and compute from a supergravity point of view, expanding on the boundary of the IIB spacetime. The latter is inspired by the back-reacted holographic solutions [34]. These are AdS 6 × M 4 solutions, which are near-horizon limit of (p, q) 5-branes webs.
What will be important for our analysis is the topology of M 4 of the near-horizon geometry, which we will be using to dimensionally reduce the topological coupling of IIB supergravity. The topology of M 4 is based on isometries as well as asymptotic charges of the semi-infinite 5-branes and it is not affected by any large (p, q) charge holographic limit, therefore will be valid in any IIB (p, q) 5-brane-setup. Our focus here is on the dimensional reduction of the IIB 10-dimensional topological coupling on M 4 where we also need to include contributions coming from (p, q) 5-brane source to determine the anomalies for 5d SCFTs for both Lagrangian and non-Lagrangian theories. This beautifully complements the geometric analysis in M-theory. At last, as another application we also derive a mixed anomaly between continuous [6] and discrete 1-form symmetries of the (1,1) little string theory (LST) engineered by NS5-branes in IIB [35] from the holographic linear dilaton background [36].

A Differential Cohomology Refinement of Dimensional Reduction
M-theory compactification on singular Calabi-Yau 2-and 3-folds gives rise to 7d super-Yang Mills (SYM) and 5d superconformal field theories (SCFTs), respectively. These theories have 1-form symmetries, and in the 5d case also 0-form symmetries. The 1-form symmetry in all these cases is discrete and is characterized in terms of the relative homology quotient of the Calabi-Yau X, with 2 One complication in the general case is that topological terms in the parent string theory are not the only source of topological terms in the lower dimensional theory. For instance, in the presence of chiral fermions the lower dimensional theory will generically include couplings proportional to the η invariant evaluated in the compactification space.
respect to its boundary ∂X [22][23][24]37] Γ (1) = H 2 (X, ∂X; Z) H 2 (X; Z) . (1.2) To derive SymTFTs for global 1-form symmetries, one will have to incorporate their backgrounds B 2 ∈ H 2 (M 5 ; Γ (1) ) into the supergravity formalism. The torsional nature of these fields introduces various subtleties in the process. We will use differential cohomology to address these subtleties. 3 Differential cohomology has seen numerous applications within quantum field theory and string/Mtheory. Some of the earlier works on the subject include [26,28,[40][41][42]. For more recent examples of differential cohomology applications in formal high-energy physics that are of some relevance to this work see [16,[43][44][45][46][47][48][49][50][51][52]. In this paper differential cohomology will be used to refine the notion of dimensional reduction (or KK-reduction) of supergravity theories, with the goal of providing a precise treatment of the effect of torsion cohomology classes in the compactification manifold.
To illustrate, without going into the mathematical intricacies of differential cohomology yet, we now give a very concrete example of how the concept of dimensional reduction is reformulated using this approach. We begin with M-theory on a 5d space L 5 (which in this paper will be a manifold linking the singular point of a non-compat Calabi-Yau three-fold) which has H 2 (L 5 ; Z) = Z n ⊕ Z = t 2 ⊕ v 2 , where t 2 is a torsional generator of the degree two cohomology group and v 2 is a free generator. The reduction for the latter is the standard KK-reduction. It is the torsion part that will most benefit from the uplift to differential cohomology. We denote the differential cohomological uplifts of t 2 and v 2 byt 2 andv 2 , the precise meaning of these will be explained below.
In the case of M-theory we need to introduce a differential refinementG 4 of G 4 as well. As in the standard KK-expansion, this has a decomposition in terms of differential cohomology classes along the internal space L 5 (torsion and free), as well as external spacetime M 6 : There is an extra term, discussed below, that we are ignoring here for simplicity. The meaning and properties of the product " " will also be explained below. The CS-term in the M-theory action is which upon inserting the decomposition (1.3) ofG 4 we integrate over L 5 × W 6 . The integration over the internal space L 5 results in the SymTFT on W 6 . In ordinary cohomology the integral on L 5 would pick out only the forms of degree 5, however that would mean the purely torsional part L5t 2 t 2 t 2 would naively not contribute. Differential cohomology works differently, and as reviewed below L5t 2 t 2 t 2 can be non-vanishing.
Reformulating the problem in terms of differential cohomology on the link of the singularity involves some additional technical complications, but the effort pays off in a number of ways: • Geometric engineering of QFTs corresponds to "compactification" of string and M-theory on non-compact spaces Y d . This can be mathematically challenging, in particular it is difficult to define in a precise mathematical sense what one means by reducing the Chern-Simons action on a non-compact space Y d . In our approach we are instead reducing the Chern-Simons action on the closed manifold X d−1 = ∂Y d , which is the boundary or link of the non-compact space Y d . This is a much better defined mathematical question, that can be clearly analysed using the formalism of differential cohomology.
• The effective field theory in 11 − d dimensions is most interesting when Y d is singular, so it becomes a non-abelian Yang-Mills theory in 7d (for d = 4) or a non-trivial interacting 5d SCFT (for d = 6). But it is precisely in this singular geometric regime that it is most difficult to pin down what one means by doing a geometric reduction of the effective action.
By contract, our formalism is entirely agnostic about the singular structure of Y d , and can be applied without issues even when Y d is singular (in fact, it is arguably at the singular cone point in moduli space where it is most natural to apply our techniques!).
• Often, the analysis of reduction on singular spaces is done by removing these singularities, e.g. in 5d going to the Coulomb branch. It is well-documented, e.g. in the set of canonical singularities realizing 5d SCFTs from isolated hypersurface singularities (see [53][54][55][56] for a discussion in the context of 5d SCFTs), that we can have terminal singularities that do not admit a Calabi-Yau (crepant) resolution. This is obviously a class of theories where many standard methods will fail. Another setting where field theory-inspired arguments (including those employed in [57]) are not extendable, is when the 5d SCFT may have a Coulomb branch, but does not admit a non-abelian gauge theory description.
In contrast, our approach of deriving the SymTFT and thereby the anomaly of the QFT in terms of the reduction on the boundary is applicable in all those instances, and we will provide some examples of non-Lagrangian 5d SCFTs and their SymTFTs below.
• The approach uniformly encodes the entire SymTFT, for all symmetries arising from the compactification. E.g. in 5d we derive both the mixed 0-1-form symmetry anomalies as well as the 1-form symmetry (B 2 ) 3 anomaly [58] (see section 4).
• This approach lends itself to applications also in the context of string and M-theory compactifications with torsional cycles are in the geometry, such as those studied in [38,39], which give rise to theories with discrete gauge symmetries in string theory.

Comparison to other Approaches
Continuous global symmetries are usually rather manifest in terms of the geometry, e.g. the Rsymmetry of a 4d N = 1 SCFT is often encoded in some geometric isometry (such as in the setup of D3-branes probing Calabi-Yau cones), or the flavor symmetry in terms of non-compact divisors in a Calabi-Yau compactification -as will be the case in this paper. Discrete and continuous higherform symmetries are encoded also in the topology of the compactification space -usually in terms of relative homology, that captures the defect operators, modulo screening [59]. In the reduction on the link, the anomalies arise in terms of topological couplings for the background fields of global symmetries. Systematic tools for treating symmetries associated to isometries and non-torsional cohomology classes in brane constructions in M-theory and Type IIB are studied in [60,61] in connection to equivariant cohomology, see also [62].
In this paper we focus instead mostly on the torsional sector. Dealing with torsion cycles in supergravity has of course a history. One particularly promising framework was put forward in [38,39]. In supergravity, form-fields are usually expanded in harmonic forms, which however do not capture the torsion parts Tor H p (X d ; Z). The key idea of these papers is to use non-harmonic forms to model classes in Tor H p (X d ; Z). More precisely, a non-trivial class in Tor H p (X d , Z) of order k is modeled by a (p − 1)-form β p−1 and a p-form α p subject to the condition Moreover, motivated by the above general remarks on the KK-expansion, β p−1 is required to be a co-exact eigenfunction of the Laplacian with a non-zero eigenvalue. (It then follows automatically that α p is also an eigenform with the same eigenvalue.) To illustrate the physical picture underlying this proposal, let us consider for example the terms in the expansion of C 3 associated to a pair (β 1 , α 2 ) satisfying (1.5). We have schematically The 11d kinetic term for G 4 induces lower-dimensional kinetic terms of the schematic form where the Hodge star and the integration are now over external spacetime in 11 − d dimensions.
The quantities g A1 A1 , g B2 B2 depend on the details of the compactification and on the eigenvalue of the forms β 1 , α 2 under the action of the internal Laplacian. We do not need to discuss them in detail for this argument. We observe that the lower-dimensional fields A 1 , B 2 participate in a Stückelberg mechanism: the 2-form field B 2 "eats" the 1-form field A 1 and gets massive (as is expected, since the eigenforms β 1 , α 2 have a non-zero eigenvalue of the internal Laplacian). The Stückelberg mechanism leaves behind an unbroken Z k gauge symmetry: in the IR, the field B 2 is a continuum description of a Z k discrete 2-form gauge field. We identify the latter with the discrete gauge field that originates from a formal expansion of G 4 of the form is the torsion class modelled by the pair of forms (β 1 , α 2 ).
This approach has been applied successfully in various compactification scenarios with torsion (co)-cycles. However, clearly, this prescription leaves various mathematical questions open, and equally importantly, in more complicated compactification settings, carrying out this approach consistently can be difficult. The differential cohomology approach that we propose here has the advantage of providing a sound mathematical framework, which unambiguously lets us implement torsion in a supergravity setting. It would be interesting to provide a precise map to the above prescription using non-closed forms.
We should also comment that the M-theory approach often has a IIA-avatar, e.g. when there is a circle-fibration in the geometry. We will see that this can be a useful complementary check of our proposal. However, obviously this only applies in a very limited set of geometric situations.
To contrast and compare to the IIA setting, we discuss the IIA counterparts to the M-theory computations in this paper in appendix B.
In the context of 5d SCFTs and 7d gauge theories based on elliptic Calabi-Yau singularities, the paper [57] has discussed in specific instances 4 the anomalies of 1-form symmetries. The analysis is based on a Coulomb branch (CB) computation in the resolved geometry. What is implemented is a combination of a field theoretic analysis, with information from the geometry about the structure of the CB. The point of view in that paper is roughly complementary to ours: they integrate the Chern-Simons term on the singular Calabi-Yau X d to obtain an anomalous coupling in the (11 − d)dimensional field theory, while we integrate the Chern-Simons coupling on the (d − 1)-dimensional link of the singular point in X d to obtain the (10 − d)-dimensional anomaly theory (together with the non-invertible sectors, which are not considered in [57]). As emphasised above, this change in perspective has many benefits, for instance it allows us to approach problems where we do not have a gauge theory description to guide us.
We also resolve a puzzle in the computation in [57] of the mixed anomaly between the 1-form symmetry and the instanton symmetry. It was observed in that paper that the results of the computation did not always agree with the field theory expectation [63]. In the present paper, we obtain the expected field theory mixed anomaly, 5 and in addition also compute the B 3 1-form symmetry anomaly, where we also find agreement with [58]. Reproducing these very involved results gives us confidence that our approach is both sound and fruitful. More generally we expect 4 The cases discussed do not have discriminant components of the elliptic fibration intersecting the boundary of the Calabi-Yau. 5 Our understanding is that the results in [57] differ because their choice of non-compact divisor dual to the instanton generator is not always the same as in the field theory, but can differ from that by torsion generators.
to be able to apply our approach to non-geometric, and also not-resolvable geometries, which would provide a substantial extension compared to both field theory and Coulomb branch approaches.
Plan. The plan of this paper is as follows: We start in section 2 with an overview of differential cohomology, providing a (hopefully physics-friendly) summary of its salient features. We then apply this to the M-theory topological couplings -the CS and C 3 ∧ X 8 terms, and give the general KK-expansion in terms of differential cohomology. This framework is put to work in the context of C 2 /Γ ADE compactifications in M-theory in section 3 and in section 4 to 5d SCFTs realized on canonical singularities in Calabi-Yau three-folds. The anomalies of the 1-form symmetry are determined in these cases, including the mixed 0-1-form symmetry anomaly in 5d. For 5d we derive the anomalies from a complementary point of view in section 5 using the 5-brane webs in Type IIB -again from the boundary of the spacetime. Finally, we determine the anomaly of the little string theory (LST) from an edge mode approach in section 6. In appendix A we address a technical point regarding G 4 -flux quantisation and in appendix B we provide a countercheck to the M-theory results, using more conventional IIA reductions, which are available in some setups.

Differential Cohomology and M-theory
In this section we discuss how to apply the language of differential cohomology to describe the topological couplings in the M-theory low-energy effective action. We then describe the dimensional reduction of these couplings on a generic internal space L including contributions originating from both free and torsion elements in the cohomology of L.

Aspects of Differential Cohomology
Differential cohomology provides a mathematical framework to describe U (1) (p − 1)-form gauge fields on arbitrary spacetimes. This formalism is particularly useful to keep track of subtler aspects that emerge when we consider spacetimes with non-trivial topology (and in particular with torsional cycles). We refer the reader to [27,28,49,[64][65][66] for reviews aimed at physicists. We also highly recommend the textbook by Bär and Becker [67] for a pedagogical discussion of most of the results below.
Characteristic class and field strength. Let us consider a closed, connected, oriented manifold M. The p-th differential cohomology groupH p (M) of M is an Abelian group furnishing a differential refinement of the ordinary cohomology group H p (M; Z). The groupH p (M) sits at the center of the following commutative diagram: where all the diagonals are short exact sequences.
The maps i, I, τ , R are natural (that is, given a smooth map f : M → M they commute with the pullback f * of f ). Let us now proceed to unpack the relevant information contained in the above diagram, and to provide some physical interpretation: • The symbol Ω q Z (M) denotes closed differential q-forms on M with integral periods. The surjective map R associates to each elementȃ ∈H p (M) a p-form R(ȃ) ∈ Ω p Z (M), which we refer to as the field strength ofȃ. Physically,ȃ models a U (1) (p − 1)-form field, up to gauge equivalences, and the p-form R(ȃ) is identified with the physical field strength of the (p − 1)-form gauge field. (The fact that R(ȃ) has integral periods encodes the fact that the gauge group is U (1) and not R.)  can be identified with globally defined (p − 1)-forms on M, up to additive shifts by closed (p − 1)-forms with integral periods. In physics term, a topologically trivial (p − 1)-form gauge field can be described globally by specifying a (p − 1)-form. The shift by closed (p − 1)-forms with integral periods is interpreted as a gauge transformation (a "large gauge transformation" if the (p − 1)-form is closed but not exact).
• Commutativity of the square on the RHS of the diagram (2.1) is the statement that, for any a ∈H p (M), r R(ȃ) = I(ȃ) . (2. 2) The short exact sequence in the lower NW-SE diagonal of (2.1) comes from the isomorphism which is a by-product of de Rham's theorem. From the physics perspective, it is well-known that information about the topological aspects of a (p−1)form gauge field configuration can be extracted from its field strength (for example, the integer charge of a monopole configuration for a U (1) 1-form gauge field on M = S 2 is extracted integrating the 2-form field strength on S 2 ). Crucially, however, the field strength encodes only (I(ȃ)) and not necessarily I(ȃ). To see this, let I(ȃ) = [a] ∈ H p (M; Z) and embed Z into [a] contains more information than [F ] dR at the differential level since [a] R can be obtained from [a] but the converse is not true. In particular, information about torsional components in I(ȃ) is lost in passing to (I(ȃ)).  Commutativity of the lower triangle in the diagram (2.1) gives us The symbol d Z in the diagram denotes the standard de Rham differential on forms, which passes to the quotient of Ω p−1 (M) by Ω p−1 Z (M). The relation (2.5) is familiar in physics: if we have a topologically trivial (p − 1)-form gauge field, described by the globally defined form ω in some gauge, its field strength is simply dω. Concretely, if we pick some originΦ for the fiber on top of I(Φ), we can write the most general elementȃ of the fiber asȃ where ω ∈ Ω p−1 (X) is a differential form representing a class [ω] in the quotient of Ω p−1 (M) by Ω p−1 Z (M). As pointed out above, a different choice for ω in the same class [ω] is simply a gauge transformation.
Torsion Classes. Let us consider a torsion cohomology class t ∈ H p (M; Z). It will be useful for us to choose a convenient originΦ for the fiber on top of t. By exactness of the long exact sequence (2.4), we have that if t ∈ H p (M; Z) is torsion then there is some (not necessarily unique) u ∈ H p−1 (M; R/Z) such that t = −β(u). Our choice for the origin of the fiber above t isΦ = i(u).
Commutativity of (2.1) ensures I(Φ) = t, confirming indeed thatΦ lies in the fiber on top of t. Moreover, the differential cohomology classΦ is flat, R(Φ) = 0, as follows from exactness of the central NW-SE diagonal in (2.1). Product structure in differential cohomology. There exists a bilinear product operation on differential cohomology classes, The product is natural and satisfies the following identities: for anyȃ ∈H p (M),b ∈H q (M), (2.10) In the above relations, ∧ is the standard wedge product of differential forms and is the standard cup product of cohomology classes.
The product of a topologically trivial (respectively flat) element inH p (M) with any element in H q (M) is again topologically trivial (respectively flat). More precisely, we have the identities Fiber integration in differential cohomology. Given a locally trivial fiber bundle M with base B and closed fiber F, we can define an integration over the fiber which we can characterize axiomatically. First, it is a natural group homomorphism that is compatible with taking the curvature and taking the characteristic class: (2.13) (On the left hand side of these expressions we are using the usual notions of fiber integration of differential forms and cohomology classes.) It is also compatible with the maps i and τ : (2.14) An important special case is when we take B = pt and we identify the fiber F with M itself.
One hasH 0 (pt) ∼ = Z,H 1 (pt) ∼ = R/Z, whileH p (pt) is trivial for p = 0, 1. We then have two 6 Using the fact that , ∧, are graded commutative, these identities can also be written in the form non-trivial integration maps. The first is integer-values and yields the so-called primary invariant of a differential cohomology class of degree dim(M), The second integration operator is valued in R/Z and yields the so-called secondary invariant of a differential cohomology class of degree dim(M) + 1, We have used the fact that any elementȃ ∈H dim(M)+1 (M) is necessarily flat for dimensional reasons, and therefore can be written asȃ = i(u) for some u ∈ H dim(M) (M; R/Z).

Differential Cohomological Formulation of M-theory
The topological terms in the M-theory low-energy effective action can be written schematically in the form where M 11 is 11d spacetime, C 3 is the M-theory 3-form gauge field, G 4 is its field strength, and The expression (2.17) for the topological couplings can only be taken literally if the 3-form is topologically trivial, in which case C 3 is a globally defined 3-form on M 11 , and the integral in (2.17) can be understood as the standard integral of an 11-form. In topologically non-trivial situations, such as those studied in this work, greater care is needed to make sense of the formal expression (2.17).
For our purposes, it will be enough to model the M-theory 3-form gauge field as a class G 4 ∈H 4 (M 11 ) in (ordinary) differential cohomology. 7,8 In particular, we are implicitly restricting ourselves to situations in which the periods of the M-theory 4-form field strength are integrally quantized. As explained in [69], on certain spacetimes the periods must be half-integrally quantized.
We argue in appendix A that this does not occur in the setups discussed in this work. 9 7 It should be noted that in the mathematical literature this element ofH 4 (M 11 ) is sometimes denotedC 3 . We prefer the notationG 4 to make manifest the degree of this differential cohomology class. 8 More subtle questions require working on some generalised differential cohomology theory, of which there are many types. The generalised cohomology theory appropriate for M-theory has been postulated in [68]. It would be interesting to see if this more refined picture leads to any interesting consequences in field theory. 9 We refer the reader to [70] for a model for the M-theory 3-form in terms of a shifted differential cohomology class, which can accommodate both integral and half-integral periods.
The topological action (2.17) is interpreted as the R/Z-valued secondary invariant of a differential cohomology classȊ 12 ∈H 12 (M 11 ), whereȊ 12 is given by In the previous expression,p i (T M 11 ) ∈H 4i (M 11 ) denotes a differential refinement of the Pontrya- Within the formalism of differential cohomology we are allowed to consider products and Zlinear combinations of differential cohomology classes, but multiplying by rational coefficientssuch as the factor of 1/6 in front of theG 4 G 4 G 4 term in (2.20) -leads to a quantity which is not well defined in general. The fact that the particular combinationȊ 12 is nonetheless well-defined stems from the analysis of [69,73], which demonstrates that the total topological action e iS CS is well-defined up to a sign, which cancels a potential sign problem in the definition of the Rarita-Schwinger determinant. This sign ambiguity arises if and only if the periods of the G 4 field strength are half-integrally quantized. As mentioned above, this does not occur for the setups discussed in this work, meaning that e iS CS is well-defined by itself.

Kaluza-Klein Reduction in Differential Cohomology
Let us consider an 11d spacetime M 11 that is the direct product of an "internal" manifold L n of dimension n, and an "external" spacetime W 11−n of dimension 11 − n, It is standard to consider the expansion of the M-theory 3-form onto harmonic forms on L n , to obtain massless U (1) gauge fields on W 11−n of various p-form degrees. Our goal is to generalise this picture, by expanding the M-theory 3-form onto all cohomology classes of L n , both free and torsional.
On a factorized spacetime such as (2.21), it is natural to start from objects (differential forms, cohomology classes, differential cohomology classes) defined on the two factors, and combine them into objects on the total space. Let be the projection maps onto the two factors of M 11 . For notational simplicity, we henceforth omit the pullback maps p * W and p * L from various factorized expressions. For example, We observe that the naturality of the products and , together with (2.10), implies For each p = 0, . . . , n, H p (L n ; Z) is a finitely generated Abelian group. We take the generators (2.25) The subscript p is a reminder that these are classes of degree p, while (α), (i) are labels that enumerate the generators. We define b p := dim H p (L n ; R), the p-th Betti number of L n . For the torsion generators, the index set I p is some finite set of labels, which can be specified more explicitly in concrete examples. Each torsional generator has a definite torsional order: the minimal positive For simplicity we take By the Künneth formula, we may then expand a generic cohomology class a 4 ∈ H 4 (M 11 ; Z) as In the above expression, σ . Recall that the map I in (2.1) is surjective. This applies both for elements inH * (W 11−n ) and H * (L n ) 10 . It follows that there exist differential cohomology classesF With the objectsF we can construct the following differential cohomology class (2.29) The salient property ofȃ 4 in (2.29) is that it represents a possible lift of a 4 in (2.27), in the sense that This is verified using (2.28), the naturality of the differential cohomology product , and the second identity in (2.10).
The differential cohomology classȃ 4 is not the most general class that reduces to a 4 under the action of I. From the discussion around (2.8), however, we know that any other class that reduces to a 4 must differ fromȃ 4 by a topologically trivial element ofH 4 (M 11 ), which can be represented by a globally defined 3-form. These considerations lead us to the following final form for the Ansatz forG 4 , The first two sums in (2.31) encode all topological information aboutG 4 , while the last term collects the topologically trivial part ofG 4 .
The differential cohomology classesF , which restricts to a background field for a U (1) (2 − p)-form symmetry on the boundary; is the torsion order of t p(ip) , which restricts to a background field for a Z n (ip ) (3 − p)-form symmetry on the boundary.
The first case is familiar, but the second one requires some additional explanation. Notice in particular the difference in the relation between the differential cohomology class degree and the degree of the higher form symmetry on the boundary. Consider two classesB,B ∈H 4−p (W 11−d ), such that I(B) = I(B ). Then I(B −B ) = 0, so by exactness of (2.1) there is some globally defined differential form b of degree 3 − p such thatB =B + τ (b). By (2.11) and naturality of τ and R we ). But such a cohomology class is a map (up to homotopy) from , which is the data that defines a principal bundle for a (3 − p)-form symmetry. For instance, when p = 3 we have an ordinary (0-form) discrete symmetry, and the backgrounds for such symmetries are maps from W 11−d to BZ n (ip ) , or equivalently elements of H 1 (W 11−d ; Z n (ip ) ).

Integration on products.
Finally, we want to integrate differential cohomology classes on prod- For the Chern-Simons coupling (2.19) we have p + q = dim(X) + dim(Y ) + 1. In this case: simply by taking into account that the integrals on the right hand side of (2.32) are only nonvanishing for very specific values of p and q, as explained above. Here we have used that in the first caseȃ is flat for degree reasons, so there is some u ∈ H dim(X) (X; R/Z) such thatȃ = i(u), and similarlyb = i(v) in the second case.
In this paper we are particularly interested in those integrals over the internal space involving torsional elementst (we omit the subindices here for notational simplicity). First note that since we have chosen these torsional generators to be flat, with no monomials involving both τ ([ω 3 ]) and the torsional classes. The second class of monomials are accessible using the ordinary formalism based on differential forms, so we will not discuss them further; both because they are well understood and because our interest in this paper is on discrete higher form symmetries, which arise from the torsional sector. We now turn to deriving these theories, which will be the SymTFTs, in various geometric engineering settings.

Symmetry TFTs from M-theory on S /Γ
Consider first the case where the theory T that we are engineering is a seven dimensional N = 1 theory with Lie algebra g Γ , obtained by putting M-theory on M 11 = M 7 × C 2 /Γ, where Γ is a discrete subgroup of SU (2), and M 7 is a (closed and torsion-free, for simplicity) manifold where T lives. The resulting seven dimensional theory has defect group [59] Z(G Γ ) the center of universal cover group G Γ . For instance, the field theory with gauge group G Γ has electric 1-form symmetry Z(G Γ ), while the theory with gauge group G Γ /Z(G Γ ) has magnetic 4-form symmetry Z(G Γ ). Other global forms for the gauge group are often possible, depending on the choice of Γ, the analysis of the possibilities is identical to the one in [15]. The seven dimensional theory has additionally a 2-form U (1) instanton symmetry, with generator the integral of the instanton density on a closed 4-surface.
In the rest of this section we will argue that reducing M-theory on S 3 /Γ = ∂(C 2 /Γ) leads to an eight dimensional TFT encoding both this choice of global form for the seven dimensional theory (equivalently, the choice of its higher form symmetries) and the anomalies of these higher form symmetries.
We expect these two sectors of the eight dimensional TFT to interact in interesting ways: recall that choosing a global form for the gauge group (which will be able to rephrase as a choice of boundary behaviour in the BF theory (3.7)) can also be understood as a gauging of the higher form symmetries [16]. In the presence of 't Hooft anomalies this gauging procedure might be obstructed, or lead to less conventional symmetry structures (see for instance [3,7] for systematic discussions).
It would be very interesting to analyse this problem from our higher dimensional vantage point, where it requires study of gapped boundary conditions of the TFTs we construct, but we will not do so in this paper.

Choice of Global Structure from 8d
We start by discussing how to see the choice of global form in terms of a choice of boundary conditions for a gapped eight dimensional theory. The basic idea is discussed in [22][23][24], where it was found that the geometric origin of the 1-form and 4-form symmetries, and the fact that there is a choice to be made, can be traced back to the non-commutativity of the boundary values of F 4 and F 7 fluxes in the presence of torsion in the asymptotic boundary ∂M 11 at infinity [27,28].
Consider M-theory compactified on R × X 10 , where we take the first factor to be the time direction. For the moment we ignore the Chern-Simons terms in the M-theory action, and view M-theory as a generalised Maxwell theory. We will discuss the effect of the Chern-Simons terms extensively below. We focus on the operators Φ(T 3 ) and Φ(T 6 ) measuring the periods of C 3 and C 6 on torsional cycles T 3 and T 6 inside X 10 (our discussion is only sensitive to the homology class of the cycles). The authors of [27,28] found that these two operators do not always commute, but rather there is a phase in their commutation relation: Here, L is the linking pairing This pairing can be defined as follows. Assume that T 6 ∈ Tor H 6 (X 10 ; Z) is of order n ∈ Z. That is, there is some chain C 7 such that ∂C 7 = nT 6 . Then One can analogously define a linking pairing Tor cohomology, which will appear below.
In our case we are putting M-theory on M 7 × C 2 /Γ, which is not quite of the form R × X 10 , but [22][23][24] argue that taking the radial direction of C 2 /Γ as time (that is, taking reproduces the answers one expects from field theory. Our task is to modify the analysis in those papers to adapt it to the viewpoint that we are advocating here. In this section we will focus on the fate of the Φ(T 3 ) and Φ(T 6 ) operators upon dimensional reduction. The Chern-Simons part of the M-theory action gives additional contributions, which we study in detail below. We will find that reducing M-theory on S 3 /Γ leads to a non-invertible theory in eight dimensions whose states naturally correspond to the possible choices of global structure for the seven dimensional theory.
The analysis goes as follows. We place M-theory on a manifold of the form where we view the R factor as time when quantising. We are assuming that Tor k (M 7 ) = 0 for all k, so by the Künneth formula where we are using the fact that the torsion in the homology of S 3 /Γ is localised on degree 1: The values for Γ ab when Γ ⊂ SU (2) are given in table 1. Looking to this table, one observes [22,59,74] that Γ ab = Z(G Γ ), which is a key fact necessary for the whole picture to be consistent.
We find, in particular, that the torsional cycles T 3 and T 6 are necessarily of the form T 3 = Σ 2 × T 1 and T 6 = Σ 5 × T 2 , with T 1 and T 2 generators of the torsional group H 1 (S 3 /Γ; Z), and . Assume for simplicity that Γ ab = Z k , for some k.
where we have defined −1 := L S 3 /Γ (T 1 , T 1 ). 11 This is precisely the operator content and commutation relations of a BF theory with the following Lagrangian (see [25] for a derivation) For | | > 1 this is a non-invertible theory, whose state space reproduces the choices of global structure expected from the field theory side [15,22,25,75].
For the problem at hand, the terms inȊ 12 in (2.20) originating from X 8 do not contribute for degree reasons. (They will play an important role in the case of 5d SCFTs below.) For simplicity of exposition, let us assume first that Γ ab has a single generator. Substituting (3.9) into the Chern- (3.10) The first term in this expression encodes a potential mixed anomaly between a (−1)-form symmetry and the 2-form instanton symmetry. The second term in (3.10) corresponds to a mixed 't Hooft anomaly between the center 1-form symmetry Z(G Γ ) = Γ ab and the instanton 2-form symmetry.
In what follows we will concentrate on this last term. To find the coefficient of this anomaly, we must evaluate the integral This R/Z-valued quantity is the spin Chern-Simons invariant, evaluated for the 3-manifold S 3 /Γ and the flat connectiont 2 ∈H 2 (S 3 /Γ; Z). In general such Chern-Simons invariants also depend on the spin structure on the manifold. In our case, by construction, we have the spin connection induced on the boundary of the supersymmetric compactification of M-theory on C 2 /Γ.
Using S 3 /Γv = 1, and neglecting the τ term according to the general discussion of section 2.3, we compute the SymTFT: The cases in which Γ ab has two generators can be treated in a completely analogous way, yielding Note that formally one would be tempted to write 14) The factor of 1 2 makes the right hand side not well defined. Luckily (but unsurprisingly, given that our starting Chern-Simons coupling in M-theory action is well-defined [69]), due to the symmetry properties (2.10) of the Cheeger-Simons product is only the sum CS[S 3 /Γ] ij + CS[S 3 /Γ] ji that enters in the anomaly theory (3.13), and this sum is well defined: (3.15) Similar remarks apply to the off-diagonal entries in the D 2n case in table 3 below. We provide a more systematic discussion of this issue at the end of this section.

Evaluation of the Chern-Simons Invariant
Let us now discuss a convenient formalism to evaluate the CS invariant (3.11) (including the 1/2 prefactor), obtained by a straightforward generalisation of a discussion in the three dimensional case by Gordon and Litherland [76].

Cohomology of the Link and Bulk Compact Divisors
Let L d−1 be a closed, connected, oriented (d − 1)-manifold, and suppose that L d−1 can be realised as boundary of a d-manifold X d . The long exact sequence in relative homology yields We now make the assumption    • To every class a 2 ∈ H 2 (L d−1 ; Z) we can associate a non-compact (d − 2)-cycle κ in X d .
• To every torsional class a 2 ∈ H 2 (L d−1 ; Z) we can associate a compact (d − 2)-cycle Z in X d via the following relations, where κ is a non-compact (d − 2)-cycle in X d .
Let us now analyse the map A in (3.18) in greater detail. By Lefschetz duality, To proceed, we make the further assumption (that holds in all the cases in this paper) The universal coefficient theorem then guarantees that We may then recast (3.18) in the form The homomorphism A can be equivalently regarded as a bilinear Z-valued pairing between H 2 (X d ; Z) and H d−2 (X d ; Z),

Evaluation for S 3 /Γ
To compute the SymTFT for the 7d theory (3.12) we need to evaluate the CS invariant. Let us now specialise to a 3d link L 3 , and fix a class t 2 ∈ Tor H 2 (L 3 ; Z) such that nt 2 = 0. Let Z ∈ H 2 (X 4 ; Z) be the compact 2-cycle in X 4 associated to t 2 . By the discussion above, the linking pairing of (the Poincaré dual to) t 2 with itself can be computed as (3.25) In the above expression, · denotes the intersection pairing among compact 2-cycles in X 4 . Our task is actually to compute a CS invariant of the form (3.11). In the Gordon-Litherland approach, this quantity is given by In particular, we apply this formalism to the case of interest L 3 = S 3 /Γ. The bulk X 4 can be chosen to be the resolved ALE space C 2 /Γ. We notice that the assumptions (3.17) and (3.21) are indeed satisfied. The intersection matrix M 2,2 representing the map A equals minus the Cartan matrix of the Lie algebra g Γ . For the ADE-singularities, the choice of central divisors which gives the center of the gauge group has been identified in [37]. We list their results in table 2. Using these compact divisors as representatives of the torsional generators in order to be able to compare easily with known field theory results, 12 we obtain the results given in 3 for the spin Chern-Simons . It is a nice check of our formalism that the resulting coefficients in the anomaly theory perfectly reproduce the answer one gets from a purely field theory analysis [48].  proven as follows. 12 Note that there is a genuine ambiguity here: if t is a generator of Zn, so is kt for any k such that gcd(k, n) = 1. We have CS[S 3 /Γ, kt 2 ] = k 2 CS[S 3 /Γ,t 2 ] mod 1, so this rescaling can potentially change the coefficient of the anomaly. This is why it is important to choose the right generator of the torsional group when comparing with field theory results, even though any choice of k is in principle physically allowed. Table 2: The center divisors Z written in terms of the compact curves associated to the simple roots for ADE-singularities. We omit the E 8 case as it has trivial center symmetry.
Sinces 2 andt 2 are flat, we can use the commutativity of (2.1), and (2.11) to writȇ where we have also used that H 1 (S 3 /Γ; R) = H 2 (S 3 /Γ; R) = 0, so the Bockstein map β in (2.1) is an isomorphism. For the integral of (3.28) we then have  The more refined statement that follows from our M-theory construction is instead: The discussion in [22] was specific to four dimensional theories on Spin manifolds, and the two statements agree on that class of manifolds (up to an overall sign that was chosen oppositely in [22]), but (3.31) gives the correct answer on non-Spin manifolds too.

Symmetry TFTs for 5d SCFTs
Placing M-theory on toric Calabi-Yau threefold singularities leads to a rich and interesting class of five dimensional SCFTs. This is a very active area of investigation, started by [78][79][80][81][82]. These theories have an intricate set of global symmetries, starting with 0-form flavor symmetries, which are enhanced at the UV fixed point, as well as discrete higher-form symmetries -both 1-form (or 2-form) symmetries [23,24,37] and 3-form symmetries [54,55]. These symmetries can have 't Hooft anomalies, and by gauging some of the symmetries one obtains theories with 2-group structure [3,83].
In this section we will determine a subset of the six dimensional symmetry TFT by reducing Mtheory on the five-dimensional Sasaki-Einstein manifold L 5 linking the singular point. In section 5 below we will present an alternative derivation based on reducing the Chern-Simons terms on the worldvolume of the dual system on (p, q) 5-branes down to six dimensions. One important omission from our analysis is the sector leading to 2-group structures, which we leave for future work.

5d SCFTs from M-theory and Higher Form Symmetries
We will start by giving a summary of the salient features of 5d SCFT engineering from M-theory that will play a role in this paper. We are interested in the dynamics of M-theory on a singular where the last isomorphism is true if all 1-cycles in L 5 trivialise in the bulk (this will be true for X simply because H 1 (X; Z) = 0). In the cases of interest to us this group will be of the form Z k for some k that depends on X.
As noted in [22][23][24], this group can be computed in any smooth resolution of X using the intersection of compact divisors and compact curves in the Calabi-Yau by where the Betti numbers are related to the rank of the CB gauge group r and flavor rank f by b 2 = r + f and b 4 = r. M 4,2 = (S i · X C k ) is the intersection matrix of compact divisors S i with compact curves C k . Alternatively, it can be computed without needing to resolve X by looking to the structure of the external points in the toric diagram [24,92].
Similarly, as explained in section 2.3, the torsional part of H 3 (L 5 ; Z) will lead to a finite abelian 1-form symmetry under which line operators are charged, while the free projection of H 3 (L 5 ; Z) leads to a continuous 0-form symmetry, which includes the instanton symmetry or U (1) I .
These higher form symmetries are not all realised simultaneously in a given field theory. As in the seven dimensional case studied above, we will obtain a generically non-invertible BF sector in the symmetry theory when reducing on L 5 , and different choices of boundary conditions for the symmetry theory will determine which higher form symmetries are actually realised. The derivation of the BF sector is very similar to the one in that case, so we will be brief. Consider the operators Φ(T 3 ) and Φ(T 6 ) wrapped on generators t 1 and t 3 of H 1 (L 5 ; Z) and Tor H 3 (L 5 ; Z). They will lead to operators Ψ(Σ 2 ) and Ψ(Σ 3 ) in the effective six dimensional symmetry theory, with a commutation relation on a spatial slice, where −1 := L L5 (t 1 , t 3 ). This is the content of a BF theory with action 13 We are using conventions where the periods of B 2 are integrally quantised (as opposed to 2π/ gcd(p, q) quantised) and the 1-form symmetry in this case is Γ (1) = Z gcd(p,q) . This coupling corresponds to a 't Hooft anomaly for the 1-form symmetry, and therefore field theoretically obstructs its gauging.
This will imply that (potentially) some asymptotic flux choice might be obstructed, and not all the global form of the gauge group are allowed, unless a more complicated structure arises, which mixes the 1-form symmetry with other symmetries present in the theory. We plan to explore deeper consequences of this coupling by using our methods in the future.
There is also a field theoretic mixed anomaly between the instanton U (1) I and 1-form symmetry, determined in [63] for the SU (2) 0 theory using field theory arguments. In the IR for SU (p) q it takes the form [58,63] (This contribution to the anomaly was also analysed in [57] using string theory methods, reaching a different conclusion. We believe that the discrepancy between their result and ours might be due to a different choice of torsional representative, see footnote 12 above.) We will now derive these anomalies from first principles using the differential cohomology approach developed in this paper, being agnostic about whether this is a UV or IR computation.
We will see that an essential contribution for these anomalies comes from the C 3 ∧ X 8 term in the M-theory effective action.

Link Reduction Using Differential Cohomology
The integral cohomology of L 5 , the base of the toric Calabi-Yau cone X, takes the form For simplicity, we assume that the Betti number b 1 of L 5 is zero. (This is true in all examples we study.) The expansion ofG 4 then reads We can now consider the reduction of the G 3 4 coupling in M-theory. Using (4.8) and collecting all relevant terms, we arrive at (4.9) In the first term, we have used = δ β α . Next, let us consider the terms that originate from the G 4 X 8 coupling in M-theory. Recall that 11d spacetime is taken to be the direct product M 11 = W 6 × L 5 . As a result, at the level of cohomology classes with integer coefficients, one has 14 (4.11) 14 The Whitney sum formula for Pontryagin classes in integral cohomology is [93] pq(A ⊕ B) = r 2q−j (A) r j (B) , r 2s = ps , r 2s+1 = Bock(w 2s ) Bock(w 2s ) + ps Bock(w 1 ) , (4.10) where A, B are O bundles, the w i 's are Stiefel-Whitney classes, and Bock is the Bockstein homomorphism associated to the short exact sequence 0 → Z 2 − → Z → Z 2 → 0. In appendix A we show w 1 (T L 5 ) = w 2 (T L 5 ) = 0, which implies that (4.11) holds at the level of cohomology with integral coefficients.

These relations imply
Promoting integral cohomology classes to differential cohomology classes (the precise representative of p 1 one chooses is not important [94]) we can write the G 4 X 8 coupling in the form (4.13) In the second step we used (4.8) and we observed that the only internal differential cohomology classes that can have a non-trivial pairing withp 1 (T L 5 ) are the degree-2 torsional classest 2(i) .
To proceed, we make use of the following congruence for integral cohomology classes [95] p 1 (T W 6 ) a 2 = 4 a 2 a 2 a 2 mod 24 for any a 2 ∈ H 2 (W 6 ; Z) . (4.14) This congruence can be derived using the Atiyah-Singer index theorem as follows [96]. We take external spacetime W 6 to be a Spin manifold. Consider an arbitrary a 2 ∈ H 2 (W 6 ; Z). There exists a line bundle with connection A on W 6 such that its first Chern class equals a 2 . Consider the Dirac operator on W 6 twisted by this line bundle. The Atiyah-Singer theorem implies where F is the curvature 2-form of the connection A, satisfying [F ] dR = (a 2 ). We conclude that W6 [4 a 2 a 2 a 2 − a 2 p 1 (T W 6 )] ∈ 24 Z, which is equivalent to (4.14).
Relation (4.14) then implies where we have used the fact that W6ȃ 6 = W6 I(ȃ 6 ) for anyȃ 6 ∈H 6 (W 6 ), together with (2.10), , and I(p 1 (T W 6 )) =p 1 (T W 6 ). The quantities M (i) are unspecified integers, encoding the ambiguity in the mod 24 congruence (4.14). There is no summation on the repeated label i in (4.16). Inserting (4.16) into (4.13), we arrive at Notice that there is an ambiguity in the definition of the differential cohomology classesv 2(α) associated to the free part of H 2 (L 5 ; Z), which can be shifted by integral multiples of the differential cohomology classest 2(i) associated to Tor H 2 (L 5 ; Z),v 2(α) →v 2(α) + m (α) Our choices are such that for the examples in the paper we have Combining (4.9) and (4.17), we obtain the following anomaly couplings in the symmetry TFT: where the R/Z-valued quantities Ω ijk , Ω ijα are CS invariants defined by (4.20) As we demonstrate in appendix A, for the setups of interest in this work G 4 is integrally quantised, and therefore the 11d couplings in the M-theory effective action are guaranteed to be well-defined.
It follows that the CS invariants (4.20) are also well-defined. Let us emphasize, however, that the two terms in Ω ijk with i = j = k are not separately well-defined, in general.
The CS invariants Ω ijk , Ω ijα are defined purely in terms of the link geometry L 5 . In order to evaluate them for a given L 5 , however, it can be convenient to resort to a computation in the bulk of the Calabi-Yau X 6 , using an extension of the Gordon-Litherland formalism discussed in section 3.3.
Let n (i) denote the torsional degree of t 2(i) ∈ H 2 (L 5 ; Z), and let Z (i) be the compact divisor in the bulk associated to t 2(i) . We also associate a non-compact divisor D (α) to the non-torsional classes v 2(α) ∈ H 2 (L 5 ; Z), which correspond to flavor symmetries. With this notation, the invariants (4.20) can be computed as , where · denotes intersection of divisors in X 6 .

Examples: SU (p) q from Y p,q
This general approach can be exemplified for all toric Calabi-Yau cones, in particular the SCFTs with SU (p) q IR description, which have from field theory analysis, the anomalies in (4.5) and (4.6).
The Sasaki-Einstein link is given by Y p,q , and the Calabi-Yau has simple toric description: the toric diagram for SU (p) q is given by figure 3. For a detailed discussion of this geometry see e.g. [97].
The external vertices that determine the toric fan are k y ) , (4.22) where the CS-level q is determined by There are linear relations among these non-compact divisors D wi , and the instanton U (1) is identified with The compact divisors are S a = (0, a) , a = 1, · · · , p − 1 . This compact divisor is also identified with the compact divisor associated to the generator of Tor H 2 (L 5 ; Z) according to the discussion in section 3.3.
We will also need an explicit expression for p 1 (T X 6 ) = −c 2 (T X 6 ⊗ C), with T X 6 ⊗ C the complexification of the tangent bundle of the toric Calabi-Yau X 6 . Since T X 6 is a complex vector bundle we have T X 6 ⊗ C = T X 6 ⊕ T X 6 , so c(T X 6 ⊗ C) = c(T X 6 )c(T X 6 ). For a toric variety X 6 with divisors D i we have [98] c(T X 6 and therefore p 1 (T With this information at hand it is straightforward to compute the anomaly coefficients using the formulae (4.21), specialised to the case of one torsion generator of order gcd(p, q) and one free generator. We have done these computations with Sage [99] for p < 20 and |q| ≤ p, and find results compatible with the empirical formulas  This condition guarantees that the terms in (4.17) not fixed by the mod 24 congruence (4.14) can indeed be safely dropped.
Assuming the validity of (4.29) it is straightforward to verify that where P = p/gcd(p, q). Plugging (4.29) in (4.21), and using (4.31), we find that the action for the symmetry TFT contains the terms terms generated by the B 3 2 term are q p (p − 1) (p − 2) 6 gcd(p, q) 3 3 gcd(p, q) (4.33) Similar remarks apply to the B 2 2 F I term.

Examples: Non-Lagrangian Toric Models
Our geometric approach becomes particularly useful when the theories in question do not have any non-abelian gauge theory description -i.e. are truly non-Lagrangian. As an illustration, we consider the toric models B N and B Theory Γ (1) Toric Fan ((0, 1, 1), (N − k, k, 1)), k = 0, · · · , N − 1 We can again compute the B 3 terms in the SymTFT for B N : For B (1) For B (2) Finally for B N the B 3 2 anomaly vanishes. In computing these, we have picked a particular central divisor associated to the generator of the 1-form symmetry. e.g. in the case of B (2) N theories, we conjecture this to be of the form (4.37) Since there is no non-abelian gauge theory description any central divisor is in fact equally acceptable, but this choice leads to simple general formulas.
Finally, let us note that the B N theories have a U (1) factor in their flavor symmetry groups [100]. We can compute the mixed anomaly between this U (1) symmetry and the 1-form symmetry. To this end, we use the general formula (4.21), specialised to the case of a single Z central divisor.
The non-compact divisor D associated to the U (1) flavor symmetry is identified with the divisor associated to the vertex with coordinates (N − 1, 0) in the toric diagram for B (1) N , see figure 4. We find the following additional term in the symmetry TFT: where F is the field strength of the background field for the U (1) flavor symmetry.
The application of our approach to this class of non-Lagrangian theories in 5d, demonstrates the flexibility and generality of the approach using our differential cohomology extension of dimensional reductions on the link. Here we focused on toric models, but any non-compact Calabi-Yau three-fold geometry that has a canonical singularity, i.e. 5d SCFT, can be studied in this way.

5d Anomalies from the Boundary of (p, q) 5-brane Webs
We now consider IIB (p, q) 5-brane webs engineering 5d SCFTs, and in particular we would like to evaluate the IIB supergravity action at the boundary of these webs to compute a 6d bulk action.
We focus on the part of this action which involves the 1-forms symmetries and corresponds to the anomalies of the 5d SCFTs.

Mixed Anomalies
In order to describe the boundary geometry, we assume that the topology of the near-horizon limit of the IIB (p, q) 5-brane webs at large N in [34] extends generically to all webs. This implies that the boundary is given by M 4 that is an S 2 fibered over a disc, Σ, with punctures at the boundary of Σ representing the (p, q) 5-brane sources. The fibered S 2 shrinks at the boundary of the disc away from the punctures, and the full space is topologically equivalent to an S 4 with punctures. The are also non-trivial 1-cycles with volume forms given by {ω }. Locally they can be thought as the hodge dual of the β , and they have to satisfy the same constraints (5.1). 15 The intersection pairing reads where self-duality of F 5 impose that * 6 f 5 = f 1 and * 6 f 4 = f 2 . In addition there is a field strength for the pair H 3 , F 3 expanded on ω corresponding to Not all of a 1 will be linearly independent, since the 1-cycles as well the 3-cycles, c , are also not all independent (5.1). The most relevant aspect is the counting of the massless vector fields which corresponds to the backgrounds of the abelian flavor symmetries, whereas the details of the intersection pairing Ω, as long as it is non-trivial, will not affect the results in any significant way.
We study now the reduction of the IIB topological coupling, which we extend to a 11-dimensional coupling to ensure gauge invariance The Bianchi identity dF 5 = H 3 ∧ F 3 imposes the following constraints in BF frame (f 5 , f 4 ), and in the Stückelberg frame (f 1 , f 2 ) BF frame. From the expansion (5.3) and (5.6), we obtain various contributions to couplings.
The first interesting coupling is given by the singleton theory, which can be recast in a SL(2R) covariant way as follows, where by suppressing the indices, using matrix multiplication, and defining C 3 = {c 1 3 , . . . , c L 3 }, we can write (5.8) as, where Q SNF = BQA is the Smith normal form of Q and A, B is the pair of matrices transforming Q, moreover we have that d C 3 = (B −1 ) T ΩdC 3 and F 3 = A −1 σF 3 . The study of boundary conditions and mutual locality of this action determines the 1 or 2-form symmetries of the theory.
We now study what happens to the vector fields a 1 with coupling This can be written in an SL(2, R) covariant way as follows S vec 7d = dc 4 ∧ Ω j Q I σ IJ F Ij 2 = dc 4 ∧ Tr(Q T ΩF T 2 σ) . (5.12) where Q and σ have been defined in (5.8) (5.13) In particular (5.12) can be rewritten as follows, where F 2 = (B −1 ) T ΩF T 2 . This is the dual of the Stückelberg mechanism, which makes a combination of the vectors massive, and the massless one satisfies, Finally, we also have additional couplings, which could potentially lead to anomalies between the U (1) flavor symmetries and the 1-form symmetries, which in SL(2, Z) coviariant form reads Stückelberg frame. The same physical consequences hold also in Stückelberg frame. The second Bianchi identity in (5.7) implies that The kinetic therm for g 2 could make some of the component C 2 discrete depending on the entries of Q SNF . The first identity in (5.7) can be rewritten such that leading to the same constraint given by (5.14). Moreover the expansion of (5.5), leads to the same expression (5.16) for the mixed anomaly between the 1-form symmetry and the U (1) flavors. We will see in the example of subsection 5.3, that this reproduce the anomaly discussed in [63].

't Hooft Anomaly for the 1-Form Symmetry
So far we have studied the bulk supergravity action, but the effective 6d anomaly theory can receive contributions from the Chern-Simons terms of the 5-brane sources. To be consistent we add these Chern-Simons actions in the SL(2, Z) covariant form [102,103], which in general reads, where the sum over the 5-branes and on each 5-brane we haveĈ =Ĉ 0 +Ĉ 2 +Ĉ 4 +Ĉ 6 +Ĉ 8 ,Â is the A-roof genus of R T , R N tangent and normal bundle curvatures respectively. The expansion in terms of Pontryagin classes reads In addition, we havê where C 6 = (b 6 , c 6 ), and theâ I correspond to the center of mass mode of the 5-brane stack and all decouple. We now expand the action and the relevant terms are where we suppressed all the indices in favour of the matrix product, and we ignored the normal bundle contributions. Moreover, there is no contribution of such kind from the bulk (5.5). In terms of C 2 = A −1 σC 2 , which is the diagonal basis where the (p, q) charges take the Smith normal form.
We get the contribution to the SymTFT from the CS couplings to be: with A defined such that Q SNF = BQA. In particular this action reproduces the cubic anomaly introduced in [58], as we will see in the examples of the next subsection.

Examples
We now apply these general results to brane-webs that realize 5d SCFTs. This complements the geometric analysis in the earlier parts of the paper, in particular the toric Calabi-Yau reductions in M-theory in section 4. The brane-webs can easily be obtained from the toric diagrams (by passing to a dual graph) and vice versa.
Example: SU (2) 0 SCFT. The asymptotic 5-branes charges for the SU (2) 0 Seiberg theory are The Smith normal form reads (5.26) which together with (5.8) encodes the Z 2 TQFT that upon a suitable choice of boundary condition determines the Z 2 1-form symmetry of the theory, together with its background field. We have 2 independent linear combination of the vector fields, and the condition (5.15) sets one of them to zero. Then we have a contribution to the anomaly coming from (5.16), and the relevant contribution is S bulk 7d = 2 dã 1 ∧c 2 ∧f 3 + . . . , (5.27) where dã 1 = (f 2 2 −h 2 2 ) is the background for the instanton U (1) I of the gauge theory which enhances to SO(3) at the conformal point [63,83].c 2 is the background field for the Z 2 1-form symmetry with Z/2 periods, therefore if we map the field to the one with integral periods,c 2 →c 2 /2, and integrate on the 6-dimensional boundary we get We can also compute the anomaly coming from (5.24). First of all we use the congruence, that tells us,c 2 p 1 (R T ) = 4c 2c2c2 mod 24 (5.29) withc 2 having integer periods. We also need Q where x i are general integer parameters. Evaluating (5.24) we get, wherec 2 is the one with integer periods. However the freedom of choosing x 1 , x 4 can be reabsorbed by adding the two terms to the action, which do not change the classical equations of motion of the low-energy anomaly theory, whereg 2 2 = dc 2 1 + 2c 2 , moreover dg 2 2 − 2f 3 is no other than of component of the Bianchi identity (5.17) defining the Z 2 1-form symmmetry in Stückelberg frame, andg 2 = (B −1 ) j g j 2 . Finally, since these are Chern-Simons terms m, m must be integers. By integrating on the 6d boundary and by plugging in the congruence (5.29), gauge invariance underc 2 →c 2 +2dλ 1 fixes 3(x 1 +x 4 ) = 4m+m , which implies S cubic Sym = S CS Sym + S add 6d = 0.
which together with (5.8) encodes the Z gcd(p,q) 1-form symmetry of the theory, and its background field. We have 2 independent linear combinations of vector fields, and the condition (5.15) sets one of them to zero. Then we have a contribution to the anomaly coming from (5.16) that is, where dã 1 = (h 2 2 ) generically corresponds to the background for the instanton symmetry U (1) I of the gauge theory that enhances to SO(3) at the superconformal point [83]. 16c 2 is the background field for the Z 2 1-form symmetry with Z/gcd(p, q) periods, therefore if we map the field to the one with integral periods,c 2 →c 2 /gcd(p, q), and integrate on the 6-dimensional boundary we get This expression is not gauge invariant underc 2 →c 2 + gcd(p, q)dλ 1 . We then add the following term which does not change the anomaly theory, where we recall that g 2 2 = dc 2 1 + gcd(p, q)c 2 , upon integrating on the 6d boundary, and rescaling c 2 →c 2 /gcd(p, q), gauge invariance fixes n to be an integer such that gcd(p, q)n = −p, such that Thec 3 2 anomaly comes from evaluating (5.24). First of all we use the congruence, that tells us (5.29) and we also need Q, that is where x i are general integer parameters. Evaluating (5.24) we get wherec 2 is the one with integer periods. We can now ignore the contribution proportional to x 1 , x 2 .
They can be reabsorbed by adding two terms to the action as in the previous example. These new terms do not change the classical equations of motion of the low-energy anomaly theory, since they are proportional to Bianchi identities. They read where g 2 2 = dc 2 1 + gcd(p, q)c 2 , and dg 2 2 − gcd(p, q)f 3 is no other than of component of the Bianchi identity (5.17) defining the Z 2 1-form symmmetry in Stückelberg frame, andg 2 = (B −1 ) j g j 2 . By integrating on the 6d boundary and by plugging in the congruence (5.29), gauge invariance under c 2 →c 2 + gcd(p, q)dλ 1 fixes m, m to be integers such that 4m + m = 2(p − 3) p 2 q gcd(p,q) 3 , which implies    where dã 1 = (h 2 2 ) its the background for the U (1) flavor symmetry of the theory, and we display only the interesting contribution.c 2 is the background field for the Z 2 1-form symmetry with Z/(N − 1) periods, therefore if we map the field to the one with integral periods,c 2 →c 2 /(N − 1), and integrate on the 6-dimensional boundary we get This expression is not gauge invariant underc 2 →c 2 + (N − 1)dλ 1 . We then add the following term which does not change the anomaly theory, where we recall that g 2 2 = dc 2 1 + (N − 1)c 2 , upon integrating on the 6d boundary, and rescaling c 2 →c 2 /(N − 1), gauge invariance fixes n = −1, such that N SCFTs. The asymptotic 5-branes charges and SNF are We can also compute the 't Hooft anomaly for the Z N 1-form symmetry form symmetry of the N . The S CS Sym has trivial 1 N contribution sufficiently implies that S cubic Sym = 0. These examples illustrate that the approach using the webs and the geometry agree nicely and reproduce the same anomalies also in non-Lagrangian theories. In the context of the webs it is possible to describe theories, which are not described by toric geometries (but so-called generalized toric polygons (GTPs) [104][105][106]). One can still compute from the web-data using a simple combinatorial prescription the 1-form symmetry [37]. It would be very interesting to generalize the above analysis to webs which are dual to such GTPs.

Little String Theory Anomaly from Linear Dilaton Holography
The final application is more holographic in spirit, but follows the same philosophy as the other parts of our analysis. We derive he anomaly of the LST from a holoraphic point of view -again using an expansion on the boundary of the space.
N NS5 branes in IIB are believed to be described by a Little String Theory [80], that is a non-gravitational theory with non-local stringy excitation [107]. The low-energy limit of this 6d theory is given by (1, 1) SYM in 6 dimensions. Many little string theories have 7d bulk gravity dual solution with linear dilaton behaviour [36]. In this case the dual is given by (string frame) where dx 2 6 is an abbreviation for a flat 6-dimensional Minkowski metric, dΩ 3 is the metric on a round 3-sphere and vol 3 its volume, whereas φ 0 is a constant.
Given this background we can now implement a reduction of the IIB topological coupling where we expand the fluxes on the cycle of the internal S 3 geometry, Because of self-duality of F 5 , satisfy f 5 = * 7 f 2 , where * 7 is the hodge dual in the seven-dimensional space spanned by Minkowski 6 and the ρ coordinate. The Bianchi identity dF 3 = dH 3 = 0 and Since self-duality of F 5 imposes f 5 = * 7 f 2 we manifestly have two frames.

BF-frame.
In this frame we choose to keep f 5 , and therefore what we get is the following coupling, 10) The first one give rise to the singleton theory, which upon choice of boundary condition, fixes whether the boundary theory has a Z N 1-form symmetry with background c 2 , such that N dc 2 = 0, or a dual 3-form symmetry, or a mixture [22-25, 27-30, 108, 109]. The background c 2 corresponds to the center 1-form symmetry of the 6d SU (N ) gauge theory low-energy limit of the LST. The second coupling is the holographic realization of part of the mixed anomaly between the continuous 1-form symmetry associated with the current J = * 6 Tr(F ∧ F ) of the low-energy 6d SU (N ) gauge theory [6], whose background is b 2 , and c 2 , the background of the discrete Z N 1-form symmetry corresponding to the center of the SU (N ), with periods such that c 2 ∈ Z N .

Stückelberg-frame.
In this frame the discrete Z N 1-form symmetry is realised via the Stückelberg mechanism that is induced by the kinetic term for f 2 = dc 1 + N c 2 . This implies that c 1 can be entirely gauged away, and c 2 is the background for a Z N 1-form symmetry. The surviving topological coupling is then, which will give rise to part of the mixed anomaly among the 2 different 1-forms symmetries of the theory. In order to evaluate the full mixed anomaly we need to take into account the invariance under large gauge transformation for the b 2 , for any integer value of N . The expression (6.11) is not invariant under these large gauge transformation on a space with a non-trivial boundary. Indeed the term, is not invariant under (6.12). The general principle is that any effective action describing the reduction of the IIB action on S 3 and in particular the correct classical equations of motion, is valid. We can indeed add the following term, where adding the second term does not change the theory and the classical configuration, since is vanishing due to the Bianchi identity (6.8). We can now pick this action and integrate on a space with a 7d boundary to obtain the SymTFT where we forget about total derivatives. We need to require that this action is gauge invariant under (6.12) and that the coefficient is well defined modulo 1. This fixes m = 1, so that the final result for the anomaly reads,

Conclusions and Outlook
The main object of study in this paper is the Symmetry Topological Field Theory (SymTFT). This is an object that encodes the choice of symmetries given the local dynamics, and the anomalies of these symmetries.
Our main result is that this SymTFT can be computed from string/M-theory by a reduction on the boundary of the compactifaction spacetime -this can be in a purely geometric engineering setup, as in the M-theory analysis in sections 4 and 3, but also brane-setups fall into this framework, as in section 5.
In the geometric engineering setup we made a case that this requires a refined notion of dimensional reduction, recasting the topological terms in the supergravity action in terms of differential cohomology classes. This differential cohomology approach is essential in order to accommodate torsion (co)-cycles and their contributions. The latter is pertinent when the symmetries under consideration are discrete, such as is commonly the case in higher-form symmetries.
There are various obvious applications to other geometric engineering setups. For instance, within the M-theory setting that we have discussed in this paper the natural extension is to consider reduction to 4d on G 2 -holonomy manifolds, such as the ones proposed by Bryant and Salamon [110], and generalizations thereof, which model the confining-deconfining transition of SYM theories. The reduction on Calabi-Yau four-and five-folds should also result in interesting anomalies in 3d and 1d. Specializing to the case of elliptic Calabi-Yau n-folds, the results in M-theory have an uplift to F-theory, and thus anomalies in the context of even-dimensional QFTs, like 6d SCFTs (for a review see [111]), 4d N = 1 SQFTs (for a review see [112]) and 2d (0,2) theories [113]. More generally, type IIB compactifications can yield supersymmetric gauge theories, which can have 1-form symmetries [22,[114][115][116][117].
In addition to the existence of 1-form (and higher-form) symmetries and resulting anomalies, supersymmetric QFTs can have higher-group symmetries. E.g. in 6d SCFTs [118] and LSTs [6,119], 5d SCFTs [83], and 4d class S [120]. Such higher-groups (continuous or discrete) are dual to mixed anomalies, and thus should have an imprint in the supergravity link reduction. Beyond this multitude of purely geometric constructions there are brane-systems, and of course the more familiar setting of holography. We hope to return to many of these applications in the future.

A Integral Quantization of G 4 -flux
In order to diagnose whether G 4 has integral or half-integral periods on a four-cycle C 4 , we can compute the integral of the fourth Stiefel-Whitney class of the tangent bundle T M 11 on C 4 [69], where the pullback to C 4 is implicit. In this work, we consider 11d spacetimes that are a direct product, M 11 = W 11−n × L n , where n = 3 and L 3 = S 3 /Γ (with Γ and ADE subgroup of SU (2)), or n = 5 and L 5 a smooth Sasaki-Einstein manifold.
The total Stiefel-Whitney class splits as w(T M 11 ) = w(T W 11−n ) w(T L n ). Possible contributions to w 4 (T M 11 ) are therefore of the form w 4−i (T W 11−n ) w i (T L n ), i = 0, 1, 2, 3, 4. We observe that all the spaces L n in this work are the base of a Calabi-Yau cone. In particular, each L n is orientable and Spin, and therefore w 1 (T L n ) = 0 = w 2 (T L n ). We also assume that external spacetime is orientable and Spin, so that w 1 (T W 11−d ) = 0 = w 2 (T W 11−n ). We conclude that terms of the form w 4−i (T W 11−n ) w i (T L n ) with i = 1, 2, 3 cannot give any non-zero contributions to w 4 (T M 11 ). The remaining potential contributions thus have i = 0 or i = 4. To kill the contribution with i = 0 we assume that external spacetime satisfies w 4 (T W 11−n ) = 0. What remains to be checked is whether w 4 (T L n ) is trivial. Clearly, we have w 4 (T L 3 ) = 0 for dimensional reasons. 17 Next, we prove that w 4 (T L 5 ) = 0 for any smooth Sasaki-Einstein five-manifold L 5 . In fact, we can prove a stronger statement: w i>0 (T L 5 ) = 0.
Let us adopt the shorthand notation w i = w i (T L 5 ). We have already observed w 1 = 0 = w 2 .
We also have (in general, from the Wu formula) which implies w 3 = Sq 1 (0) = 0 in our case. We also have that the Wu class necessarily vanishes on a 5-manifold for degree reasons (as it represents Sq 4 acting on H 1 (L 5 ; Z 2 ), which vanishes by general properties of the Steenrod squares), so we conclude w 4 = 0. Finally, again from the Wu formula which implies, given w 1 = w 2 = w 3 = 0, that w 5 = 0.

B Type IIA Analysis
We have seen that the differential cohomological approach is very effective in computing the SymTFTs starting from M-theory. To cross-check and to complement this with more familiar methods in this appendix we compute the same SymTFTs using the IIA background, which is given by not only geometry but also RR-flux. This setups is however more canonical in that there are no torsion cycles. The results will agree (to the extent that the more limited IIA analysis below is applicable, not every M-theory background that we analyse above has a well-behaved IIA limit), but this more conventional approach will be a countercheck and contrast to the much more efficient differential cohomology approach that we are proposing.

B.1 Type IIA Background for 7d SYM
The Einstein metric on S 3 /Z N that descends from the quotient of the round unit-radius metric on S 3 can be written as The coordinate θ has range [0, π] and the angles φ, ψ have period 2π. We identify the Hopf fiber S 1 ψ in (B.1) with the M-theory circle. It then follows that in type IIA the link geometry consists simply of a round S 2 (described by the coordinates θ, φ in (B.1)) threaded by N units of F 2 R-R 2-form flux. Let us stress that in the type IIA setup there are no torsional cohomology classes. As a result, the reduction can be performed according to the standard paradigm, including only the modes associated to harmonic forms in the internal space. In the present situation, the latter is S 2 and the only non-trivial harmonic forms are 1 and where F 2 is the field strength of the RR 1-form field, obeying dF 2 = 0.
As already noted above, we have a non-zero F 2 flux threading S 2 The expansions of the field strengths H 3 , F 4 take the form The quantities H 3 , f 1 , γ 4 , F 2 are external field strengths, whose Bianchi identities follow from (B.4), 18 Our normalizations for the NS-NS 2-form B 2 , the R-R 1-form C 1 , and the R-R 3-form C 3 are such that their minimal couplings to a fundamental string with worldsheet W 2 , a D0-brane with worldline W 1 , and a D2-brane with worldvolume W 3 introduce factors e in the path integral, respectively. In these conventions H 3 = dB 2 , F 2 = dC 1 , F 4 = dC 3 − C 1 ∧ dB 2 .
Upon plugging (B.6) into I 11 in (B.3) and fiber integrating along S 2 , we obtain the following 9-form expression in the external field strengths, (B.8) The key subtlety in the following is to note that we have the freedom of adding an integer multiple of the Bianchi identity to this anomaly Adding these contributions Applying an anti-derivative, and dropping total derivatives, we get These relations are written in the supergravity normalization. We work with integral fluxes We reduce of X 4 = S 2 × S 2 . The volume forms on the two S 2 factors are denoted ω 2,a=1 , ω 2,a=2 .
The intersection pairing in this basis is 14) The IIA field strengths are given by  We are free to choose n to make the term well-defined under gauge transformations. The mixed anomaly between the instanton U (1) I and the 1-form symmetry is invariant for n = 1 Similarly the B 3 anomaly has an ambiguity, however as we know from the M-theory computation this also acquires contributions from C 3 X 8 . The match with the M-theory computation is more subtle and might require a detailed analysis of the higher derivative couplings studied in [122]. We leave this for the future and focus our endeavours on the differential cohomology approach.