Spin(7)-Manifolds as Generalized Connected Sums and 3d N=1 Theories

M-theory on compact eight-manifolds with $\mathrm{Spin}(7)$-holonomy is a framework for geometric engineering of 3d $\mathcal{N}=1$ gauge theories coupled to gravity. We propose a new construction of such $\mathrm{Spin}(7)$-manifolds, based on a generalized connected sum, where the building blocks are a Calabi-Yau four-fold and a $G_2$-holonomy manifold times a circle, respectively, which both asymptote to a Calabi-Yau three-fold times a cylinder. The generalized connected sum construction is first exemplified for Joyce orbifolds, and is then used to construct examples of new compact manifolds with $\mathrm{Spin}(7)$-holonomy. In instances when there is a K3-fibration of the $\mathrm{Spin}(7)$-manifold, we test the spectra using duality to heterotic on a $T^3$-fibered $G_2$-holonomy manifold, which are shown to be precisely the recently discovered twisted-connected sum constructions.


Introduction
Geometric engineering of supersymmetric gauge theories is fairly well-understood in theories with at least four real supercharges. Most prominently in recent years, F-theory has established a precise dictionary between geometric data (and fluxes) and 6d and 4d theories with N = 1 supersymmetry, and dually, M-theory of course has a well-established dictionary between 3d and 5d theories and Calabi-Yau geometries. For 4d N = 1 the realization in terms of M-theory compatifications on G 2 -holonomy manifolds is already much less well understood, in particular due to the scarcity of compact geometries of this type. Even less is known about M-theory on Spin(7)-holonomy compactifications, which yield 3d N = 1 theories.
The goal of this paper is to provide a new construction of compact Spin(7)-holonomy manifolds, which have an interesting field theoretic counterpart in 3d. This is motivated on the one hand to build large sets of examples of 3d N = 1 vacua in M-theory/string theory, and we relate these constructions in certain instances to heterotic string theory on G 2 -holonomy manifolds. A second motivation is to study Spin (7) compactification in the context of M/Fduality and uplifting these to supersymmetry breaking vacua in F-theory [1] based on the observations that 3d N = 1 supersymmetry can be related to the circle-reduction of a not necessarily supersymmetric theory in 4d [2,3].
The construction that we propose, and refer to as generalized connected sum (GCS) construction of Spin(7)-manifolds is motivated by a recent development for compact G 2 -holonomy manifolds in Mathematics, where a large class of compact G 2 -holonomy manifolds were constructed using a twisted connected sum (TCS) construction [4,5,6]. This construction relies on the decomposition of the G 2 -holonomy manifold in terms of two asymptotically cylindrical (acyl) Calabi-Yau three-folds -for a sketch see figure 1. Following these mathematical developments, there has been a resurgence in interest in the string and M-theory compactifications, which for TCS-manifolds have been studied in [7,8,9,10,11,12,13,14]. For a review of M-theory on G 2 and Spin(7)-holonomy manifolds related to earlier results in the '90s and early '00s see [15].
One of the nice features of TCS G 2 -manifolds is that the connected sum gives rise to a field theoretic decomposition in terms higher supersymmetric subsectors [11,13]: the asymptotic regions that are acyl Calabi-Yau three-folds (times a circle), give rise to 4d N = 2 subsectors, whereas the asymptotic neck region where the two building blocks are glued together is by S 1 x Z + \S + 0 S -S 1 x Z -\S -0 HKR S + S -K3 Figure 1: The twisted connected sum (TCS) construction for G 2 -manifolds: The two acyl Calabi-Yau three-fold building blocks are X ± = Z ± \S 0 ± , where S ± denotes the K3-fibers of each building block. Unlike the Spin (7) case, here the gluing is done with a twist (or a Donaldson matching): a hyper-Kähler rotation (HKR) on the K3-fibers, and an exchange of the circles as shown above. The asymptotic region where the gluing is performed is K3×S 1 × S 1 × I. Topologically the resulting G 2 -manifold is K3-fibered over an S 3 .
itself K3 times a cylinder times S 1 , and corresponds to a 4d N = 4 subsector, which is present in the decoupling limit (infinite neck limit). The theory breaks to N = 1, when the asymptotic region is of finite size and the states of the N = 4 vector multiplet become massive, leaving only an N = 1 massless vector. This observation may be used to study non-perturbative corrections, e.g. M2-brane instantons [16], which has been initiated for TCS-manifolds in [7,8,14].
For Spin(7)-manifolds the type of constructions have thus far has been rather limited.
There are (to our knowledge) two constructions of compact manifolds with Spin(7)-holonomy due to Joyce: either in terms of a Joyce orbifold T 8 /Γ [17] or a quotient by an anti-holomorphic involution of a Calabi-Yau four-fold (CY 4 ) [18]. There are variations of these constructions using CY 4 with Spin(7)-necks in [19]. In both constructions, the quotient has singularities, which get resolved and shown to give rise to an eight-manifold with Spin(7)-holonomy. Although it is relatively straight forward to construct string and M-theory compactifications on the Joyce orbifolds, understanding the second class of constructions is far less straight-forward.
Recent attempts were made, in the context of M/F-theory duality for Spin(7)-manifolds [1] in [20,21]. One of the challenges in this is determining the 3d N = 1 theory, which is crucially related to the singular loci of the Calabi-Yau quotient, and not only the sector that is 3d  (7)-manifolds proposed in this paper: the left hand building block is an the acyl Calabi-Yau four-fold building block. We require this to be asymptotically approach a Calabi-Yau three-fold times a cylinder. One way to realize this is in terms of a CY 3 fibration over an open P 1 . The right hand building block is a circle times a G 2 -manifold, which asymptotes to a CY 3 × I times a circle. The simplest way to realize this is in terms of G 2 = (CY 3 × R)/Z 2 where Z 2 acts as an anti-holomorphic involution. The asymptotic region is CY 3 ×cylinder.
In the present paper we provide a different approach to constructing Spin(7)-manifolds, which is closer in spirit to the TCS construction. We will motivate a generalized connected sum (GCS)-construction where the building blocks are an open asymptotically cylindrical (acyl) Calabi-Yau four-fold (CY 4 ) and a open asymptotically cylindrical (acyl) G 2 -holonomy manifold (times S 1 ), which asymptote to a Calabi-Yau three-fold (times a cylinder/line). This is shown in figure 2. Explicitly, we construct the acyl CY 4 as CY 3 → CY 4 → C , (1.1) and the G 2 = (CY 3 × R)/Z 2 , where Z 2 acts as a reflection on the circle and anti-holomorphic involution on the CY 3 . In case the anti-holomorphic involution acts freely, these are so-called 'barely G 2 -manifolds' [16], which have holonomy group SU (3) Z 2 .
Field theoretically we will show that each of the building blocks will give rise to a 3d N = 2 subsector, and the asymptotic region CY 3 × cylinder results in the limit of an infinite neck region in a 3d N = 4 sector. At finite distance, some of the modes of the vector multiplet of 3d N = 4 become massive and the theory is broken to 3d N = 1.
We motivate this construction in section 3 by considering a simple Spin(7) Joyce orbifold, T 8 /Γ, which is known to have a related G 2 Joyce orbifold, where the precise connection is through M-theory on K3 to Heterotic on T 3 duality applied to this toroidal setting [22]. It  .1) and the picture shows the geometry after the quotient. Away from the boundary circle, which is the fixed locus of the involution, we have a CY 3 fibered over a disc, which is locally a CY 4 . Above the boundary circle (dark green) the quotient acts non-trivially on the CY 3 × interval, shown in light green, which results in a G 2 . Pulling the base apart, results in the GCS-decomposition of the resulting Spin(7)-manifold.
is known that such Joyce orbifolds have a TCS-decomposition [12], which we apply to the G 2 Joyce orbifold, and subsequently uplift the decomposition to the Spin(7) Joyce orbifold.
The TCS-building blocks then map precisely to an open CY 4 and G 2 × S 1 , respectively. This motivates our general construction of GCS Spin(7)-manifolds, which will be given in section 4.
One may ask whether these constructions in fact globally fit together to (resolutions of) CY 4 quotients, like in Joyce's constructions in [18]. Whenever the acyl G 2 manifold is realized as (the resolution of) a quotient of X 3 × R, this is indeed the case and we can write our GCS Spin(7)-manifolds globally as resolutions of quotients of CY 4 by an anti-holomorphic involution. Here, one starts with a Calabi-Yau four-fold fibered by a Calabi-Yau three-fold CY 3 as in (1.1), and acts on it with an anti-holomorphic involution on the CY 3 -fiber combined with an action on the base P 1 given by z i ↔ −z j , i = j, where z i are the homogeneous coordinates. The fixed locus of this on the base is a circle and the quotient space is half-CY 4 , which is a CY 3 -fold fibered over a disc, and in the vicinity of the boundary circle we obtain (CY 3 ×R)/Z 2 which is a (barely) G 2 -manifold, times an S 1 . Pulling the base half sphere apart, gives the decomposition into the GCS building blocks. This is sketched in figure 3. The key difference to the examples and constructions in [18] is however that the singularities are either absent or occur over a locus of real dimension at least 1. In Joyce's construction via CY 4 /σ the singularities are only point-like and get resolved using an ALE-space. It seems likely that the analog to this procedure in our construction is gluing in the open, acyl G 2 -manifold.
The paper is structured as follows: we begin with some background material on Spin(7)holonomy and M-theory compactifications to 3d N = 1 in section 2. In section 3 we motivate our construction by first considering a Joyce orbifold example. Section 4 contains our proposal for the general construction of GCS Spin(7)-manifolds. We also comment on how this construction relates to and differs from other known setups and provide an example of a new Spin(7)-manifold. For some examples of GCS constructions there is a K3-fibration of the Spin(7)-manifold, and we utilize this to apply M-theory/heterotic duality and construct the dual heterotic on G 2 compactifications in section 5, and match the spectra of the theories. We conclude with a discussion and outlook in section 6. For reference we include a brief summary of the TCS-construction of G 2 -manifolds in appendix A.
2 Spin(7)-Holonomy, M-theory and 3d N = 1 Theories As a preparation for this work and to introduce some notation, this section reviews some basic aspects of manifolds with holonomy group Spin (7), see [23] for a detailed discussion.

Spin(7)-Holonomy
A compact orientable 8-dimensional manifold Z which has a Ricci-flat metric g with holonomy group contained in Spin(7) supports a closed, self-dual four-form Ψ = * g Ψ, which can be expressed in local coordinates (in which g is the euclidean metric) as (2.1) Here, we use dx ijkl as a shorthand for dx i ∧ dx j ∧ dx k ∧ dx l . This four-form is in the stabilizer of the action of the holonomy group Spin (7). Such a form defines a Spin(7) structure, which is called torsion free if Ψ is closed and self-dual.
Having this structure in place does not necessarily mean that the holonomy group is exactly Spin (7) and, as long as Z is simply connected, we may discriminate between different cases by computing theÂ genus, which tells us the number of covariantly constant spinors: All of these manifolds have b 1 (Z) = 0 and the remaining independent Betti numbers are related to theÂ genus by where b 4 ± are the dimensions of the (anti)self-dual subspaces of H 4 (Z, Q). As we are interested in compactifications of M-theory which preserve N = 1 supersymmetry in three dimensions, we will only be interested in the caseÂ = 1.
For eight-manifolds Z with holonomy contained in Spin(7) andÂ(Z) = 1, a necessary and sufficient condition for the holonomy group to be all of Spin (7) is that Z is simply connected, π 1 (Z) = 0 [17,23]. This still allows for cases with non-trivial subgroups of Spin (7), and we will see examples of such spaces later on. We will refer to manifolds Z withÂ(Z) = 1 and a metric g with hol(g) ⊆ Spin(7) as barely Spin (7)-manifolds.
The dimension of the moduli space of Ricci-flat metrics on a Spin(7)-manifold is given by b 4 − + 1. Together withÂ = 1, this number is already determined by the Euler characteristic and the two Betti numbers b 2 and b 3 by using (2.3) Calibrated submanifolds of Spin (7)-manifolds must be of real dimension four and are called Cayley submanifolds. The dimension of the moduli space of such a Cayley submanifold N is [24,23] where τ (N ) is the Hirzebruch signature and χ(N ) is the Euler characteristic. Note that this expression evaluations to for a Cayley submanifold which is a K3 surface, where τ (N ) = 16 and χ(N ) = 24. It hence seems sensible to assume there exist Spin(7)-manifolds fibered by K3 surfaces over a four-dimensional base. Using duality to heterotic strings, it is precisely the existence of such fibrations which we will conjecture and exploit. In contrast, note that for a Cayley submanifold with the topology of a four-dimensional torus T 4 , the same computation gives so that Spin(7)-manifolds cannot possibly be fibered by calibrated four-tori.

M-theory on Spin(7)-Manifolds
M-theory on a Spin(7)-manifold gives rise to 3d N = 1 supersymmetric theory and the spectrum is encoded in the topological data of the Spin(7)-manifold as follows: Volume modulus (2.8) where ω (2) , ρ (3) are a basis of harmonic two and three-forms, and ξ I are a basis of harmonic anti-self-dual four-forms. For 3d N = 1 the scalar multiplet has only a real scalar as its bosonic component and the vector multiplet is just a 3d vector. As we can dualize the 3d vectors to real scalars, compactifications of M-theory on Spin(7)-manifolds hence give rise to massless real scalars at the classical level. Anomaly cancellation furthermore requires the introduction of space-time filling M2-branes [25,26]. As such M2-branes can freely move on Z, they each contribute a further 8 real degrees of freedom.
In the absence of G 4 -flux, the effective action for a smooth Spin(7)-manifold is a 3d N = 1 field theory with b 2 abelian vectors, and scalars with the following kinetic terms The theory for a smooth Spin(7)-manifold is an abelian gauge theory. With singularities this can enhance to non-abelian gauge symmetries. With G 4 -flux, additional Chern-Simons terms and scalar potential are generated -for an in depth discussion of the effective theory see [27,28,29,30,31,32,33,20]. In this paper we will not consider fluxes but focus on the geometric constructions. Of course it is interesting to consider this in the future and study the effects of these on supersymmetry breaking and potential obstructions to dualities, both M/F-duality [34], where in particular 4d Poincaré invariance could be broken, as well as obstructions to M-theory/weakly-coupled heterotic duality [35,36].

Setup and Motivation
Our goal is to construct new classes of Spin (7)-manifolds -which we have motivated from various points of view in the introduction. The construction which we will end up with is inspired by combining two ideas: 1. The recent construction of G 2 -holonomy manifolds as twisted connected sums (TCS), with each building block an asymptotically cylindrical Calabi-Yau three-fold. We review this construction in appendix A.
Combining these ideas will lead us to consider a generalized connected sum (GCS) construction, where -as we will show -the building blocks are acyl CY four-folds and G 2 -manifolds, respectively. To motive this we start with a well-known construction by Joyce of both G 2 and Spin(7)-manifolds and a well-known duality, between M-theory and heterotic strings, which will be discussed in more detail in section 5. which is both the Narain moduli space of heterotic string theory on T 3 and the moduli space of Einstein metrics on K3 [37] (for a recent exposition in the context of fiber-wise application see [12]). The string coupling on the heterotic side is matched with the volume modulus of the K3. We can fiber this over a four-manifold M 4 in such a way that there is a duality of 3d where Z 8 is K3-fibered over M 4 and J 7 has T 3 -fibers. This duality has been tested in the case when both manifolds are Joyce orbifolds in [22]. More generally, fiberwise application of this duality will lead us to a correspondence between two realizations of a 3d N = 1 theory, in terms of a Spin(7)-compactification of M-theory, and a T 3 -fibered G 2 -holonomy manifold 1 . It is this setup which will motivate our construction of GCS Spin(7)-manifolds: it is the analog of the TCS-decomposition for G 2 -manifolds on the heterotic side, mapped to M-theory using the duality. In this way we obtain a dual pair of connected sums: M-theory on GCS Spin(7)-manifold ←→ Heterotic on TCS G 2 -manifold (3.4) As a warmup we now show how this works for a simple G 2 Joyce orbifold, which has a TCS-decomposition and determine what this decomposition corresponds to on the M-theory side. This will give a first hint as to what the general connected sum construction will be for Spin(7)-manifolds.
3.2 Joyce Orbifolds: Spin(7)-and G 2 -Holonomy We will start the construction of a Spin(7)-manifold Z 8 by considering a Joyce orbifold T 8 /Γ, where each generator of the order two subgroups acts as follows [23]: The entries 1 2 − denote x → −x + 1 2 . The singular sets are locally given by whereas the 4 singularities of the form C 2 /Z 2 × T 4 yield  The key observation to make is that this Spin(7)-manifold gives rise naturally to a G 2manifold as follows: consider the T 7 given by x 1 , x 2 , x 3 , x 5 , x 6 , x 7 , x 8 , and act with Z β 2 ⊕Z γ 2 ⊕Z δ 2 . This is a Joyce construction of G 2 -manifolds by orbifolds as already observed in [22].

TCS-decomposition of the Joyce G 2 -Manifold
Our goal is to first identify the TCS description of this Joyce G 2 -orbifold and to then lift this to a connected sum description, GCS, for the Spin(7)-manifold. First note that T 7 / β, γ, δ is fibered over an interval which identifies this as x 6 ∼ −x 6 ∼ −x 6 + 1 2 , and thus x 6 ∈ I = [0, 1/4]. The generator δ is the stabilizer of x 6 and so (3.11) The notation T 3 (2,6,8) indicates the three-torus along the coordinates x 2 , x 6 , x 8 . We now need to identify the action of β and γ on this space. Again, there will be two components, which we denote by X 0 and X 1/4 , located at x 6 = 0, 1/4, which are given by (3.12) Each of these halves is an open, K3-fibered CY three-fold. To see this introduce the coordinates The K3s are along z 1 , z 2 , where δ acts non-trivially. The remaining coordinates combine with the interval coordinate x 6 . At   (8) . These are glued together along the x 6 coordinate. Both asymptote to X s times a cylinder, where X s is the Schoen Calabi-Yau three-fold. In this particular example each building block contains a K3, which is also present in the neck region of the Schoen Calabi-Yau.

Uplift to GCS-decomposition of the Spin(7)-Manifold
Adding back the circle along the x 4 coordinate, as well as the additional orbifold generator α, α acts trivially on x 6 , so that the fibration over the interval remains intact, but α : z 0 1 → −z 0 1 , so that the Ω (3,0) form is not invariant any longer. Let us now again define two building blocks along the x 6 interval, close to either boundary. The interesting observation is that the two halves behave quite differently: first consider x 6 = 0. Define another complex coordinate with the action of α Together with the action of α on z 0 i , we see that Ω is invariant, and we obtain a building block which is an open Calabi-Yau four-fold At the other end of the x 6 interval we have where M 7 will be shown to be a G 2 holonomy manifold. The action of α is α : From this we can define the G 2 -form which is invariant, as follows from the action of α and the coordinates z 1/4 .
In summary we obtained a GCS-construction of a Spin(7) Joyce orbifold Z, which has two building blocks, an open CY 4 and an open G 2 , W times a circle, respectively. The geometry in the middle of the x 6 internal is where the X s is a CY 3 along the directions 1, 2, 3, 4, 5, 7. Not too surprisingly, this is in fact the Schoen Calabi-Yau three-fold. The action along these coordinates, written in terms of the complex coordinates of Z 0 is α : This is precisely the action required to get the Schoen Calabi-Yau as a Joyce orbifold (see [12] for a detailed discussion of that). Inside X s , there is a K3 surface along z 0 1 , z 0 2 . On the other hand the CY 4 Z 0 is fibered by X s over x 6 , x 8 .
Note that we may also think about this Spin(7) orbifold as a quotient of a Calabi-Yau four-fold X 4 by an anti-holomorphic involution. The action of α, β, δ respects the holomorphic coordinates z 0 i on T 8 and leaves Ω (4,0) 0 invariant, so that it produces a Calabi-Yau orbifold.
The full Spin (7) orbifold is then formed by acting with γ, which acts as an anti-holomorphic involution. We expect this structure to persist after resolution.
The Joyce orbifold (3.5) has several generalizations, which can be parametrized in terms of the following data: where the shift vectors can take the following values

(3.22)
Case I is the orbifold studied above. Instead of the fibration over the interval (3.10), in this general setting, we consider pulling these orbifolds apart along x 3 , with the interval There is an asymptotic middle-region, where the space is X s = T 6 (1,2,5,6,7,8) / β, γ , which in fact is again the Schoen Calabi-Yau three-fold for all of these Joyce orbifolds. The asymptotic manifolds at the fixed points x 3 = 0, 1 4 are again open a Calabi-Yau four-fold and G 2 -manifold M 7 , respectively, given by (3.23) The analysis will follow very much along the lines of the first example we studied and we will instead now move to generalize this construction beyond Joyce orbifolds.
We have seen that the Joyce orbifold Z 8 (3.5) has a natural decomposition in terms of a GCS-construction, where one building block is an open Calabi-Yau four-fold, and the other is a G 2 -manifold times a circle, where both geometries asymptote to a Calabi-Yau three-fold X s times a cylinder. With this example in mind, we now turn to providing a generalization of this construction in the next section.

The Construction
In the discussion of the last section, we were led to think about Spin (7)  We are now going to propose a generalization of this procedure in which both the two building blocks Z ± and the resulting Spin (7)-manifold Z are no longer (resolutions of) orbifolds. To start this discussion, let us first propose the following definitions: Definition: An asymptotically cylindrical (acyl) Calabi-Yau four-fold Z + is a non-compact algebraic fourfold, which admits a Ricci-flat metric, is simply connected, and is diffeomorphic (as a real manifold) to the product of a cylinder I × S 1 and a compact Calabi-Yau three-fold X 3 outside of a compact submanifold κ. The Ricci-flat metric of Z + exponentially asymptotes to the Ricci flat metric on the product I × S 1 × X 3 ∼ = X 4 \ κ.
Asymptotically Calabi-Yau manifolds were discussed in [38]. Similar to the explicit construction of acyl three-folds in [5,6], we expect to able to construct Z + by excising a fiber X 3 from a compact four-foldZ + with c 1 (Z + ) = [X 3 ], which is fibered by Calabi-Yau threefolds. This implies that we can think of producing Z + by appropriately cutting a compact, CY 3 -fibered Calabi-Yau four-fold in half, or via toric methods as in [9]. Likewise we define: Definition: An asymptotically Calabi-Yau (acyl) G 2 -manifold Z − is a non-compact manifold of G 2 holonomy, which is diffeomorphic to the product of an interval I with a compact Calabi-Yau threefold X 3 outside of a compact submanifold of Z − , and the Ricci-flat metric on Z − exponentially asymptotes to the Ricci-flat metric on the product manifold I × X 3 .
Such manifolds were discussed in [39]. Note that such G 2 -manifolds are easily constructed as (resolutions of) orbifolds (X 3 × R t )/Z 2 with the Z 2 acting as t → −t and as an antiholomorphic involution on the Calabi-Yau three-fold X 3 . The quotient by such involutions can be thought of as a Calabi-Yau three-fold X 3 fibered over an half-open interval which undergoes a degeneration at one end. In particular, it is not surprising if the Ricci-flat metric on such a resolution of (X 3 × R)/Z 2 asymptotes to the Ricci-flat metric on X 3 × R far away from the origin of R. A particularly simple case is given by freely acting anti-holomorphic involutions. In this case no resolution is required, and the Ricci flat metric on ( is simply the quotient of the Ricci-flat metric on X 3 × R. The simplicity of this examples comes at a price, however, as the holonomy group of such acyl G 2 -manifolds is not the full G 2 [40,39], but only SU (3) Z 2 , so that we can call them acyl barely G 2 -manifolds. Such acyl barely G 2 -manifolds in particular have a nontrivial fundamental group Z 2 .
We are now ready to propose our construction of what we will call generalized connected sum (GCS) Spin (7)-manifolds. Take an acyl Calabi-Yau four-fold Z + with asymptotic neck Then Z + and Z − × S 1 θ − can be glued as topological manifolds to a manifold Z by identifying the neck regions for a biholomorphic map φ. If the Ricci-flat metrics on Z ± asymptote to the Ricci-flat metrics on X ± 3 , this takes us close to a Ricci-flat metric on Z. We conjecture that for long enough neck regions (l large enough) there exists a Ricci-flat metric g associated with a torsion free Spin(7) structure on Z, which is found by a small perturbation of the Ricci-flat metrics on Z + and Z − × S 1 .
If Z ± are both simply connected, it follows from the Seifert-van Kampen theorem that Z is a simply connected eight-manifold. This means that the holonomy group of Z must be equal to Spin(7) (and not a subgroup) if Z has a torsion-free Spin(7) structure.
Before exploring the consequences of our proposed construction further, let us briefly discuss the mathematical work needed to put our proposal on firm ground. The work [40,39,38] on acyl Calabi-Yau four-folds and acyl G 2 -manifolds and their deformation theory, together with clear criteria when the asymptotic Calabi-Yau three-folds X ± 3 allow a biholomorphic map φ, should clarify under which circumstances a gluing can be found for a given pair of such manifolds. In our examples, we can easily find such diffeomorphisms by realizing X ± 3 as hypersurfaces in toric varieties, so that a diffeomorphic pair can be simply constructed by writing down identical algebraic equations. The crucial task is then to show the existence of a Ricci-flat metric with holonomy Spin(7) on the resulting topological manifold Z. By making the neck regions very long, we expect that the torsion introduced when gluing Z + and Z − × S 1 can be made sufficiently small for a torsion-free Spin(7) structure to exist nearby.
As we shall discuss in more detail in section 4.5, there are instances of GCS Spin (7)

Calibrating Forms
Both the acyl Calabi-Yau four-fold Z + and the product of the acyl G 2 -manifold Z − with a circle have a torsion free Spin (7) structure. In this section we show how the identification (4.1) between the asymptotic neck regions produces a globally defined Spin (7) structure.
Let us denote the Kähler form and holomorphic four-form of Z + by ω + and Ω + . The Cayley four-form defining a torsion-free Spin(7) structure on Z + is then given by In the asymptotic neck region, we can introduce coordinates t + and θ + and decompose the SU (4) structure as where ω 3,+ and Ω 3,+ are the Kähler form and holomorphic three-form on X + 3 . This means that the Spin(7) structure in the neck region is Similarly, the Spin(7) structure on Z − × S 1 θ − is given by in terms of the G 2 structure ϕ − on Z − . In the neck region, this is further decomposed as in terms of the Kähler form and holomorphic three-form on X − 3 . We hence find that the Spin(7) structure on the neck region of Z − × S 1 can be written as This means that the two Spin (7) structure Ψ ± are consistently glued together under a diffeomorphism which identifies the neck regions as which is noting but the map (4.1) proposed earlier.

Topology of GCS Spin(7)-Manifolds
The easiest topological number to determine for the GCS Spin (7)-manifold Z is given by the Euler characteristic χ(Z). As it is additive, we have where we have used the fact that the Euler characteristic vanishes for any manifold with an S 1 factor. Two copies Z + , Z + of the acyl Calabi-Yau four-fold Z + can be glued to a compact Calabi-Yau four-fold X 4 = Z + ∪ Z + such that Z + ∩ Z + = I × S 1 × X 3 for a Calabi-Yau three-fold X 3 . It follows that χ(X 4 ) = 2χ(Z + ) = 2χ(Z). As the Euler characteristic of any Calabi-Yau four-fold is divisible by six [41], this implies that χ(Z) is divisible by 3, so that b − 4 in (2.4) is always an integer.
As we have defined an acyl Calabi-Yau four-fold to be simply connected, but left room for the possibility of an acyl G 2 -manifold to have a non-trivial fundamental group, the Seifert-van Kampen theorem tells us that Z is simply connected if an only if Z − is simply connected. If Z − is not simply connected, Z can have a non-trivial fundamental group, which signals that the holonomy group of Z is smaller than Spin(7) [17,23].
The easiest way to construct acyl G 2 -manifolds is by a free quotient X 3 × R by Z 2 , in which case the fundamental group of Z − is not trivial but equal to Z 2 . This works as follows.
Consider two points identified by the anti-holomorphic involution on X 3 over the origin of R. Any path connecting two such points becomes a closed loop in the quotient and, as the involution acts freely, cannot be homotopic to a point. If X 3 is simply connected, all such loops are homotopic, so that π 1 ((X 3 ×R)/Z 2 ) = Z 2 , which implies that Z − only has holonomy group SU (3) Z 2 , i.e. is a barely G 2 manifold. On Z, such loops give rise to a non-trivial fundamental group, which is Z 2 as well, so that Z does not have the full holonomy group Spin (7). From the point of view of physics, compactification on such 'barely' Spin(7)-manifolds does not give rise to extended supersymmetry as there is still only one covariantly constant spinor. In fact, it is not hard to see that Z has holonomy group SU (4) Z 2 in this case: the holonomy group SU (4) of Z + and the holonomy group SU (3) Z 2 of Z − share a common SU (3), the holonomy group of X 3 .
We can compute the cohomology groups of Z in terms of the cohomology groups of Z ± and the pull-backs to X 3 by using the Mayer-Vietoris exact sequence, which implies that can be expressed in terms of the restriction maps Note that both β i − and β i−1 − feature in γ i due to the product S 1 . Under a few assumptions, which are met in the examples to be discussed later, we can now explicitly work out the various contributions to H • (Z). First of all, let us assume that the acyl Calabi-Yau four-fold Z + can be constructed from an algebraic four-foldZ which is fibered by Calabi-Yau three-folds by excising a fiber X 3 . Furthermore, let us assume that the images of β 2 + and β 4 + are surjective and that b 3 (Z + ) = b 5 (Z + ) = 0. As noted above, we assume that we can construct Z − as (a resolution) of the quotient (X 3 ×R t )/Z 2 by an anti-holomorphic involution. The resolution must be such that it preserves a metric of holonomy G 2 on Z − and the property of Z − being asymptotically cylindrical. We will denote the numbers of even/odd classes of X 3 under this Z 2 by b i e and b i o , respectively. Finally, the kernels of the restriction maps β i ± and their ranks are abbreviated as (4.13) With this notation and under the assumptions we have made, (4.10) implies that (4.14) This can be seen as follows. For H 1 (Z), the unique class in H 1 (X 3 × S 1 ) is in the image of γ 1 and solely originates from Let us now consider b 2 (Z). As the unique class in H 1 (Z − × S 1 ) is in the image of γ 1 , we find that coker(γ 1 ) = 0 so that H 2 (Z) = ker(γ 2 ) follows. In turn, ker(γ 2 ) has three different contributions: As β 2 + is surjective by assumption, the last term is simply given b 2 e and we recover the expression in (4.14).
The computation for b 3 (Z) is made particularly simple by our assumptions. As | coker(γ 2 )| = 0, it only receives a contribution from ker(γ 3 ). As b 3 (Z + ) = 0, the only contribution to b 3 (Z) , all potential contributions are non-trivial. First of all, there is a contribution n 4 + from ker β 4 + , as well as n 3 − + n 4 − from ker β 3 − + ker β 4 − . Furthermore, there are b 2 o classes in coker(γ 3 ), which correspond to three-forms with one leg on the product S 1 of the cylinder region, and b 3 o classes in coker(γ 3 ) which correspond to three-forms purely on X 3 . Finally, there is a contribution which simply has dimension b 4 e + b 3 e as β 4 + is surjective. Together, all of these contributions give the expression for b 4 (Z) in (4.14).
The computation of b 5 (Z) again only receives a contribution from ker γ 5 as coker γ 4 is empty by assumption. Furthermore, γ 5 is only non-trivial on Z − ⊗ S 1 , so that | ker γ 5 | = n 4 − + n 5 − . The computation of b 6 (Z) is similar to b 2 (Z), it receives a term b 4 o from coker(γ 5 ), as well as terms n 5 − and n 6 + from ker(γ 6 ). Note that N 6 − = 0 as Z − is a G 2 manifold. The number of deformations of the Ricci-flat metric is given by b 4 − + 1 for manifolds of Spin(7)-holonomy. Using (2.4), this can be written as which becomes in our GCS-construction. The Euler characteristic of Z is given by and only depends on the data of Z + , as expected.
Even though the topology of the resolutions of Joyce's Spin (7)-orbifolds discussed in section 3 is most conveniently computed otherwise, one can also use their decomposition as GCS Spin (7)-manifolds together with the relations of this sections to find their topology from resolutions of two building blocks Z ± .

3d Field Theory and Sectors of Enhanced Supersymmetry
In compactifications of M-Theory on a smooth Spin(7)-manifold, the low energy effective theory at the classical level gives a 3d N = 1 supergravity theory with n v = b 2 (Z) massless U (1) vector multiplets and n r = b 3 (Z) + b 4 − (Z) + 1 massless uncharged real scalar multiplets. As the Spin(7)-manifolds considered here are glued from pieces with holonomy SU (4) and G 2 , respectively, we expect to find subsectors of enhanced supersymmetry in our spectrum, similar to the observations made in [11] regarding TCS G 2 manifolds, to arise from localized forms in the building blocks.
Consider first the multiplets arising from localized forms on Z − × S 1 . By (4.14), each two-form in N 2 − gives rise to both a two-form and a three-form on Z, which combine to form the bosonic field content of a 3d N = 2 vector multiplet. Furthermore, each three-form in − gives rise to a real scalar due to its appearance in the formula for b 3 (Z) in (4.14). As for compactifications of M-Theory on a G 2 -manifold times a circle, the moduli associated to these three-forms pair up with deformations of the metric to form 3d N = 2 chiral multiplets: this can be seen from (4.18), which shows that each three-form in N 3 − corresponds to an anti-self-dual four-form, i.e. a deformation of the metric.
Let us now turn to multiplets which arise from localized forms on Z + . Each two-form of Z + in N 2 + gives rise to a 3d N = 1 vector multiplet due to its contribution to b 2 (Z). As for compactifications of M-Theory on Calabi-Yau four-folds, we expect this degree of freedom to pair up with a real scalar to the bosonic degrees of freedom of a 3d N = 2 vector multiplet.
This degree of freedom must originate from an anti-self-dual four-form in N 4 + . Furthermore, we expect the remaining anti-self-dual four-forms in N 4 + to appear pairwise, so as to combine into 3d N = 2 chiral multiplets.
To see how this comes about, we need to exploit the fact that Z + and its compactificatioñ Z + are Kähler manifolds and carry a Lefschetz SU (2) action. The upshot is that the antiself-dual four-forms onZ + are given by the four-forms in H 3,1 (Z + ) ⊕ H 1,3 (Z + ), together with four-forms of the type ω k ∧ J, where J is the Kähler form onZ + and the ω k are (1, 1) forms onZ + such that ω k ∧ J 3 = 0. As every two-form in N 2 + gives a four-form in N 4 + upon wedging with J, we can hence associate an anti-self-dual four-form in N 4 + on Z + to every element of N 2 + (modulo the (1, 1) forms in the image of β 2 + ). The remaining anti-self-dual four-forms in N 4 + then have to appear pairwise, as they must correspond to H 3,1 (Z + ) ⊕ H 1,3 (Z + ). As we can think about the forms in N 4 + as being localized on the acyl Calabi-Yau four-fold Z + far away from the gluing region, (anti) self-duality on Z + will imply (anti) self-duality of the corresponding forms on Z.
Note that this argument precisely reflects how these degrees of freedom originate in Physics.
For two-forms in N 2 + and self-dual four-forms in N 4 + , there are deformations of the Ricci-flat metric of Z + (Kähler and complex structure deformations), which do not alter the cylindrical region in which Z + is glued to Z − × S 1 . As we expect the Ricci-flat metric of Z to be well approximated by the Ricci-flat metrics of Z + far away from the gluing region, these become deformations of the Spin (7)-manifold Z if we stretch the neck region sufficiently long. The metric deformations associated with N 2 + and N 4 + , together with the 3d vectors originating from the C-field on N 2 + , only see the geometry of a Calabi-Yau four-fold and hence give rise to a subsector with 3d N = 2 supersymmetry. As the property of forms being (anti) self-dual is a topological constraint, the counting and identifications we have performed will persists throughout the moduli space of Z, so that the subsectors with enhanced N = 2 supersymmetry in 3d will persist. We have summarized the result of this discussion in table 1 3d Multiplets Table 1: Subsectors with enhanced supersymmetry and their topological origin. The N = 4 vector multiple from the neck region is only massless in the infinite neck limit and away from this becomes an N = 1 vector.
The remaining moduli of M-Theory on the GCS Spin (7)

GCS Spin(7)-Manifolds as Quotients of CY 4
Our construction has a natural connection to Spin(7)-manifolds constructed as quotients of Calabi-Yau four-folds by anti-holomorphic involutions, however we will see that there are key difference to the previous constructions in [18,20]. Consider a compact Calabi-Yau four-fold X 4 , which is fibered by Calabi-Yau three-folds over a base P 1 with homogeneous coordinates [z 1 : z 2 ] and let z = z 1 /z 2 . Let us denote the fiber over a point p of the P 1 base by X 3 (p).
For appropriate fibrations, we may then tune the complex structure (defining equation) of X 4 such that For appropriate anti-holomorphic involutions σ acting on X 3 (p) for all p, we may then construct an anti-holomorphic involution acting in X 4 . This will produce a singular Spin (7)-manifold Z s = X 4 /Σ, and we will assume that it can be resolved into a smooth Spin(7)-manifold Z.
In the P 1 base the fixed locus of this involution will be given by the circle S 1 f = {|z| 2 = 1} and the quotient effectively truncates the base P 1 to a disc with boundary. We may make this disc very large and furthermore confine all of the singular fibers of the fibration of X 3 (p) near the origin. If the fibration of X 3 (p) over the fixed S 1 f is trivial, the fibers over S 1 f all become identical and we may simply denote them by X 3 . Cutting along a circle of fixed radius now produces one half near the boundary of the disc which may be described as In the limit in which the disc is very large, a resolution of Z s to Z is equivalent to a resolution of (X 3 ×R)/σ to a smooth acyl G 2 manifold Z − . The other half of Z near the origin of the disc does not require resolution and becomes an acyl Calabi-Yau four-fold Z + which asymptotes to X 3 . In the middle region, the two spaces Z + and Z − × S 1 f are glued along the product of an interval and X 3 × S 1 f . This is precisely the gluing construction we have given above. An illustration is given in figure 3.
Let us highlight two aspects of the topology of Z which are immediately recovered from this point of view. The resolution of Z s to Z only introduces new cycles sitting over a circle, so that the topological Euler characteristic of Z is the same as that of Z s . But the topological Euler characteristic of Z s is simply given by half of the Euler characteristic of X 4 . In fact, we may think of X 4 as being glued from two copies of the acyl Calabi-Yau four-fold Z + , which implies that the Euler characteristic of Z is equal to that of Z + , which is the same result found from our gluing construction earlier. Furthermore, we have observed that we produce a barely Spin(7)-manifold with fundamental group Z 2 , whenever Z − is constructed from a free quotient of X 3 × R by σ. This means that we may write Z as the free quotient X 4 /Σ, so that we immediately recover π 1 (Z) = Z 2 .
As we have explained, a GCS Spin(7)-manifold can be described as a resolution of a quotient of a Calabi-Yau four-fold by an anti-holomorphic involution whenever the acyl G 2 manifold Z − can be described as a resolution of (X 3 × R)/σ for an anti-holomorphic involution σ. In the absence of a different construction for acyl G 2 manifolds, our GCS construction is hence equivalent to forming quotients of a specific class of Calabi-Yau four-folds. In this context, it implies that a resolution of singularities of the quotient by Σ is already captured by resolving quotients (X 3 × R)/σ. Furthermore, it gives a completely new point of view on such geometries; it shows how to distinguish subsectors of enhanced supersymmetry and, as we shall see, allows to construct dual heterotic backgrounds on TCS G 2 manifolds. Finally, the realization that many GCS Spin(7)-manifolds are simply (resolutions of) quotients X 4 /Σ in fact proves that our construction produces eight-dimensional manifolds with Ricci-flat metrics with holonomy group Spin(7) (or SU (4) Z 2 ).
In [18] Joyce proposed another construction of Spin(7)-manifolds base on non-free quotients of Calabi-Yau four-folds with orbifold singularities was given. Crucially, the resulting singular Spin(7) orbifolds are taken to only have isolated singularities here, which can be resolved by gluing in appropriate ALE spaces. Even though similar in spirit, the anti-holomorphic involution appearing here are either free or have fixed loci of real dimension at least one, so these constructions are distinct. It seems likely, however, that there are examples for which the resolutions can be described by gluing in a acyl G 2 manifold times a circle, so that there will be many Spin (7)-manifolds which can be found using both constructions.

A Simple Example
We now construct a new Spin(7)-holonomy manifold using the GCS construction. There will be further examples in the next section, which are geared toward the application of Mtheory/Heterotic duality.
Consider a smooth anti-canonical hypersurface in weighted projective space P 11114 . The resulting space is a Calabi-Yau three-fold X 1,149 with Hodge numbers h 1,1 = 1 and h 2,1 = 149.
We now show how to construct an acyl G 2 -manifold Z − which asymptotes to X 1,149 × R and an acyl Calabi-Yau four-fold Z + which asymptotes to X 1,149 × S 1 × R, and then glue Z + and Z − × S 1 to a Spin(7)-manifold Z as described above. Let us first describe the acyl G 2 -manifold Z − . The Calabi-Yau three-fold X 1,149 admits a freely acting anti-holomorphic involution, which is obvious by choosing a Fermat type hypersurface where [x 1 : x 2 : x 3 : x 4 : x 5 ] are the homogeneous coordinates of P 11114 . Letting x i →x i then acts as a fixed-point free anti-holomorphic involution, as the above equation has no solutions purely over the reals. The odd/even Hodge numbers are and (X 1,149 × R)/Z 2 is an acyl G 2 -manifold. Note that n i − = 0 for all i as no resolution of singularities is required on Z − .
Note that we may choose an element of the algebraic familyZ + such that the fiber over some chosen point in the base P 1 is described by (4.23), so that Z + manifestly asymptotes to a Calabi-Yau three-fold which is isomorphic to the Calabi-Yau three-fold in the neck region of Z − . For the resulting GCS Spin (7) Using (2.4), this is most efficiently calculated by using χ(Z) = 1680. One can also work out that n 4 + = 1374, which reproduces b 4 (Z) = 1676 using (4.14). Due to the free action of the anti-holomorphic involution on X 1,149 , this example produces a barely Spin(7)-manifold. By considering more general anti-holomorphic involutions and resolving the resulting singularities, similar examples with full holonomy Spin (7) can be constructed from this procedure.
In agreement with our general discussion in section 4.5, Z may also be found as the free quotient of a Calabi-Yau four-fold X 4 realized as a hypersurface in P 11114 × P 1 . The Hodge numbers of X 4 are and χ(X 4 ) = 3360. The free involution Σ acts as where [z 1 : z 2 ] are homogeneous coordinates on the base P 1 and x i are the homogeneous coordinates on P 11114 . We may choose a smooth invariant hypersurface on which Σ acts without fixed points as  and an open acyl G 2 -manifold Z − (times a circle), which are both asymptote to a cylinder times the same Calabi-Yau three-fold X 3 . The additional input in this section is that we will consider these in the context of the duality to Heterotic strings theory on G 2 -manifolds, which we show to be TCS-manifolds, with building blocks X ± .
Our strategy in finding dual compactifications will be similar to the one we used in [12], i.e. using fiberwise duality between M-theory and heterotic string theory, we will identify dual pairs of building blocks, which can then be glued to find the dual Spin (7) and G 2 -manifolds.
As this requires to carefully dissect geometries on both sides, the following discussion will unfortunately be rather technical. After motivating our construction for a dual pair of geometries, we will summarize our results in Section 5.4 before checking the spectra. In our first example, the

Heterotic String Theory on G 2 -Manifolds
Before we begin with a detailed discussion of dual pairs, we should explain how to construct 3d N = 1 theories from heterotic string theory on a G 2 -holonomy manifold together with a vector bundle. Here we will discuss briefly heterotic compactifications on G 2 -holonomy manifolds.
A compact orientable 7-dimensional manifold J which has a Ricci-flat metric g with holonomy group contained in G 2 supports a closed three-form Φ, the Hodge- * -dual of which is closed as well, which can be expressed in local coordinates (in which g is the euclidean metric) as This three-form and its Hodge- * -dual are preserved by the action of G 2 . Besides having a Ricci-flat metric, G 2 -manifolds support a single covariantly constant spinor.
In contrast to Spin(7)-manifolds, a necessary and sufficient condition for the holonomy group of a manifold J with a Ricci-flat metric g and hol(g) ⊆ G 2 ,to be equal to G 2 is that the fundamental group of J is finite rather than trivial. We will refer to manifolds which support a single covariantly constant spinor, but have holonomy group SU (3) Z 2 as barely Let us now discuss compactification of heterotic E 8 × E 8 strings on G 2 -manifolds. Earlier studies of this in the context of resolutions of Joyce orbifolds T 7 /Γ have appeared in [22,42,43].
The most in depth analysis of the conditions that such a compactification needs to satisfy were derived in [44] and are given in terms of a heterotic G 2 -system. This is comprised of a G 2holonomy manifold and its tangent bundle (J, T J) as well as an E 8 × E 8 -vector bundle with connection (V, A). The curvatures R and F of T J and V satisfy Furthermore there can be NS5-branes wrapped on associative three-cycles, subject to the where dH =[NS5] the Poincaré dual of the homology class corresponding to associatives wrapped by NS5-branes.  (7) holonomy, we expect the resulting geometry to be in this class.

GCS Spin(7)-Manifolds and M-theory/Heterotic Duality
We already discussed the 7d duality between M/K3 and Het/T 3 in section 3.1. We now apply this fiberwise to construct dual pairs of TCS and GCS constructions, in cases when there is a K3-fibration on the GCS Spin(7)-manifold and dually on the heterotic side there is a T 3 -fibered TCS G 2 -manifold. In this section we make some initial observations about the general features of these compactifications.
If J is a TCS G 2 -manifold, mirror symmetry for Type II strings implies the existence of a calibrated T 3 -fibration, which realizes a mirror map in the spirit of Strominger-Yau-Zaslow (SYZ) [45] via three T-dualities [13]. Such a fibration exists if one of the two building blocks, say X + , used in the TCS-construction of J is fibered by K3 surfaces S + , which in turn admit an fibration by an elliptic curve E E → S + → P 1 . • X + : we replace the T 3 -fibration given by E × S 1 e,+ on X + × S 1 e,+ , with a K3 surface S + . Then only two out of the three forms of the hyper-Kähler structure of S + have a nontrivial fibration over the base. In particular, we may choose to described the resulting K3 fibration algebraically and end up replacing the product of the acyl Calabi-Yau three-fold X + with S 1 e,+ by an acyl Calabi-Yau four-fold Z + .
• X − : here, the SYZ-fiber of X − is fibered such that application of Heterotic/M-theory duality leads to a K3-fibration in which all three (1,1)-forms of the hyper-Kähler structure have a non-trivial variation over the base. As in the 4d N = 1 duality between heterotic strings and M-theory, we replace a Calabi-Yau three-fold by a G 2 -manifold.
In the present setup, this means replacing an acyl Calabi-Yau three-fold X − with an acyl G 2 -manifold Z − . In the cylinder region of J, where J is diffeomorphic to a product S × S 1 × S 1 × I for a K3-surface S with a calibrated T 2 -fibration 2 , we simply find X 3 × S 1 × I by application of Heterotic/M-theory duality.
We hence find the statement that M-theory duals of heterotic strings on TCS G 2 -manifolds are compactified on Spin(7)-manifolds, which allows a construction as proposed in our GCSconstruction: an acyl Calabi-Yau four-fold Z + and an acyl G 2 -manifold Z − times an S 1 are glued along a cylinder times a Calabi-Yau three-fold.
In our discussion, we have so far ignored that the heterotic compactification comes equipped with a vector bundle on J, and we need to identify what this is mapped to on the dual Mtheory geometry. For X + , where only E varies non-trivially, replacing X + × S 1 e,+ by Z + is simply a non-compact version of the 3d duality between heterotic strings and M-theory on Calabi-Yau four-folds. We can hence apply the usual logic of how bundles constructed from spectral covers are translated in F-theory [46]. Restricting J to X − × S 1 e,− , holomorphic vector bundles on a (non-compact) Calabi-Yau three-fold get translated to the geometry of a (noncompact) G 2 -manifold, similar to discussed in [12]. Gluing Z + and Z − × S 1 to a compact Spin(7)-manifold Z then implies that the geometry of Z determines a heterotic G 2 -system on the G 2 -manifold J. It is hence tempting to construct such vector bundles on twisted connected sum G 2 -manifolds by appropriately gluing holomorphic vector bundles on the building blocks X ± , see [47] for a concrete realization of this idea. In section 5.5 we will use this logic to count the number of bundle moduli on a TCS G 2 -manifold and confirm a matching with the degrees of freedom of the dual M-theory compactification on a GCS Spin(7)-manifold.

Duality for the Building Blocks
Before heading to the full construction of M-theory/Spin (7) and Heterotic/G 2 we first explore the dual pairs that are relevant for each building block. Recall: • GCS Spin(7)-manifold Z is built out of an acyl CY 4 building block Z − and an acyl G 2 -building block Z + .
• TCS G 2 -manifold J is build out of two acyl CY 3 , X ± .
We will now construct dual pairs for CY 4 and G 2 and correspondingly on the heterotic side CY 3 in sections 5.3.2 and 5.3.3. These will then be used as building blocks to construct the GCS/TCS dual pairs. As a preparation we first need to construct pairs of G 2 and CY 3 and duals for M/het duality.

Geometric Preparation: Dual Pairs of G 2 and CY 3
In this section we discuss a well-suited example for building the G 2 building block of the GCSmanifold, by consider M-theory on G 2 -manifold and the heterotic on CY 3 compactification.
Similar to the strategy used in [16], we can find such a pair by forming an appropriate quotient  On the type IIA side we act with a free involution Z o 2 to produce a G 2 -manifold 3 M as On the dual heterotic side, the corresponding involution Z h 2 acts as a combination of z → −z (on the T 2 h factor) together with the Enriques involution 5 on the K3 surface to produce a Calabi-Yau three-fold with Hodge numbers h 1,1 (X 11,11 ) = h 2,1 (X 11,11 ) = 11. Furthermore, it acts on the bundle data by twisting the two Wilson lines such as to break E 8 × E 8 to U (1) 8 . We can think about the surviving bundle data as an extension of a line bundle associated with a Wilson line breaking E 8 × E 8 to U (1) 8 , and there are eight complex degrees of freedom specifying this bundle.
The orbifold furthermore identifies the 24 NS5-branes on T 2 h pairwise 6 , so that the number of 4d N = 1 U (1) vector and chiral multiplets on the heterotic side is given by n c = 1 + 2 · 11 + 3 · 12 + 8 = 67 (5.13) in perfect agreement with the dual M-theory on the G 2 -manifold M . With this preparation in hand we can now study the duality for the building blocks. 4 Another way to see this is as follows: the three-fold X43,43 is fibered by elliptic K3 surfaces with two II * over P 1 and this involution acts as the antipodal map of the base of this elliptic fibration (another way to see why it acts freely) in particular identifying the two II * fibers. It furthermore acts (non-freely) on the P 1 base of the K3 fibration and identifies the 24 reducible K3 fibers on X3 pairwise. This reproduces why there are 8 + 12 = 20 even classes in b 2 (X3). 5 The Enriques involution is the unique fixed-point free non-symplectic involution on a K3 surface and is identified by the invariants (r, a, δ) = (10, 10, 0) in the classification of [48]. 6 Equivalently, one can compute that ch2(X11,11) = 12[T 2 h ].

The acyl G 2 -Manifold and its dual acyl CY 3
We can turn X 11,11 and M in the last subsection into a dual pair of an acyl Calabi-Yau threefold and an acyl G 2 -manifold by cutting each of them in half. Let us start with M in (5.7).
By cutting along the middle of the interval remaining of the circle S 1 u , M is turned into an acyl (barely) G 2 -manifold Z − which we may represent by where On the heterotic side, Calabi-Yau three-fold X 11,11 (5.12) carries a natural (almost trivial) where the base P 1 h is found as the quotient of T 2 h . Over four points on this P 1 h , the K3 fiber is truncated to an Enriques surface, but is constant otherwise. Note that this way of thinking reproduces the fact that the Euler characteristic of X 11,11 vanishes from χ(X 11,11 ) = 24 · (2 − 4) + 4 · 12 = 0 , (5. 16) where we have excised the four fibers which are Enriques surfaces from the base P 1 h in the first term and added them back in for the second. If we now cut P 1 h into two halves, each of which is C and contains two of the Enriques fibers over it, we produce an acyl Calabi-Yau three-fold X − as the TCS G 2 building block where L(2) denotes a lattice L with its inner form rescaled by a factor of two. The expression for N (X − ) is nothing but the even sublattice of H 2 (K3, Z) under the Enriques involution.
Furthermore, X − may be compactified to a compact building blockX − in the sense of [5] by gluing in a single K3 fiber. It now follows from a computation as above that χ(X − ) = 24, which together with b 2 (X − ) = 11 (and b 1 (X − ) = 0 implies that b 3 (X − ) = 0. This is all of the data we will need in using X − as a building block for a G 2 -manifold.
Note that we have indeed decomposed M and X 11,11 by cutting along the same S 1 . This becomes particularly clear, by noting that these dual compactifications descend from a compactification to five dimension, in which M-theory is put on X 43,43 and heterotic string on K3×S 1 . The dual models we have constructed are then found by compactifying both side on a further S 1 u and quotienting by Z 2 . It is the quotient of S 1 u , which becomes one of the two circles on T 2 h , which is cut in half to produce Z − and X − , respectively.

The acyl CY 4 and its dual acyl CY 3
We now construct a dual pair of an acyl Calabi-Yau four-fold Z + (as a building block for the Spin(7) GCS) and the heterotic dual acyl Calabi-Yau three-fold X + (as a building block for the G 2 TCS). As we want Z + to asymptote to X 43,43 so that we can glue Z + with Z − × S 1 , we construct so that the algebraic three-foldZ + satisfies Note that we may also construct Z + by cutting a Calabi-Yau four-fold X 4 , which is a fibration of X 43,43 over P 1 in two halves. This four-fold is a toric hypersurface defined by the pair so that χ(X 4 ) = 2208. As a generalization of [9] to four-folds,Z + can also be constructed from one of the two isomorphic tops in ∆ • corresponding to the fibration by X 43,43 . Either way, it follows that the Euler characteristic of Z + is χ(Z + ) = 1104, half of the Euler characteristic of This will be all the data from Z + we need in the following.
Let us now find the dual geometry for the heterotic string. As Z + and X 4 are fibered by elliptic K3 surfaces, this is fairly straightforward, as we simply need to apply the known rules of heterotic-F-theory duality. In particular, Z + and X 4 are fibered by elliptic K3 surfaces with Picard lattice U ⊕ (−E 8 ) 2 , so that the dual heterotic theory has no E 8 × E 8 vector bundles turned on. Furthermore, Z + and X 4 are elliptic fibrations over (blow-ups of) C × P 1 × P 1 . These blowups capture the presence of the NS5-branes needed on the heterotic side to satisfy the Bianchi identity. Hence the dual heterotic geometry dual to is X 4 given by an elliptically fibered Calabi-Yau X 3,243 with base P 1 × P 1 times a circle S 1 . The three-fold X 3,243 is constructed from a pair of reflexive polytopes with vertices and it follows that X 3,243 has Hodge numbers (h 1,1 , h 2,1 ) = (3,243). The dual acyl CY building block X + of the acyl Calabi-Yau four-fold Z + is constructed using by cutting X 3,243 along an S 1 in the base of its K3 fibration. Equivalently, we may glue in a K3 fiber to compactify X + to a building blockX + which can be constructed using the methods of [9] from a pair of projecting tops

First Example of Dual Pairs: New Spin(7)-and its dual G 2 -Manifold
Above we have constructed a pair of an acyl Calabi-Yau four-fold Z + and an acyl G 2 -manifold Z − which both asymptote to (a cylinder/an interval times) the same Calabi-Yau three-fold X 43,43 , together with the two dual acyl Calabi-Yau three-folds X + and X − together with the corresponding bundle data on the heterotic side. We will now glue Z + with Z − × S 1 to a GCS Spin(7)-manifold Z and X ± × S 1 to a TCS G 2 -manifold J and verify that the light fields in the effective field theories indeed agree.

The M-theory on GCS Spin(7)-Manifold
As detailed in Section 4, the Spin (7)-manifold Z is constructed as Alternatively, this Spin(7)-manifold can also be obtained as a free quotient of the Calabi-Yau four-fold X 4 specified as a toric hypersurface by (5.20).

The Dual Heterotic Model on TCS G 2 -Manifold
The dual heterotic side has various components: geometry, vector bundle and NS5-branes. We will determine all the dual data, and compare with the spectrum obtained in the M-theory compactification -and find agreement.
Geometry M-theory on Z is dual to heterotic on a G 2 -manifold J which can be obtained by fiber-wise duality as where X ± are the acyl Calabi-Yau three-folds introduce in sections 5.3.3 and 5.3.2 above.

Bundle data
Recall from section 5.3.1 that the heterotic compactification on X 11,11 = X − ∪ X − dual to M-theory on M = Z − ∪ Z − has eight complex bundle degrees of freedom, associated with a bundle which is an extension of appropriate Wilson lines on the double cover of X 11,11 . The origin of the corresponding degrees of freedom in the dual M-theory suggests to associate 4 complex bundle degrees of freedom with each of the S 1 s in the K3 × T 2 h double cover of X 11,11 . For X − , one of these two S 1 s has disappeared, so that we are left with 8 real bundle degrees of freedom there. In particular, these give the non-trivial holonomies in E 8 × E 8 along S 1 b,− in the base C of the K3 fibration on X − . In the TCS G 2 -manifold, these get mapped to Wilson lines on S 1 e, . Note that this is precisely what is expected for the duality between M-theory on the Calabi-Yau four-fold X 4 and heterotic strings on X 3,243 × S 1 e : the non-zero volumes of the divisors on Z + associated with the (−E 8 ) ⊕2 in the Picard lattice of the K3 fibers of Z + are mapped to Wilson lines on S 1 e+ on the heterotic side. We hence conclude that J carries a vector bundle V inherited from a bundle on X − which breaks E 8 × E 8 → U (1) 8 and has eight real moduli. The data of this bundle is captured by the volumes of the curves in (−E ⊕2 8 ) of the K3 fibers of Z ± .

NS5-branes
Furthermore, our compactifications of the heterotic string require the addition of NS5-branes for consistency as the gauge bundle V does not have a second Chern character. We will infer the NS5-branes which we need to include by exploiting piecewise duality. In particular, we can think about a representative of the first Pontrjagin class of J as being glued from representatives of c 2 (X ± ).
The second Chern class ofX + is c 2 (X + ) = 46HĤ + + 46H + σ + + 23Ĥ + σ + + 11σ 2 + = 46H +Ĥ+ + 24H + σ + + 12Ĥ + σ + , (5.33) whereĤ + is the divisor class corresponding to fixing a point on the base of the K3 fibration onX + and H + is the class corresponding to fixing a point on the other P 1 in the P 1 ×P 1 base of the elliptic fibration onX + . The divisor class σ + is represented by the section of this elliptic fibration and these classes satisfy the relation σ + (σ + + 2H + +Ĥ + ) = 0. Note that The 46 NS5-branes on the curve HĤ are wrapped on the elliptic fiber of X + times the auxiliary S 1 +,e multiplying X + in the construction of J. As such, they are wrapped precisely on the conjectured fiber of the T 3 fibration of J used for the duality and we can associated them with the 46 M2 branes present on the M-theory side. Let us confirm this by counting the number of moduli. Each of these NS5-branes has 4 real degrees of freedom by displacing the elliptic fiber on X + . Furthermore, there is one real degree of freedom from the scalar φ in the tensor multiplet (the position along the 11th direction in heterotic M-theory) and three real degrees of freedom from the self-dual two-form B on the b 1 (T 3 ) = b 2 (T 3 ) = 3 cycles of the T 3 they are wrapped on. This makes 8 degrees of freedom matching the counting for the M2 branes.
The 12 NS5-branes onĤ + σ + are points on the P 1 base of the K3 fibration ofX + so that they are wrapped on a three-manifold with the topology S 2 × S 1 in J. Within X + , each such NS5-brane has 2 real moduli associated with displacement 7 , one modulus associated with the worldvolume scalar φ, and one modulus from the self-dual two-form B. At least in the Kovalev limit, we are hence led to associate 4 real moduli with each such NS5-brane. 7 The associated holomorphic curves which lift to associatives are fixed to lie on the section of the elliptic fibration of X+ Finally, there are the 24 NS5-branes in the class H + σ + ofX + × S 1 e,+ . On X + =X + \ S 0 + , they are wrapped on the whole of the open base C × S 1 e,+ of the K3 fibration on X + × S 1 e,+ . Similarly, there are 12 NS5-branes wrapped on a double cover the open base C×S 1 e,− of the K3 fibration on X − × S 1 e,− as we have seen in Section 5.3.1. Note that this sector of NS5-branes becomes 24 copies of the T 2 factor times the interval in (X + ×S 1 e,+ )∩(X − ×S 1 e,− ) = I ×T 2 ×K3. On the G 2 -manifold, these NS5-branes can hence be joined to form 12 irreducible NS5-branes.
This has several effects. First of all, the relative positions of the 24 branes on X + × S 1 +,e , which are points on the K3 fiber S + of X + ×, are pairwise fixed to be symmetric under the Enriques involution acting on S − . Second, each such pair of NS5-branes only has two real moduli of deformation. Because they are wrapped on the C-base of the K3 fibration on X + and the elliptic fibration of X + is non-trivial there, they cannot be displaced in the direction of the elliptic curve of X + . Third, the two branches of each pair of NS5-branes are swapped when encircling two special points in the base of the K3 fibration on X − , so that we should think of each such pair of NS5-branes as a single NS5-brane wrapped on a three-manifold L which is a double cover of S 3 branched along two unlinked S 1 s. We conclude that there are 12 NS5-branes wrapped on three-manifolds L inside J with two real deformations each. It is not hard to see that b 1 (L) = b 2 (L) = 1, so that each of these NS5-branes is associated with 4 real degrees of freedom: 2 real moduli of displacement together with 2 real moduli from φ and B adding up to 4 real moduli each.

Counting of Degrees of Freedom
We are now ready to count the number of degrees of freedom on the heterotic side. To do so, we can neglect the 46 NS5-branes wrapped on the T 3 fiber of J, as these are mapped to M2 branes in M-theory and we have already matched their degrees of freedom. Summarizing the different contributions discussed above, the remaining light fields for heterotic strings on

Second Example of Dual Pairs
Let us now make consider a variation of the previous example and work out the topology of the Spin(7) associated with putting an E 8 × E 8 vector bundle on J. To describe such a situation, we intend to replace Z − and Z + by an acyl G 2 -manifolds and and acyl CY 4 originating from a K3 fibration with Picard lattice U instead of U ⊕ (−E 8 ) ⊕2 .

The acyl Calabi-Yau four-fold Z +
Following the usual rules of heterotic-M-theory (F-theory) duality Z + is now found as one half of a compact Calabi-Yau four-fold X 4 described by a generic Weierstrass elliptic fibration over P 1 × P 1 × P 1 and Z + ∩ (Z − × S 1 ) = X 3,243 × S 1 × I. Here, X 3,243 has already appeared in Section 5.3.3 and X 4 is found from the pair of reflexive polytope: and χ(X 4 ) = 17568. We can write Z + ∪ Z + = X 4 with Z + ∩ Z + = X 3,243 × S 1 × I and the data of Z + relevant for its use in the construction of Z are and χ(Z + ) = 8784.

The acyl
The three-fold X 3,243 was constructed as a toric hypersurface associated with the pair of reflexive polytopes shown in For X + , the number of bundle moduli can be inferred as follows. Consider heterotic strings on X 3,243 = X + ∪ X + . By a straightforward application of duality to F-theory (which is compactified on the manifold X 4 discussed in Section 5.5.1) immediately gives that a generic vector E 8 × E 8 model W , which is flat on the elliptic fiber of X 3,243 , has m W = 5344 real moduli [46,49]. An E 8 × E 8 vector bundle on X + ∩ X + = I × S 1 × K3 has 4 · 112 real moduli, which gives the number of conditions when matching two generic E 8 × E 8 bundles W ± on X ± .
We hence expect As furthermore m W + = m W − , we find that m W + = 2896.
On the G 2 -manifold J, we can hence construct a suitable vector bundle V by letting V | X + ×S 1 e,+ = W + . Such a bundle restricts to a vector bundle on S + which then has to be appropriately restricted to match V | X − ×S 1 e,− . As we have seen V S − is not a generic vector bundle on a K3 surface, but there are 224 conditions arising at it must be symmetric under the Enriques involution. Once this condition is met, V S − uniquely defines V | X − ×S 1 e,− . As we have seen, the Enriques involution gives 224 real restrictions, so that we conclude that

Discussion and Outlook
We proposed a new construction of eight-manifolds with Spin(7)-holonomy, based on a generalized connected sum (GCS), where two building blocks -a Calabi-Yau four-fold and a G 2 -holonomy manifold times S 1 -are glued together along an asymptotic region that is a Calabi-Yau three-fold times a cylinder. This construction is in part inspired by the recent twisted connected sum (TCS) realization of G 2 -holonomy manifolds, which has resulted in a multitude of new examples of compact G 2 -manifolds, and thereby a resurgence of interest in the string/M-theory context. Likewise the GCS-construction that we propose, provides an avenue to construct large classes of compact Spin(7)-manifolds systematically. In particular the construction of acyl Calabi-Yau three-folds using semi-Fano three-folds [5] or tops [9] which is useful for TCS-constructions, has an obvious generalization to acyl Calabi-Yau four-folds, which is useful for expanding the set of examples of GCS-constructions.
We gave an alternative description of the GCS Spin(7)-manifolds in terms of a quotient by an anti-holomorphic involution of a Calabi-Yau three-fold fibered Calabi-Yau four-fold in section 4.5, which is similar in spirit to the constructions of Joyce, however the key difference is that instead of gluing in ALE-spaces at point-like orbifold singularities, we glue in G 2manifolds with suitable asymptotics.
There is a multitude of future directions to consider: 1. M/F-Duality for Spin (7): M-theory on Spin(7) results 3d N = 1 theories, and we have seen that there are subsectors of the effective field theory, which in the limit of infinite asymptotic region enjoy enhanced supersymmetry, as discussed in section 4.4.
It would clearly be very interesting to apply M/F-duality to the GCS Spin(7)-manifolds and determine how the supersymmetry breaking in the 4d F-theory vacuum is realized [1]. Needless to say it is not difficult to construct GCS-examples that have an elliptic fibration and we will return to this shortly elsewhere. Again key for this will be also to understand the fluxes in the GCS Spin(7)-manifolds.
2. M/Heterotic-Duality and Heterotic G 2 -systems: As was exemplified in section 5, some GCS-constructions have a K3-fibration, so that M-theory compactification on these has a dual description in terms of Heterotic on G 2 -manifolds with a T 3 -fibration with vector bundle. We have seen that the GCS-decomposition of the Spin(7)-manifold maps in the dual G 2 -compactification to a TCS-decomposition. Heterotic on G 2 -manifolds has been studied only very sparsely, and this approach may very well provide further insight into the construction and moduli of vector bundles for so-called heterotic G 2 -systems.
Furthermore, this duality also gives evidence for the existence of associative T 3 -fibrations on G 2 -manifolds as conjecture from G 2 mirror symmetry in [13], and these may indeed be less rare as was proposed in [23] despite the obstructions of associatives.
3. Non-abelian gauge groups: Furthermore we expect, again through the duality to heterotic, that the GCS-Spin(7) manifolds can give rise to non-abelian gauge symmetries.
Locally, these will have a description in the form of an ADE singularities over Cayley four-cycle (for a discussion of such non-compact examples see [29]). It would indeed be interesting to develop this further and understand e.g. a Higgs bundle description of the effective theories and its relation to the compact Spin(7) manifolds -much like what has been done in recent years for M/F-theory on CY 4 . 4. Mirror Symmetry: Mirror symmetry for TCS-G 2 manifolds as studied in [10,13] by applying the mirror map to the Calabi-Yau three-fold building blocks. It would be interesting to see whether there is a similar way to study mirror symmetry for Spin (7) manifolds (as proposed in [50]) by applying the mirror map to the building blocks. What will be most important for this paper is how to determine the cohomologies of the G 2 -manifold in terms of data of the TCS gluing. For each building block we define the restriction map on the second cohomology as where we used the cohomology of the K3-surfaces, where U is the hyperbolic lattice of rank 2. The cohomology of the G 2 -manifold, can then be written in terms of the following lattices We will determine a similar relation for the Spin(7) GCS-construction.