Effective action from M-theory on twisted connected sum G_2-manifolds
Thaisa C. da C. Guio, Hans Jockers, Albrecht Klemm, Hung-Yu Yeh
BBONN–TH–2017–01
Effective action from M-theory on twisted connected sum G -manifolds Thaisa C. da C. Guio , Hans Jockers , Albrecht Klemm , Hung-Yu Yeh , Bethe Center for Theoretical Physics, Physikalisches Institut der Universit¨at Bonn,Nussallee 12, D-53115 Bonn, Germany Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany [email protected]@[email protected]@mpim-bonn.mpg.de
Abstract
We study the four-dimensional low-energy effective N = 1 supergravity theoryof the dimensional reduction of M-theory on G -manifolds, which are constructed byKovalev’s twisted connected sum gluing suitable pairs of asymptotically cylindricalCalabi–Yau threefolds X L/R augmented with a circle S . In the Kovalev limit theRicci-flat G -metrics are approximated by the Ricci-flat metrics on X L/R and weidentify the universal modulus — the Kovalevton — that parametrizes this limit. Weobserve that the low-energy effective theory exhibits in this limit gauge theory sectorswith extended supersymmetry. We determine the universal (semi-classical) K¨ahlerpotential of the effective N = 1 supergravity action as a function of the Kovalevtonand the volume modulus of the G -manifold. This K¨ahler potential fulfills the no-scale inequality such that no anti-de-Sitter vacua are admitted. We describe geometricdegenerations in X L/R , which lead to non-Abelian gauge symmetries enhancementswith various matter content. Studying the resulting gauge theory branches, we arguethat they lead to transitions compatible with the gluing construction and providemany new explicit examples of G -manifolds.February, 2017 a r X i v : . [ h e p - t h ] N ov ontents G -manifolds 4 G -manifolds . . . . . . . . . . . . . . . . . 6 G -manifolds 14 G -manifolds . . . . . . . . . . . . . . . . . . . . . 19 G -manifolds . . . . . . . . . . . . . . . . . . . . . . . 325.3 Examples of G -manifolds from orthogonal gluing . . . . . . . . . . . . 34 N = 2 gauge sectors on twisted connected sums 44 N = 2 Abelian gauge theory sectors . . . . . . . . . . . . . . 466.2 Phases of N = 2 non-Abelian gauge theory sectors . . . . . . . . . . . . 516.3 Examples of G -manifolds with N = 2 gauge theories . . . . . . . . . . 556.4 Transitions among twisted connected sum G -manifolds . . . . . . . . . 61 A.1 Definitions and useful relations . . . . . . . . . . . . . . . . . . . . . . 66A.2 G -representations and the Rarita–Schwinger G -bundle . . . . . . . . 67A.3 The massless four-dimensional fermionic spectrum . . . . . . . . . . . . 68A.4 The flux-induced holomorphic superpotential . . . . . . . . . . . . . . . 70i lossary Symbol Description PagesGeometric spaces: M , eleven-dimensional Lorentz manifold (M-theory space-time) 5 M , four-dimensional Minkowski space 5,7 Y compact seven-dimensional compactification manifold 5,7,18,25 Y ······ twisted connected sum G -manifold from orthogonal gluing 37,40,44 M moduli space of Ricci-flat metrics of G -metrics 8,13 M C semi-classical moduli space of M-theory on G -manifolds 13∆ cyl complex one-dimensional open cylinder { x ∈ C | | z | > } X ∞ cylindrical Calabi–Yau threefold 15 X ∞ L/R left/right cylindrical Calabi–Yau threefold 16 X L/R left/right asymptotically cylindrical Calabi–Yau threefold 16 X L/R ( T ) left/right truncated Calabi–Yau threefold X L/R K L/R left/right compact complement of asymp. cyl. of X L/R ( T ) 15,18 S ∗ L/R left/right circle of the asymptotic region of X L/R S L/R left/right circles in Kovalev’s twisted connected sum 16,17 Y ∞ L/R left/right asymptotic seven-manifolds X ∞ L/R × S L/R Y L/R left/right non-compact seven-manifolds X L/R × S L/R Y L/R ( T ) left/right truncated non-compact seven-manifolds Y L/R S K3 surface 14,20 S L/R left/right polarized K3 surface 16,17(
Z, S ) building block for asymptotically cylindrical Calabi–Yau threefold 22( Z sing , S ) singular building block with N = 2 gauge theory sector 48,51 dP (cid:96) del Pezzo surface of degree (9 − (cid:96) ) 42 P weak Fano, semi-Fano or Fano threefold 30,31 P Σ toric weak Fano, toric semi-Fano or toric Fano threefold 31,35Σ toric fan of a toric variety 31,35∆ / ∆ ∗ toric and dual toric polytopes 31,35Forms and tensors: η µν four-dimensional Minkowski metric 7 g mn Riemannian metric tensor of compactification manifold Y g metric tensor of eleven-dimensional Lorentz manifold M , ϕ G -structure (three-form) 5,8 g ϕ G -structure metric 5 ω (2) I basis of harmonic two-forms of G -manifold Y ρ (3) i basis of harmonic three-forms of G -manifold Y ρ sym i, ( mn ) basis of zero modes of the Lichnerowicz Laplacian on Y η covariantly constant spinor of G -manifold Y ω ∞ K¨ahler form of cylindrical Calabi–Yau threefold X ∞ ∞ holomorphic three-form of cylindrical Calabi–Yau threefold X ∞ g X ∞ Ricci-flat metric of cylindrical Calabi–Yau threefold X ∞ g L/R left/right Ricci-flat metric of Calabi–Yau threefold X L/R ii ymbol Description PagesForms and tensors (continued) : ω I,J,KL/R left/right triplet of hyper K¨ahler two-forms of K3 surface S L/R g S Ricci-flat metric of K3 surface S ϕ canonical G -structure of Calabi–Yau threefold times circle 15 ϕ ∞ L/R left/right asymptotic torsion-free G -structure 16 (cid:101) ϕ L/R ( γ, T ) left/right interpolating G -structure 18 ϕ ( γ, T ) torsion-free G -structure in Kovalev’s twisted connected sum 19Coordinates and quantum fields: x µ coordinates of Minkowski space M , y m local coordinates of compact seven-manifold Y S i local coordinates of moduli space M P i three-form scalar fields 9,11 φ i local coordinates of moduli space M C & chiral scalar fields 13 χ iα four-dimensional chiral fermions of N = 1 chiral multiplets 11Φ i four-dimensional N = 1 chiral multiplets 11,13 A Iµ four-dimensional vector bosons 9,11 λ Iα four-dimensional gauginos 11 V I four-dimensional N = 1 vector multiplets 11 ν four-dimensional N = 1 chiral overall volume modulus 26,28 κ Kovalevton (four-dimensional N = 1 chiral gluing modulus) 26,28Parameters and coupling constants: κ / κ four-dimensional/eleven-dimensional Planck constant 12 V Y constant reference volume of manifold Y V Y moduli-dependent volume of manifold Y λ moduli-dependent dimensionless volume factor of manifold Y γ constant reference radius 20 γ moduli-dependent radius 15,20 λ inverse length scale of Calabi–Yau threefolds X ∞ L/R λ dimensionless inverse length scale of Calabi–Yau threefolds X ∞ L/R λ S inverse length scale of K3 surface S R dimensionless volume modulus 20,21 T dimensionless Kovalev parameter 17,18,21˜ S dimensionless non-universal moduli 21,25 t L/R left/right K¨ahler moduli of Calabi–Yau threefold X L/R z L/R left/right comp. struct. moduli of Calabi–Yau threefold X L/R L two-form lattice of K3 surfaces S L/R , L = H ( S L , Z ) = H ( S R , Z ) 23 N L/R left/right Picard lattices of polarized K3 surface S L/R T L/R left/right transcendental lattices of polarized K3 surface S L/R k L/R left/right kernels of two-form cohomology 23 W orthogonal pushout lattice 32 W L/R left/right orthogonal complement to the intersection lattice R R intersection lattice 32 iii ymbol Description PagesCohomology groups and lattices (continued) : N L ⊥ R N R orthogonal pushout of lattices N L/R at the intersection lattice R N L ⊥ N R perpendicular gluing of lattices N L/R (cid:104)· , ·(cid:105) · lattice intersection pairing 33 κ lattice intersection matrix 37,39,41∆ κ discriminant of the lattice intersection matrix 37,39,41Gauge theory data:( · ) (cid:91) superscript for Higgs branch quantity 49,53( · ) (cid:93) superscript for Coulomb branch quantity 49,53 H (cid:91) N = 2 Higgs branch 49,53 C (cid:93) N = 2 Coulomb branch 49,53 h (cid:91) complex dimension of the Higgs branch 49,53 c (cid:93) complex dimension of the Coulomb branch 49,53 G gauge group 48,52 adj adjoint representation 52,56,57 k fundamental representation of SU ( k ) 52,56,57Miscellaneous:∆ Laplacian of compact seven-manifold Y L Lichnerowicz Laplacian of compact seven-manifold Y /D Dirac operator of compact seven-manifold Y /D RS Rarita–Schwinger operator of compact seven-manifold Y K K¨ahler potential of N = 1 supergravity action 13,28,29 f IJ gauge kinetic coupling functions of N = 1 supergravity action 13 W superpotential of N = 1 supergravity action 14( · ) L/R left/right subscript in Kovalev’s twisted connected sum 16,17 F Λ gluing diffeomorphism in Kovalev’s twisted connected sum 16 r hyper K¨ahler rotation mapping polarized K3 surface S L to S R ρ reference to rank ρ Fano threefold in the Mori–Mukai classification 32K iv Introduction
M-theory compactifications on seven-dimensional manifolds with G holonomy offerthe opportunity to geometrically study the properties of N = 1 effective theories in asetting that is non-perturbative from the superstring point of view [1–4]. As M-theoryis conjectured to be the non-perturbative extension of type IIA theory, it is natural tocompare it to F-theory compactifications on elliptically-fibered Calabi–Yau fourfolds.It leads to effective N = 1 theories in four dimensions and is the geometrization ofnon-perturbative type IIB compactifications on the complex three-dimensional baseof the fibration. It also includes a varying axio-dilaton background due to space-time-filling seven-branes.While standard techniques of complex algebraic geometry provide immediatelyhundreds of thousands of elliptically-fibered Calabi–Yau fourfolds — for instance re-alized as hypersurfaces and complete intersections in weighted projective spaces ormore generally in toric ambient spaces [5, 6] — for a long time there were only abouta hundred examples of G -manifolds constructed by the resolution of special orb-ifolds of seven-dimensional torus T [7]. Likewise, the holomorphic terms in the four-dimensional low-energy effective N = 1 supergravity action obtained from Calabi–Yau fourfolds are computable in the underlying algebraic setting. In particular, theflux-induced superpotential is essentially determined by the integral periods of theCalabi–Yau fourfolds [8], which for compact fourfolds have systematically been deter-mined in refs. [9, 10]. Furthermore, the holomorphic gauge kinetic coupling functionsare in principle accessible in this setting as well as threshold corrections to the gaugekinetic terms [11, 12]. M-theory, however, has the clear advantage that — at least in the supergravitylimit — we expect an explicit description in terms of eleven-dimensional supergrav-ity, for which a unique eleven-dimensional supergravity action exists [14]. There-fore, by simply studying Kaluza–Klein reductions of this action on compact seven-dimensional manifolds one obtains four-dimensional low-energy effective theories thatcapture already many of the physical properties of the associated M-theory compact-ifications [1, 2, 16–20]. The resulting semi-classical four-dimensional effective action isthen further corrected by non-perturbative effects specific to M-theory, such as M2-and M5-brane instantons wrapping internal cycles of the seven-dimensional compact-ification manifold [21, 22].More recently, a new construction of G -manifolds has been proposed by Ko-valev [23], which we loosely refer to as Kovalev’s twisted connected sum. The es-sential idea is to consider suitable pairs of non-compact asymptotically cylindrical The gauge kinetic coupling functions are holomorphic and one loop exact. However, physicalgauge kinetic terms receive further corrections in N = 1 effective theories, see for instance ref. [13]for a thorough discussion. The maximal dimension that allows this supersymmetry representation was previously pointedout in ref. [15]. G holonomy. Further explorations of this construction show thatKovalev’s twisted connected sum offers again a large number of explicit examples of G -manifolds [24–26]. In fact, the origin of this multitude is similarly based on toricgeometry as the asymptotically cylindrical Calabi–Yau threefolds — which furnish thedistinct summands in Kovalev’s twisted connected sum — can for instance be con-structed using the toric weak Fano threefolds, defined by the 4 319 reflexive polytopesin three dimensions, by blowing up a suitable curve and removing the anti-canonicalclass [27]. Currently, there is no systematic gluing prescription for the whole classavailable, but admissible gluing conditions can be straightforwardly established forbuilding blocks constructed from 899 toric varieties of the semi-Fano type definedin [24]. They admit in general different K¨ahler cones and — as discussed further inthis work — they can be degenerated and resolved to yield different building blocksrealizing distinct branches of both Abelian and non-Abelian gauge theories. An es-timate — even accounting for the possibility that the homeomorphism class of theconstructed examples occur multiple times — yields nevertheless a factor of ten foreach admissible building block. This leads to an estimated number of 10 × m g dif-ferent G examples, where m g ≥ hypersurfaces in toric varieties. The still huge difference in the ordersof magnitude is maybe due to the fact that Kovalev’s twisted connected sum onlyrealizes a particular class of G -manifolds. Namely, Crowley and Nordstr¨om define a(non-trivial) Z -valued homotopy invariant for G -manifolds that takes the value 24for any twisted connected sum G -manifold [29].One goal of this paper is to study the four-dimensional N = 1 low-energy effectiveaction that arises from M-theory compactifications on G -manifolds that are of thetwisted connected sum type. In order to determine the defining data of the resulting N = 1 supergravity theory — such as the K¨ahler potential, the gauge kinetic couplingfunctions, and the superpotential — an important question is to which extent theharmonic analysis of the asymptotically cylindrical Calabi–Yau threefold summandswith their Ricci-flat Calabi–Yau metrics approximates the one of the Ricci-flat G -metric of the resulting compact G -manifold. In a certain limit — to be referred toas the Kovalev limit in the following — the corrections to the G -metric expressed interms of the Calabi–Yau data become exponentially suppressed [23, 24, 30, 27]. Thus,it is the Kovalev limit that allows us to reliably deduce from the geometry of theCalabi–Yau summands the resulting low-energy effective action. To some extent wecan think of the Kovalev limit of M-theory on G -manifolds as the analog of the large This is based on the observation that the number of d -dimensional reflexive polytopes growsat least exponentially in d . Mark Gross’s proof for the finiteness of elliptically-fibered Calabi–Yauthreefolds in ref. [28] suggests that the estimated number of elliptically-fibered Calabi–Yau fourfoldsis again finite and the order of magnitude independent of the construction. G -manifold is constructedfrom the (relative) cohomology of the asymptotically cylindrical Calabi–Yau threefoldsummands [23, 24]. This cohomological data determines the N = 1 vector and chiralmultiplets of the resulting four-dimensional theory. In the Kovalev limit we find thatthe N = 1 vector multiplets furnish gauge theory sectors of extended supersymmetry.Specifically, N = 1 vector multiplets attributed to the interior of the asymptoti-cally Calabi–Yau threefolds combine with N = 1 chiral multiplets to N = 2 vectormultiplets realizing N = 2 gauge theory sectors, whereas N = 1 vector multipletsassociated to the mutual asymptotic region of the Calabi–Yau summands enhance to N = 4 gauge theory sectors.In the spectrum of twisted connected sum G -manifolds, we identify two universal N = 1 chiral fields ν and κ . The real part of the chiral field ν furnishes the overallvolume modulus of the G -manifold. The chiral field κ is specific to Kovalev’s twistedconnected sum construction, as its real part parametrizes the Kovalev limit. In thesequel we refer to this multiplet as the Kovalevton κ . Restricting the dynamics ofthe two universal chiral moduli fields in the vicinity of the Kovalev limit, we arriveat the universal expression for the K¨ahler potential of the four-dimensional effectivesupergravity theory K = − log (cid:2) ( ν + ¯ ν ) ( κ + ¯ κ ) (cid:3) . (1.1)This simple semi-classical K¨ahler potential — only capturing the dynamics of the twouniversal chiral multiplets — fulfills the no-scale inequality implying a manifest non-negative F-term scalar potential, such that no (supersymmetric) anti-de-Sitter vacuacan occur.From a physics point of view, the most interesting question is whether the twistedsum construction can accommodate singularities that in the four-dimensional effectivetheory lead to enhanced non-Abelian gauge symmetries, to geometrically engineerablematter content, to a chiral spectrum and to transitions within the class N = 1 effectivetheories connecting topologically inequivalent G -manifolds. The first three questionshave been addressed in refs. [31–35] and the last one in refs. [36,37,25], however, mainlyin the context of local models. The Kovalev construction and the above describeddecoupling into sectors with different amount of supersymmetry allow us to discusssome of these questions in particular the first two and the last one in the context ofglobal G -manifolds. More explicitly, we find that, by degenerating certain algebraicequations in the description of the building blocks and blowing up the correspondingsingularities, we can achieve various Abelian and non-Abelian gauge symmetries — forexample including the standard model gauge group — as well as matter in the adjoint,bi-fundamental and fundamental representations. Moreover, following refs. [38–43]we analyze the Higgs and Coulomb branches of the N = 2 gauge theory sectors torealize transitions in the building blocks of twisted connected sum G -manifolds. The3emarkable fact is now that the predicted spectra in the various gauge theory branchesagree with the changes among the N = 1 supergravity spectra of the correspondingcompact G -manifolds. This leads us to propose geometric transitions among G -manifolds that are physically connected via branches of N = 2 gauge theory sectors.The paper is organized as follows: in Section 2 we review the geometry of the G -manifolds and the Kaluza–Klein reduction of eleven-dimensional supergravity onthese spaces. We focus on the moduli space of such G -compactifications, on thefour-dimensional low-energy effective spectrum and action, and on the resulting four-dimensional N = 1 supergravity description in terms of the K¨ahler potential, the(flux-induced) superpotential, and the gauge kinetic coupling functions. Section 3 re-views Kovalev’s twisted connected sum construction. Firstly, we introduce the asymp-totically cylindrical Calabi–Yau threefolds, and secondly we summarize the twistedconnected sum constructed from a suitable pair of such Calabi–Yau threefolds. Dueto the importance to our analysis, a particular emphasis is put on the Kovalev limit.In Section 4 we describe M-theory compactifications on twisted connected sum G -manifolds. We start with a description of the low-energy effective N = 1 spectrumas deduced from the cohomology of the Calabi–Yau summands. We analyze the uni-versal properties of the low-energy effective theory attributed to Kovalev’s twistedconnected sum. In Section 5 we apply the method of orthogonal gluing to explicitlyconstruct novel examples of twisted connected G -manifolds. We argue that in theKovalev limit these examples directly relate to Abelian N = 4 gauge theory sectors.In Section 6 we study the emergence of Abelian and non-Abelian N = 2 gauge the-ory sectors. For both the Abelian and non-Abelian N = 2 gauge theory sectors weestablish a correspondence between N = 2 Higgs and Coulomb branches of the gaugetheory and the associated phases of twisted connected G -manifolds. We illustratethe different physical aspects of the proposed correspondence with explicit examplesof G -manifolds. Our conclusions are presented in Section 7. For the convenience ofthe reader, we collect in the glossary on pages ii to iv our notational conventions of theused mathematical symbols together with a reference to their appearence in the maintext. In Appendix A we give further technical details on the G -compactifications offermionic terms, supplementing the material presented in Section 2. G -manifolds An eleven-dimensional Lorentz manifold M , together with a four-form flux G ofan anti-symmetric three-form tensor field ˆ C describe the geometry of the low-energyeffective action of M-theory. Firstly — due to fermionic degrees of freedom in M-theory— the Lorentz manifold M , must be spin. Secondly — consistency of the effectiveaction at one loop — imposes the cohomological flux quantization condition [44] G π − λ ∈ H ( M , , Z ) , λ = p ( M , )2 . (2.1)4he class λ is integral since the first Pontryagin class p is even for seven-dimensionalspin manifolds.In this work we study compactifications of M-theory to four-dimensional Minkowskispace M , with N = 1 supersymmetry. That is to say that, for the eleven-dimensionalLorentz manifold M , , we consider the compactification ansatz M , = M , × Y (2.2)with the seven-dimensional compact smooth manifold Y . In the absence of backgroundfluxes such a four-dimensional N = 1 Minkowski vacuum implies that the internalspace Y must be a G -manifold [1, 3, 4].A G -manifold Y is a seven-dimensional Ricci-flat Riemannian manifold with G holonomy and not a proper subgroup thereof. Furthermore, the manifold Y is spinwith a single globally defined covariantly constant spinor [45]. Note that for G -manifolds the characteristic class λ is always even [22,17], which is consistent with thequantization condition (2.1) for the considered compactification ansatz of vanishingbackground flux.Let us briefly recall some relevant aspects of G -manifolds. First of all, the excep-tional Lie group G is a fourteen-dimensional simply connected subgroup of SO (7),and we can think of G in the following way. A three-form ϕ on R gives rise to acanonical symmetric bilinear form on R B ϕ ( X, Y ) = −
16 ( X (cid:121) ϕ ) ∧ ( Y (cid:121) ϕ ) ∧ ϕ (2.3)with values in Λ ( R ) ∗ . For a generic three-form ϕ , the bilinear form B ϕ yields a non-degenerate pairing of some signature ( p, q ) with p + q = 7 (with respect to an orientedvolume form on R ) [46,47]. In particular, there is an open set Λ ( R ) ∗ in the space ofthree-forms Λ ( R ) ∗ such that B ϕ is a positive definite bilinear form for ϕ ∈ Λ ( R ) ∗ .Then GL(7 , R ) acts on the three-form ϕ and G is its fourteen-dimensional stabilizersubgroup. Since G leaves the positive definite pairing B ϕ invariant, it is actually asubgroup of SO (7).Since the Lie group G is the stabilizer group of the described three-form, a seven-dimensional oriented manifold Y together with the three-form ϕ in Ω ( Y ) — whichis the space of smooth three-forms oriented-isomorphic to Λ T ∗ p Y (cid:39) Λ ( R ) ∗ for any p ∈ M — becomes a G -structure manifold. Thus we call ϕ a G -structure on Y .Furthermore, the positive definite pairing (2.3) defines a Riemannian metric g ϕ on Y .Namely, at any point p ∈ M and for any basis ∂ | p , . . . , ∂ | p at T p Y , we obtain thepositive definite inner product g ϕ ( X p , Y p ) = B ϕ ( X p , Y p )( ∂ | p , . . . , ∂ | p )vol ϕ ( ∂ | p , . . . , ∂ | p ) , vol ϕ ( ∂ | p , . . . , ∂ | p ) = det [ B ϕ ( ∂ i | p , ∂ j | p )( ∂ | p , . . . , ∂ | p )] , (2.4) Note that Λ ( R ) ∗ is a convex open set in Λ ( R ) ∗ . Thus — with a partition of unity — we canconstruct a G -structure on any smooth paracompact seven-dimensional manifold Y . X p and Y p in the tangent space T p Y .The remarkable and important theorem for the following is that a G -structuremanifold has a subgroup of G as its holonomy group if and only if the three-form ϕ is harmonic with respect to the constructed G -metric g ϕ [45], i.e., dϕ = 0 , d ∗ g ϕ ϕ = 0 , (2.5)in terms of the Hodge star ∗ g ϕ of the metric g ϕ . Such a harmonic three-form in Ω ( Y )is called torsion-free. Requiring in addition a finite fundamental group π ( Y ) ensuresthat Y has G holonomy and not a proper subgroup thereof. Thus these manifolds— that is to say with finite fundamental group π ( Y ) and torsion-free G -structure— are referred to as G -manifolds.The system of partial differential equations (2.5) for torsion-free G -structures arehighly non-linear due to relation (2.4) between the metric g ϕ and the G -structure ϕ .Nevertheless, given a torsion-free G -structure of a G -manifold, the local structureof the moduli space M of G -manifolds is known due to Joyce [7]. In particular theBetti number b ( Y ) is the dimension of M . We will discuss these aspects of the localstructure in the next section. G -manifolds Eleven-dimensional N = 1 supergravity compactified on a seven-dimensional manifold Y without four-form background fluxes to four-dimensional supergravity has beenfirst discussed in ref. [1]. A possible warp factor in this compactification has beenconsidered in ref. [2], where it was shown that the warping breaks supersymmetry. The structure of the massless four-dimensional N = 1 multiplets that arise from suchcompactification was shown in ref. [4] to possess b ( Y ) abelian U (1) vector fields and b ( Y ) neutral chiral fields Φ i (see also ref. [3]). The inclusion of background fluxes G —generating a superpotential W for the neutral chiral fields Φ i and thereby genericallybreaking supersymmetry — has been analyzed in refs. [16, 17].Let us now review the Kaluza–Klein reduction of eleven-dimensional N = 1 super-gravity, which furnishes the low-energy effective description of M-theory. The masslessspectrum of this maximally supersymmetric supergravity theory consists only of theeleven-dimensional gravity multiplet. Its bosonic massless field content is given bythe eleven-dimensional space-time metric tensor ˆ g MN , the three-form tensor ˆ C [ MNP ] ,whereas the fermionic massless field content is given by the eleven-dimensional grav-itino ˆΨ αM . The degrees of freedom of the massless component fields in the gravitymultiplet transform in the following irreducible representations of the little group SO (9): While these authors give the criteria for the compactification manifold Y to yield four-dimensional N = 1 supergravity, they do not refer to the G -manifolds in Berger’s classificationof special holonomy manifolds [48], likely because compact examples of G -manifolds were onlyfound much later in ref. [7]. The metric ˆ g MN in the traceless symmetric representation . • The three-form ˆ C [ MNP ] in the anti-symmetric three-tensor representation . • The gravitino ˆΨ αM in the spinorial representation s .We now perform the Kaluza–Klein reduction to four-dimensional Minkowski space M , with the compactification ansatz (2.2). To solve Einstein’s equations in theabsence of background fluxes, we consider the block diagonal metric ˆ g ( x, y ) = η µν dx µ dx ν + g mn ( y ) dy m dy n , (2.6)where x µ and y m furnish (local) coordinates of the four-dimensional Minkowski space M , with the flat space-time metric η µν and the seven-dimensional G -manifold Y with the Ricci-flat Riemannian metric g mn , respectively. Notice that we use upper-case latin letters for eleven-dimensional indices, lower-case latin letters for seven-dimensional indices, and greek letters for four-dimensional indices.The first task is to deduce the massless spectrum of the effective four-dimensionallow-energy theory. We start with the gravitational degrees of freedom, which in-finitesimally describe the fluctuations of the metric background (2.6), i.e., ˆ g → ˆ g + δ ˆ g .Firstly, we obtain the four-dimensional metric fluctuations δg µν , which corresponds tothe gravitational degrees of the four-dimensional low-energy effective theory. Secondly,since the fundamental group of G -manifolds is finite, there are no massless gravita-tional Kaluza–Klein vectors. Finally, we determine the gravitational Kaluza–Kleinscalars S i , which furnish coordinates on the moduli space of G -metrics. At a givenpoint S i in the moduli space we fix a reference metric, and consider its infinitesimaldeformation under δS i , i.e., g mn ( S i ) dy m dy n → g mn ( S i ) dy m dy n + (cid:88) i δS i ρ sym i, ( mn ) ( S i ) dy m dy n . (2.7)Then, solving Einstein’s equations to linear order in the symmetric metric fluctuations ρ sym i, ( mn ) , we obtain Ric (cid:16) g + (cid:88) δS i ρ sym i (cid:17) = 0 ⇒ ∆ L ρ sym i = 0 , (2.8)in terms of the Lichnerowicz Laplacian ∆ L for the symmetric tensor fields. Using the G -structure ϕ on Y , we construct the anti-symmetric three-form tensors ρ (3) i, [ mnp ] = g rs ρ sym i,r [ m ϕ np ] s . (2.9) In the presence of background fluxes in the internal space Y , the ansatz for the metric is gener-alized to a warped product ˆ g ( x, y ) = e A ( y ) η µν dx µ dx ν + e − A ( y ) g mn dy m dy n in terms of the function A ( y ) on Y called the warped factor [16], which generically breaks four-dimensional N = 1 super-symmetry [2, 16]. G -manifolds the symmetric tensor ρ sym i is a zero mode of the Lichnerowicz Lapla-cian operator if and only if the above constructed three-form ρ (3) i is harmonic [49],namely ∆ L ρ sym i = 0 ⇔ ∆ ρ (3) i = 0 . (2.10)Thus, the massless Kaluza–Klein scalars S i arise from harmonic three-forms ρ (3) i ,which represent a basis for the vector space H ( Y ) of dimension b ( Y ). Accordingto eq. (2.5) the harmonic three-forms ρ (3) i are the first order deformations to thetorsion-free G -structure ϕ ( S i ) → ϕ ( S i ) + (cid:88) i δS i ρ (3) i ( S ) . (2.11)At a given point S i in moduli space the harmonic three-forms ρ (3) i of Y fall intorepresentations of the structure group G , and H ( Y ) splits as [7] H ( Y ) = H ( Y ) ⊕ H ( Y ) , dim H ( Y ) = 1 , dim H ( y ) = b ( Y ) − , (2.12)where the three-form representatives transform in the representations and of G ,respectively. The harmonic torsion-free G -structure ϕ corresponds to the unique sin-glet, and the associated deformation simply rescales the volume of the G -manifold Y .The remaining harmonic forms in the representation infinitesimally deform thetorsion-free G -structure such that the volume of Y remains constant at first orderapproximation. Analogously, the symmetric tensors ρ sym i solving the LichnerowiczLaplacian ∆ L split into a unique singlet — given by the metric tensors g — and b ( Y ) − of the G -structuregroup.Above we have seen that the infinitesimal deformations can be identified withharmonic three-forms. In ref. [7] Joyce shows that these infinitesimal deformationsare actually unobstructed to all orders. That is to say that the vicinity U ϕ ( S i ) ⊂ M ofa given torsion-free G -structure ϕ ( S i ) ∈ M — at a given point S i , i = 1 , . . . , b ( Y ),in the moduli space — is locally diffeomorphic to the de Rham cohomology H ( Y ), i.e., P ϕ ( S i ) : U ϕ ( S i ) ⊂ M → H ( Y ) , ϕ (cid:55)→ [ ϕ ] . (2.13)Hence, the Betti number b ( Y ) is indeed the dimension of M , and the scalar fields S i furnish local coordinates on M with the infinitesimal deformations δS i spanningthe tangent space T S i M . This local structure implies that the massless infinitesi-mally metric deformations ρ sym i — or alternatively the first order deformations ρ (3) i tothe torsion-free G -structure — extend order-by-order to unobstructed finite defor-mations, which therefore describe locally the moduli space M of G -manifolds. While We assume that the scalars S i describe a generic point in M . First of all, the associated G -manifold Y should be smooth. Furthermore, it should not be a special symmetric point correspondingto an orbifold singularity in M . ρ (3) i themselves depend (non-linearly) on the moduli spacecoordinates S i , we can – due to eq. (2.13) – locally expand the cohomology class [ ϕ ]of the torsion-free G -structure ϕ as[ ϕ ( S i )] = (cid:88) i S i [ ρ (3) i ] , (2.14)which is a useful local description of the moduli space of Y . Massless four-dimensional modes arise from the coefficients in the decompositionof the eleven-dimensional anti-symmetric three-form tensor ˆ C asˆ C ( x, y ) = (cid:88) I A I ( x ) ∧ ω (2) I ( y ) + (cid:88) i P i ( x ) ∧ ρ (3) i ( y ) , (2.15)in terms of the harmonic two-forms ω (2) I and three-forms ρ (3) i identified with non-trivial cohomology representatives of H ( Y ) and H ( Y ) of dimension b ( Y ) and b ( Y ),respectively. Thus, as there are no dynamical degrees of freedom in four-dimensionalanti-symmetric three-form tensor fields and due to the absence of harmonic one-formson the internal G -manifolds, the four-dimensional vectors A I , I = 1 , . . . , b ( Y ), andthe four-dimensional scalars P i , i = 1 , . . . , b ( Y ), are the only massless modes obtainedfrom the dimensional reduction of the eleven-dimensional anti-symmetric three-formtensor field ˆ C .Let us now turn to the dimensional reduction of the eleven-dimensional gravitinoˆΨ, which geometrically is a section of T ∗ M , ⊗ SM , , where SM , denotes a spinbundle of M , . Upon dimensional reduction the gravitino ˆΨ enjoys the expansionˆΨ( x, y ) = (cid:0) ψ µ ( x ) dx µ + ψ ∗ µ ( x ) dx µ (cid:1) ζ ( y ) + ( χ ( x ) + χ ∗ ( x )) ζ (1) n ( y ) dy n . (2.16)Here ( ψ µ , ψ ∗ µ ) and ( χ, χ ∗ ) are four-dimensional Rarita–Schwinger and four-dimensionalspinor fields of both chiralities in M , . ζ is a section of the (real) spin bundle SY of the compact G -manifold Y . Furthermore, ζ (1) is a section of the (real) Rarita–Schwinger bundle T ∗ Y ⊗ SY , which locally takes the form θ (1) ⊗ ˜ ζ in terms of thelocal one-form θ (1) and the spinorial section ˜ ζ .On the G -manifold the spin bundle splits as SY (cid:39) T ∗ Y ⊕ R [7] such that thesection ζ decomposes accordingly ζ = (cid:88) m a m ( y ) γ m η + b ( y ) η . (2.17) Note, however, that a given cohomology class [ ϕ ( S i )] is not necessarily represented by a uniquetorsion-free G -structure ϕ ( S i ). Mathematically, not much is known about the global structure ofthe moduli space M and, in particular, about the global map P : M → H ( Y ). In our conventions the fermionic fields ψ µ , χ and ψ ∗ µ , χ ∗ are chiral and anti-chiral, respectively,such that ψ µ + ψ ∗ µ and χ + χ ∗ become Majorana fermions. As the G -structure group of Y — a subgroup of SO (7) — is simply connected, it defines acanonical spin structure on Y . η is the covariantly constant Majorana spinor of the G -manifold and γ m arethe seven-dimensional gamma matrices. Similarly, we analyze the Rarita–Schwingersection ζ (1) of T ∗ Y ⊗ SY . It decomposes as ζ (1) = (cid:88) n,m a nm ) ( y ) dy n ⊗ γ m η + (cid:88) n,m a nm ] ( y ) dy n ⊗ γ m η + (cid:88) n b n ( y ) dy n ⊗ η . (2.18)The superscripts in the symmetric tensor a nm ) ( y ), the anti-symmetric tensor a nm ] ( y ),and the vector b n ( y ) indicate the dimension of their respective representations withrespect to the structure group G . While the anti-symmetric tensor a nm ] ( y ) and thevector b n ( y ) transform in the irreducible representations and , the symmetrictensor a nm ) ( y ) further decomposes into the trace and the traceless symmetric part,which respectively correspond to the irreducible representations and .The massless four-dimensional fermionic spectrum results from the zero modes ofthe seven-dimensional Dirac operator /D and Rarita–Schwinger operator /D RS , i.e, /Dζ = 0 , /D RS ζ (1) = 0 . (2.19)The zero modes of these operators on G -manifolds are discussed in Appendix A.3 andare also determined in ref. [50]. For the spinorial section ζ , the covariantly constantspinor η — i.e., b ( y ) ≡ ζ (1) , the one-form tensor b ( y ) = b n ( y ) dy n does not contribute anyzero modes. All zero modes arise from the zero modes of the Lichnerowicz Laplacianand the two-form Laplacian acting respectively on the symmetric tensors a ( y ) = a nm ) ( y ) dy n ⊗ dy m and the anti-symmetric tensors a ( y ) = a nm ] ( y ) dy n ∧ dy m , i.e.,∆ L a ( y ) = 0 , ∆ a ( y ) = 0 . (2.20)The zero modes of the Lichnerowicz Laplacian on G -manifolds are again identifiedwith harmonic three-forms according to eqs. (2.9) and (2.10) — with a single zero modeand b ( Y ) − G -representations and , respectively. Therefore, the zero modes of the Rarita–Schwinger bundleon Y are in one-to-one correspondence with non-trivial cohomology elements of both H ( Y ) and H ( Y ), and we arrive at the expansion of the four-dimensional chiralfermions χ ( x ) ζ (1) ( y ) = b ( Y ) (cid:88) i =1 χ i ( x ) ρ sym i, ( nm ) dy n ⊗ γ m η + b ( Y ) (cid:88) I =1 λ I ω (2) I [ nm ] dy n ⊗ γ m η , (2.21)in terms of the bases of zero modes ρ sym i of the Lichnerowicz Laplacian and of theharmonic two-forms ω (2) I . A priori, the constructed zero modes furnish elements of H ( Y ), H ( Y ) and H ( Y ) thattransform in the specified representations of the G -structure group. However, on G -manifolds allnon-trivial three- and two-form cohomology elements can respectively be represented in the repre-sentations , , and , which justifies the identification of zero modes with H ( Y ) and H ( Y ) [7],cf. also Appendix A. N = 1 multiplets1 metric g µν gravitino ψ µ , ψ ∗ µ gravity multiplet i = 1 , . . . , b ( Y ) scalars ( S i , P i ) spinors χ i , χ ∗ i chiral multiplets Φ i I = 1 , . . . , b ( Y ) vectors A Iµ gauginos λ Iα vector multiplets V I Table 2.1: This table summarizes the massless four-dimensional low-energy effective N = 1 supergravity spectrum that is obtained from the dimensional reduction of M-theory — or rather of eleven-dimensional supergravity — on a smooth G -manifold Y .Now, we can spell out the massless four-dimensional spectrum in terms of N =1 supergravity multiplets as obtained from the dimensional reduction of M-theoryupon the G -manifolds Y . It consists of the four-dimensional supergravity multiplet, b ( Y ) (neutral) chiral multiplets Φ i , and b ( Y ) (Abelian) vector multiplets V I , assummarized in detail in Table 2.1.To specify the four-dimensional low-energy effective N = 1 supergravity actionfor the determined spectrum of the massless fields, we insert the mode expansions forthe metric (2.7), the anti-symmetric three-form tensor (2.15), and the gravitino (2.16)into the eleven-dimensional supergravity action [14], which in terms of the eleven-dimensional Hodge star ∗ and the eleven-dimensional gamma matrices ˆΓ M reads S d = 12 κ (cid:90) (cid:18) ∗ ˆ R S − d ˆ C ∧ ∗ d ˆ C − ∗ i ¯ˆΨ M ˆΓ MNP ˆ D N ˆΨ P (cid:19) − κ (cid:90) ∗ ¯ˆΨ M ˆΓ MNP QRS ˆΨ N ( d ˆ C ) [ P QRS ] − κ (cid:90) d ˆ C ∧ ∗ ˆ F − κ (cid:90) d ˆ C ∧ d ˆ C ∧ ˆ C + . . . , (2.22)where we denote ˆ F [ MNP Q ] = 3 ¯ˆΨ [ M ˆΓ NP ˆΨ Q ] . The first line contains the kinetic termsof the eleven-dimensional supergravity multiplet, i.e., the Einstein–Hilbert term interms of the Ricci scalar ˆ R S , the kinetic term for the anti-symmetric three-form ten-sor ˆ C , and the Rarita–Schwinger kinetic term for the gravitino ˆΨ. The second linecomprises the interaction terms and the third line is the Chern–Simons term of theeleven-dimensional supergravity action. There are additional four-fermion interac-tions denoted by ‘ . . . ’ [14]. The coupling constant κ relates to the eleven-dimensionalNewton constant ˆ G N , the eleven-dimensional Planck length ˆ (cid:96) P and Planck mass ˆ M P according to κ = 8 π ˆ G N = (2 π ) ˆ (cid:96) P π ) M P . (2.23)11o perform the Kaluza–Klein reduction let us introduce the moduli-dependentvolume V Y ( S i ) of the G -manifold Y given by V Y ( S i ) = (cid:90) Y d y (cid:112) det g ( S i ) mn . (2.24)Furthermore, we introduce a reference G -manifold Y with respect to some back-ground expectation values S i = (cid:104) S i (cid:105) , upon which we carry out the dimensional re-duction. This allows us to introduce the dimensionless (but yet moduli-dependent)volume factor λ ( S i ) = V Y ( S i ) V Y = 17 (cid:90) Y ϕ ∧ ∗ g ϕ ϕ , (2.25)in terms of the reference volume V Y = V Y ( S i ). Here the choice of Y fixes via theresulting volume factor V Y the normalization of the three-form ϕ .Then — using eqs. (2.7) and (2.15) — the dimensional reduction of the Einstein–Hilbert term and the three-form tensor ˆ C yields the four-dimensional action [17] S bos4 d = 12 κ (cid:90) (cid:104) ∗ R S + κ IJk (cid:0) S k F I ∧ ∗ F J − P k F I ∧ F J (cid:1) − λ (cid:90) Y ρ (3) i ∧ ∗ g ϕ ρ (3) j (cid:0) dP i ∧ ∗ dP j + dS i ∧ ∗ dS j (cid:1) (cid:105) . (2.26)in terms of the four-dimensional Hodge star ∗ , the Ricci scalar R S with respect tothe metric g µν , the reference volume V Y , and the seven-dimensional Hodge star ∗ .Here we have performed the Weyl rescaling of the four-dimensional metric accordingto g µν → g µν λ ( S i ) , (2.27)such that the four-dimensional coupling constant κ — relating the four-dimensionalNewton constant G N , the four-dimensional Planck length (cid:96) P and the Planck mass M P — becomes κ = κ V Y , κ = 8 πG N = 8 π(cid:96) P = 8 πM P . (2.28)Furthermore, the couplings κ IJk arise from the topological intersection numbers κ IJk = (cid:90) Y ω (2) I ∧ ω (2) J ∧ ρ (3) k . (2.29)We can now bring the (bosonic) action (2.26) into the conventional form of four-dimensional N = 1 supergravity [51]. To identify the chiral multiplets — that is to The structure of the four-dimensional effective N = 1 action from type II and M-theory dimen-sional reduction including Kaluza–Klein modes — which we do not consider here — has recentlybeen discussed in refs. [52, 53]. φ i = − P i + iS i . (2.30)Hence, due to holomorphy of the N = 1 superpotential, the complex fields φ i furnishcomplex coordinates of the K¨ahler target space and thus represent the complex scalarfields in the N = 1 chiral multiplets Φ i in Table 2.1. This allows us to quicklyread off from the action (2.26) the K¨ahler potential and the gauge kinetic couplingfunctions [17, 18] K ( φ, ¯ φ ) = − (cid:18) (cid:90) Y ϕ ∧ ∗ g ϕ ϕ (cid:19) , (2.31) f IJ ( φ ) = i (cid:88) k φ k (cid:90) Y ω (2) I ∧ ω (2) J ∧ ρ (3) k = i (cid:88) k κ IJk φ k . (2.32)Note that the holomorphy of the gauge kinetic coupling functions is in accordancewith the complex chiral coordinates (2.30). The moduli space metric is then given by g i ¯ j = ∂ i ∂ ¯ j K = 14 λ (cid:90) Y ρ (3) i ∧ ∗ g ϕ ρ (3) j . (2.33)Thus, we see that in the physical theory the real scalar fields S i and P i com-bine to the complex chiral scalars φ i according to eq. (2.30). These complex scalarfields parametrize locally the (semi-classical) M-theory moduli space M C of the G -compactification on Y of complex dimension b ( Y ), where the real subspace Re( φ i ) = 0of real dimension b ( Y ) is the geometric moduli space M of G -metrics on Y . Note,however, that the derived moduli space M C merely arises from the semi-classical di-mensional reduction of eleven-dimensional supergravity on the G -manifold Y . Forthe resulting four-dimensional N = 1 supersymmetric theory, one expects on gen-eral grounds that the flat directions of M C are lifted at the quantum level due tonon-perturbative effects in M-theory [22] — even in the absence of background fluxes.Finally, let us remark that the presence of non-trivial four-form background fluxes G of anti-symmetric three-form tensor fields ˆ C supported on the G -manifold Y gener-ates a flux-induced superpotential [16,17,20]. While the superpotential enters quadrat-ically in the bosonic action, it appears linearly in the fermionic action generating agravitino mass term M ψ [51] L M ψ d = 12 κ e K/ (cid:0) ¯ W ( ¯ φ ) ψ Tµ γ µν ψ ν + W ( φ ) ¯ ψ µ γ µν ψ ∗ ν (cid:1) . (2.34) For corrections to the semi-classical moduli space M C see ref. [54]. W allows us to directly derive the superpotential from thedimensional reduction of the gravitino terms, as carried in detail in Appendix A.4,where we determine the holomorphic superpotential to be W ( φ i ) = 14 (cid:90) Y G ∧ (cid:18) −
12 ˆ C + iϕ (cid:19) . (2.35)Our result yields the flux-induced superpotential in refs. [16, 17, 20, 54]. As explainedin ref. [17], in order to obtain the chiral combination − δ ˆ C + iδϕ in the variation δW ofthe superpotential W , it is necessary to introduce the relative factor between ˆ C and ϕ in formula (2.35). Note that — both in the presence and in the absence of back-ground fluxes G — we expect generically additional non-perturbative superpotentialcontributions arising from membrane instanton effects [4, 22]. G -manifolds In this section we focus on the construction of G -manifolds as put forward by Ko-valev [23] and further developed by Corti et al. [27, 24]. These G -manifolds are ob-tained from a certain twisted connected sum of two asymptotically cylindrical Calabi–Yau threefolds times an additional circle S . In the Kovalev limit the Ricci-flat metricof the obtained G -manifold can be approximated by the metrics of the two Calabi–Yau summands. In order to set the stage for the derivation of the low-energy effectiveaction for M-theory compactified on such G -manifolds, we identify the Kovalev limitin the moduli space of the constructed G -manifolds. A (complex) three-dimensional Calabi–Yau cylinder X ∞ is the product of a compactCalabi–Yau twofold — which we take to be a compact K3 surface S — with an opencylinder ∆ cyl , here given as the complement of the unit disk in the complex plane C ,i.e., ∆ cyl = { z ∈ C | | z | > } . The K¨ahler form ω ∞ and the holomorphic three-formΩ ∞ of X ∞ read ω ∞ = γ ∗ i d z ∧ d¯ z z ¯ z + ω S = γ ∗ d t ∧ d θ ∗ + ω S , Ω ∞ = − γ ∗ i d zz ∧ Ω S = γ ∗ (d θ ∗ − i d t ) ∧ Ω S , (3.1) Non-vanishing four-form fluxes induce a gravitational back-reaction to the eleven-dimensionalmetric. This requires a warped metric ansatz [16] that breaks supersymmetry [2,16]. As the presentedderivation neglects such back-reactions, the resulting effective action becomes more accurate thesmaller the effect of warping. Similarly as argued in ref. [20] this is the case for a small number offour-form flux quanta.
14n terms of the K¨ahler form ω S and the holomorphic two-form Ω S of the K3 surface S ,the complex coordinate z = e t + iθ ∗ and the length scale γ ∗ of the cylinder ∆ cyl . Then— with the metric g S of the K3 surface — the product metric g X ∞ of the Calabi–Yaucylinder X ∞ becomes g X ∞ = γ ∗ (cid:0) d t + d θ ∗ (cid:1) + g S . (3.2)The length scale γ ∗ furnishes the radius of the cylindrical metric on ∆ cyl , whereasthe map ξ : X ∞ → R + with ξ = log | z | projects on the longitudinal direction of thecylinder such that ξ − ( R + ) = X ∞ .As defined in refs. [23,27,30], an asymptotically cylindrical Calabi–Yau threefold X is a non-compact Calabi–Yau threefold with SU (3) holonomy and a complete Calabi–Yau metric g X with the following properties. There exists a compact subspace K ⊂ X such that the complement X \ K is diffeomorphic to a three-dimensional Calabi–Yaucylinder X ∞ and such that the K¨ahler and the holomorphic three-form of X approachfast enough ω ∞ and Ω ∞ of the cylindrical Calabi–Yau threefold X ∞ , as given ineqs. (3.1). More precisely, given the diffeomorphism η : X ∞ → X \ K , we require thatin the limit ξ → + ∞ and for any positive integer k [23, 27, 30] η ∗ ω − ω ∞ = d µ with |∇ k µ | = O ( e − λγ ∗ ξ ) ,η ∗ Ω − Ω ∞ = d ν with |∇ k ν | = O ( e − λγ ∗ ξ ) , (3.3)for certain choices of µ and ν with the norm | · | and Levi–Civita connection ∇ ofthe metric g X ∞ . The scale λ has inverse length dimension and is determined by the(inverse) length scale of the asymptotic region X ∞ . To be precise [23] λ = min (cid:110) γ ∗ , λ S (cid:111) , (3.4)where λ S is the square root of the smallest positive eigenvalue of the Laplacian of theK3 surface S in the asymptotic Calabi–Yau cylinder X ∞ . From any Calabi–Yau threefold X with SU (3) holonomy we can always construct theseven-manifold X × S with the torsion-free G -structure ϕ = γ d θ ∧ ω + Re(Ω) , ∗ ϕ = 12 ω − γ d θ ∧ Im(Ω) , (3.5)in terms of the K¨ahler form ω , the holomorphic three-form Ω of X and the coordinate θ of S with radius γ . However, the resulting seven-dimensional manifold with ϕ stillhas SU (3) holonomy, which is a subgroup of G . With the conventional mutual normalization ( − n ( n − (cid:0) i (cid:1) n Ω ∧ ¯Ω = ω n n ! between the K¨ahlerform ω and the holomorphic n -form Ω of Calabi–Yau n -folds, we note that — assuming this normal-ization for ω S and Ω S of the K3 surface S — the K¨ahler form ω ∞ and the holomorphic three-formΩ ∞ of X ∞ are conventionally normalized. G -manifolds Y [23], the essential idea is to firstconstruct two suitable asymptotically cylindrical Calabi–Yau threefolds X L/R (referredto as left and right). Then for each of them one takes a direct product with S L/R in order to obtain two seven-manifolds Y L/R with torsion-free G -structures ϕ L/R and SU (3) holonomy as above. To obtain a genuine compact G -manifold Y , theasymptotic regions of type Y ∞ L/R = X ∞ L/R × S L/R are glued together in such a way thatthe obtained manifold Y admits a torsion-free G -structure resulting in G holonomyand not a subgroup thereof.To obtain the full G holonomy, it is necessary to reduce the infinite fundamen-tal groups π ( Y L/R ) to a finite fundamental group π ( Y ) by gluing their asymptoticregions appropriately. For suitable choices of Y L/R this can be achieved by Kovalev’stwisted connected sum construction [23]. Recall that the asymptotic regions of Y L/R are given by Y ∞ L/R = X ∞ L/R × S L/R = S L/R × ∆ cyl L/R × S L/R , (3.6)with the K3 surfaces S L/R and the cylinders ∆ cyl
L/R (cf. Section 3.1). Let us denoteby ω IL/R , ω
JL/R and ω KL/R the triplets of mutually orthogonal hyper K¨ahler two-forms,satisfying the relations ( ω IL/R ) = ( ω JL/R ) = ( ω KL/R ) . Then the SU (2) structures ofthe asymptotic polarized K3 surfaces S L/R determine their K¨ahler two-forms ω ∞ S L/R and their holomorphic two-forms Ω ∞ S L/R according to ω ∞ S L/R = ω IL/R , Ω ∞ S L/R = ω JL/R + i ω KL/R . (3.7)With eqs. (3.1) and (3.5) this explicitly specifies the asymptotic torsion-free G -structures ϕ ∞ L/R = γ L/R d θ L/R ∧ (cid:16) γ ∗ L/R d t L/R ∧ d θ ∗ L/R + ω ∞ S L/R (cid:17) + γ ∗ L/R d θ ∗ L/R ∧ Re(Ω ∞ S L/R ) + γ ∗ L/R d t L/R ∧ Im(Ω ∞ S L/R ) . (3.8)Following Kovalev [23], let us now assume that the asymptotically cylindrical Calabi–Yau threefolds X L/R are chosen such that the resulting asymptotic polarized K3 sur-faces S L/R are mutually isometric with respect to a hyper K¨ahler rotation r : S L → S R obeying r ∗ ω IR = ω JL , r ∗ ω JR = ω IL , r ∗ ω KR = − ω KL . (3.9)Then there is a family of diffeomorphisms F Λ : Y ∞ L → Y ∞ R with constant Λ ∈ R givenby F Λ : ( θ ∗ L , t L , u αL , θ L ) (cid:55)→ ( θ ∗ R , t R , u αR , θ R ) = ( θ L , Λ − t L , r ( u αL ) , θ ∗ L ) , (3.10) These conditions impose rather non-trivial constraints on the pair of asymptotically cylindricalCalabi–Yau threefolds X L/R , which — at least for certain classes of pairs — have been studiedsystematically in ref. [24]. In this section we assume that these conditions on X L/R are met, andwe come back to this issue in Section 5.3, where we explicitly construct asymptotically cylindricalCalabi–Yau threefold X L/R fulfilling these constraints. X L/R ( T ) arethe truncated asymptotically cylindrical Calabi–Yau threefolds together with theircompact subspaces K L/R . Their Cartesian products with the circles S L/R yield thetwo seven-dimensional components Y L/R ( T ) combined to form the G -manifold Y .There are two essential aspects in the gluing procedure. Firstly — as indicated bythe red horizontal arrows — the circles S L/R are identified with the asymptotic cir-cles of X L/R ( T ) here denoted by S ∗ R/L . Secondly — as depicted by the blue verticalarrows — the asymptotic polarized K3 surfaces S L/R must be matched with a cer-tain hyper K¨ahler rotation. Finally, the diagram highlights the interpolating regions, t L/R ∈ ( T − , T ], and the asymptotic gluing regions, t L/R ∈ ( T, T + 1), important forthe construction of the G -structure ϕ ( γ, T ) of Y .in terms of the local coordinates ( θ ∗ L/R , t
L/R ) of ∆ cyl
L/R , u αL/R of S L/R , and θ L/R of S L/R .Now it is straightforward to check that if and only if the radii are equal γ := γ L = γ R = γ ∗ L = γ ∗ R , (3.11)this asymptotic diffeomorphism is also an asymptotic isometry because it leaves theasymptotic G -structures ϕ ∞ L/R — and hence the asymptotic metric — invariant, i.e., F ∗ Λ ϕ R = ϕ L . (3.12)As schematically depicted in Figure 3.1, the G -manifold Y is now obtained bygluing the asymptotic ends of Y L/R with the help of the diffeomorphism F Λ . Let17 L/R ( T ) and Y L/R ( T ) be the truncated asymptotically cylindrical Calabi–Yau three-folds and the truncated seven-manifolds given by cutting off their asymptotic regionsat t L/R = T + 1 for some (large) T , i.e., X L/R ( T ) = K L/R ∪ η L/R ( R 1] be a smooth functioninterpolating between 0 and 1 within the interval ( − , α ( s ) = 0 for s ≤ − α ( s ) = 1 for s ≥ 0. Then we endow the truncated asymptotically cylindricalCalabi–Yau threefolds X L/R with the forms [23, 27] (cid:101) ω TL/R = ω L/R − d (cid:0) α ( t − T ) µ L/R (cid:1) , (cid:101) Ω TL/R = Ω L/R − d (cid:0) α ( t − T ) ν L/R (cid:1) , (3.15)in terms of the forms µ L/R and ν L/R of eqs. (3.3). By construction the forms (cid:101) ω TL/R and (cid:101) Ω TL/R smoothly interpolate between the corresponding Calabi–Yau cylinder forms (3.1)and the asymptotically cylindrical Calabi–Yau forms (3.3). At the interpolating re-gions t L/R ∈ ( T − , T ) the symplectic forms ω TL/R fail to induce a Ricci-flat metricand the three-forms Ω TL/R cease to be holomorphic. Analogously to eq. (3.5), theinterpolating G -structures (cid:101) ϕ L/R ( γ, T ) on Y L/R read (cid:101) ϕ L/R ( γ, T ) = γ d θ ∧ (cid:101) ω TL/R + Re( (cid:101) Ω TL/R ) , (3.16)which according to eq. (3.12) glue together to a well-defined G -structure (cid:101) ϕ ( γ, T ) onthe seven-manifold Y .Note that the constructed G -structure (cid:101) ϕ ( γ, T ) is closed but not torsion-free. Thetorsion of ϕ ( γ, T ) is measured by d ∗ (cid:101) ϕ ( γ, T ). It is only non-vanishing at the interpo-lating regions t L/R ∈ ( T − , T ), where it is of order O ( e − γλT ) due to eq. (3.3) [23]. Hence, it is plausible that we can view (cid:101) ϕ ( γ, T ) as an order O ( e − γλT ) approximation toa torsion-free G -structure ϕ ( γ, T ), which equips the seven-manifold Y with a Ricci-flat metric. Indeed, Kovalev shows that, for sufficiently large T , there exists in the Note that — due to relation (3.11) and the hyper K¨ahler isometry (3.12) — we have the identi-fications λ = λ L = λ R among inverse length scales. (cid:101) ϕ ( γ, T ) a torsion-free G -structure ϕ ( γ, T ) suchthat, for any positive integer k , [23] ϕ ( γ, T ) = (cid:101) ϕ ( γ, T ) + d ρ ( γ, T ) with (cid:12)(cid:12) ∇ k ρ ( γ, T ) (cid:12)(cid:12) = O ( e − γλT ) , (3.17)in terms of the norm | · | and the Levi–Civita connection ∇ of the metric inducedfrom the asymptotic G -structure (3.12).Finally, the relationship π ( Y ) = π ( X L ) × π ( X R ) among the fundamental groupsin the twisted connected sum implies that the torsion-free G -structure ϕ ( γ, T ) indeedgives rise to a genuine G -manifold Y with G holonomy [23].This summarizes the main result of ref. [23] — clarified and further developed inrefs. [24,30,27] — namely the analysis proof that the G -structure (cid:101) ϕ ( γ, T ) in Kovalev’stwisted connected sum construction furnishes an approximation to the torsion-free G -structure ϕ ( γ, T ) in the same three-form cohomology class, which gives rise to acompact seven-dimensional Ricci-flat Riemannian manifold Y with G holonomy. G -manifolds As discussed in the last section, the torsion-free G -structure ϕ ( γ, T ) in Kovalev’stwisted connected sum is approximated via eq. (3.5) in terms of the holomorphicforms Ω L/R and the K¨ahler forms ω L/R of the asymptotically cylindrical Calabi–Yauthreefolds X L/R . According to eqs. (3.3) and (3.17) this approximation is of order O ( e − γλT ).We now discuss the torsion-free G -structure ϕ ( γ, T ) as a function of the param-eters γ and T . Except for the overall volume modulus, we keep all other moduli ofthe asymptotically cylindrical Calabi–Yau threefolds X L/R fixed. That is to say, weconsider the moduli dependence of the two metrics g L/R of X L/R as g L/R ( z L/R , t L/R ) = γ R ˜ g ( z L/R , ˜ t L/R ) , (3.18)where z L/R and t L/R are the (dimensionless) complex structure moduli and the K¨ahlermoduli of X L/R , respectively. The constant γ has dimension of length such that themetrics ˜ g L/R become dimensionless. We split the K¨ahler moduli t L/R further into theoverall volume modulus R and the remaining K¨ahler moduli ˜ t L/R — measuring ratiosof volumes of subvarieties in X L/R — such that t aL/R = (cid:40) R a = 1 R ˜ t aL/R a (cid:54) = 1 . (3.19)In order to obtain the G -manifold Y from the seven-dimensional building blocks Y L/R ,the hyper K¨ahler compatibility condition (3.9) constrains the explicit values of the For an asymptotically cylindrical Calabi–Yau threefold with a single K¨ahler modulus t the volumemodulus R relates to this K¨ahler modulus as R = t without the presence of any further moduli ˆ t . z L/R and ˜ t L/R . Furthermore, the required identification (3.11) of the radii ofall circles in the asymptotic region of Y L/R determines the volume modulus R as thedimensionless ratio R = γγ , (3.20)and it justifies to introduce a mutual volume modulus R for both threefolds X L/R .In the Kovalev limit of large RT , the volume V Y of the constructed G -manifold Y becomes V Y ( ˜ S, R, T ) = V Y L ( T ) ( z L , t L ) + V Y R ( T ) ( z R , t R ) − (2 π ) γ R V S ( ˜ ρ S , R ) + O ( e − ˜ λRT ) . (3.21)The resulting volume depends on the moduli R and T and the remaining moduli of the G -manifold Y — collective denoted by ˜ S singling out those moduli fields ˜ ρ S deformingthe K3 surface S . The volumes V Y L/R ( T ) of the truncated building blocks Y L/R ( T )are calculated with the metrics of the (truncated) asymptotically cylindrical Calabi–Yau threefolds X L/R ( T ) using the expressions (3.1) and the volume formula (2.25)(with the dimensionful reference volume V Y = γ ). As the sum of the first twoterms counts the volume of the overlapping region twice, we need to subtract thiscontribution again once. It is given by the product of the volumes of the overlappinginterval, the asymptotic two-torus S L × S ∗ L ≡ S R × S ∗ R , and the asymptotic K3-surface S L ≡ S R ≡ S according to V Y L ( T ) ∩ Y R ( T ) = (2 π ) γ R V S ( ˜ ρ S , R ) = (2 π ) γ R V ˜ gS ( ˜ ρ S ) . (3.22)In the last equality, the volume of the K3-surface S is expressed in terms of thedimensionless (asymptotic) metric ˜ g . Due to approximation (3.3) of the metrics ofthe building blocks Y L/R ( T ) by the limiting metrics of Y ∞ L/R ( T ) in the interpolationregion t L/R ∈ ( T − , T ) and due to the overall correction (3.17) to the torsion-free G -structure ϕ ( γ, T ), the computed volume of the G -manifold Y receives exponentiallysuppressed corrections in ˜ λRT for large RT , where — because of eq. (3.4) — thedimensionless constant ˜ λ reads ˜ λ = λγ .Note that — up to exponentially correction terms suppressed for large RT — thevolume (3.22) is entirely determined by the (relative) periods and the K¨ahler formsof the asympotically cylindrical Calabi–Yau threefolds X L/R . However, due to non-compactness of X L/R , the relative periods and the K¨ahler volumes of non-compactcycles diverge linearly in T . Therefore, in order to obtain the finite periods and finitevolumes of the truncated asymptotically cylindrical Calabi-Yau threefolds X L/R ( T ),a suitable regularization scheme must be employed to extract the required geometricdata from the diverging periods and infinite volumes. This analysis is beyond thescope of this work, but we plan to get back to this issue elsewhere.Thus, instead of deriving the entire moduli dependence of the volume of the G -manifold Y , we focus on the moduli dependence of the two fields R and T — viewing20he remaining moduli fields ˜ S as parameters. First, we compute the volumes V Y L/R ( T ) in eq. (3.22) as V Y L/R ( T ) = γ (cid:90) Y L/R ( T ) ϕ L/R ∧ ∗ ϕ L/R = γ (cid:90) Y L/R ( T ) \ K L/R ϕ ∞ L/R ∧ ∗ ϕ ∞ L/R + V K L/R + (2 π ) γ R ∆ L/R ( ˜ S, T )= (2 π ) γ R (cid:104) V ˜ gS ( ˜ ρ S ) (cid:16) ( T + 1) + D L/R ( ˜ S ) (cid:17) + ∆ L/R ( ˜ S, T ) (cid:105) . (3.23)Here we split the integration by performing the integral over the compact parts K L/R and the asymptotic regions Y L/R ( T ) \ K L/R . The former part factors into the volumeof the K3 surface S and a contribution D L/R ( ˜ S ) to be discussed in greater detailmomentarily. The latter part is evaluated with respect to the asymptotic G -structure ϕ ∞ L/R , which introduces the correction term ∆ L/R ( ˜ S, T ) such that∆ L/R ( ˜ S, T ) = R (cid:90) T +10 dt f L/R ( ˜ S, Rt ) e − ˜ λRt = C L/R ( ˜ S ) + O ( e − ˜ λRT ) , (3.24)in terms of the function f L/R ( ˜ S, Rt ) determined by eq. (3.3). Thus, taking the cor-rection terms ∆ L/R into account, we arrive at V Y ( ˜ S, R, T ) = (2 π ) γ R V ˜ gS ( ˜ ρ S ) (cid:16) T + α ( ˜ S ) (cid:17) + O ( e − ˜ λRT ) , (3.25)with α ( ˜ S ) = (cid:16) D L ( ˜ S ) + C L ( ˜ S ) (cid:17) + (cid:16) D R ( ˜ S ) + C R ( ˜ S ) (cid:17) + 1 . (3.26)As discussed, the moduli-dependent contributions D L/R ( ˜ S ) + C L/R ( ˜ S ) — and hencethe moduli dependent function α ( ˜ S ) — are in principle computable from the (rel-ative) periods and the K¨ahler forms of the asymptotically cylindrical Calabi–Yauthreefolds X L/R .In the context of M-theory compactification on the G -manifold Y , the volume V Y determines the four-dimensional Planck constant κ according to eq. (2.28). Werefer to the Kovalev limit as the approximation in which the four-dimensional Planckconstant κ — and hence the volume V Y — remains constant, while the exponentialcorrection terms become sufficiently small. Namely, for fixed moduli ˜ S we requirethat the dimensionless quantity χ , that is given by χ = R (2 T + α ) , (3.27)remains constant. Requiring a constant four-dimensional Planck constant yields thefunctional dependence for the modulus RR ( T ) = χ √ T + α , (3.28)21uch that corrections terms scale as O ( e − ˜ λRT ) = O ( e − ˜ λχT √ T + α ) . (3.29)Thus, for large T , with R = R ( T ) — as in eq. (3.28) — the corrections for thevolume V Y in eq. (3.25) are exponentially suppressed. Furthermore, for T (cid:29) ˜ λχ —loosely referred to as the Kovalev limit — the torsion-free G -structure of Y is wellapproximated in terms of the geometric data of the asymptotically cylindrical Calabi–Yau threefolds X L/R for a given four-dimensional Planck constant κ . However, takingliterally the limit T → ∞ with R = R ( T ) does not yield a limiting Riemannianmanifold but instead yields only a Hausdorff limit in the sense of Gromov–Hausdorffconvergence of compact metric spaces. In the context of M-theory, the Kovalev limitimplies that — while for large T with R = R ( T ) the Ricci-flat G -metric gets moreand more accurately approximated in terms of the Ricci-flat Calabi–Yau metrics of X L/R — the discussed semi-classical dimensional reduction on the G -manifold Y interms of the Kaluza–Klein zero modes becomes less accurate due to the emergenceof both light Kaluza–Klein modes and substantial non-perturbative membrane andM5-brane instanton corrections. In this section we analyze the four-dimensional low-energy effective action of M-theorycompactifications on G -manifolds that are of the twisted connected sum type. Wefocus on the Kovalev limit, in which the asymptotically cylindrical Calabi–Yau metricsof the summands furnish a good approximation to the G -metric. Our first task isto analyze the four-dimensional N = 1 spectrum of such compactifications, which —according to Table 2.1 — amounts to expressing the de Rham cohomology groupsof the resulting G -manifolds in terms of the cohomology groups of the Calabi–Yausummands. In particular, we explicitly identify the chiral modulus governing theKovalev limit, referred to as the Kovalevton κ , and we discuss the effective actionand its physical properties in this limit. To deduce the four-dimensional N = 1 supersymmetric spectrum of M-theory com-pactified on a G -manifold of the twisted connected sum type, we should analyzethe de Rham cohomology of Y as arising from the cohomology of the asymptoticallycylindrical Calabi–Yau summands. A partial answer to this question has already beengiven in Kovalev’s paper [23]. In ref. [24] Corti et al. have presented a systematicanalysis of the cohomology of Y , which we summarize and use here.Following refs. [27, 24], let us first introduce the notion of a building block ( Z, S ),which allows us to construct an asymptotically cylindrical Calabi–Yau threefold X .22n this work a building block ( Z, S ) is a pair consisting of a smooth K3 fibration π : Z → P together with a smooth K3 fiber S = π − ( p ) for some p ∈ P . We require thatthe anti-canonical class − K Z is primitive and that S is linearly equivalent to the anti-canonical class, i.e., S ∼ − K Z . Due to the fibrational structure, the self-intersectionof S is trivial, which implies that the manifold X = Z \ S has the topology of anasymptotically cylindrical Calabi–Yau threefold that admits a Ricci-flat K¨ahler metric[23, 27, 24]. As in refs. [27, 24], we further impose two technical assumptions on thebuilding block ( Z, S ). Namely, we demand that H ( Z, Z ) is torsion-free and that theintegral two-form cohomology H ( X, Z ) embeds primitively into the K3 lattice L = H ( S, Z ) via the pullback map of the inclusion ρ : S (cid:44) → X (which is well-defined upto homotopy).To construct a G -manifold Y we now consider a pair of building blocks ( Z L/R , S L/R )such that the polarized K3 surfaces S L/R are isometric and fulfill the hyper K¨ahlermatching condition r : S L → S R . From the asymptotically cylindrical Calabi–Yauthreefolds X L/R = Z L/R \ S L/R we can then construct the G -manifold Y with Ko-valev’s twisted connected sum construction as detailed in Section 3.2. Under theassumptions in the definition of the building blocks ( Z L/R , S L/R ), Corti et al. derivethe cohomology of the G -manifold Y from the building blocks [24] π ( Y ) = H ( Y, Z ) = 0 ,H ( Y, Z ) (cid:39) ( k L ⊕ k R ) ⊕ ( N L ∩ N R ) ,H ( Y, Z ) (cid:39) H ( Z L , Z ) ⊕ H ( Z R , Z ) ⊕ k L ⊕ k R ⊕ N L ∩ T R ⊕ N R ∩ T L ⊕ Z [ S ] ⊕ L/ ( N L + N R ) . (4.1)Here [ S ] is the Poincar´e dual three-form of a K3 fiber S in the building blocks( Z L/R , S L/R ), and L denotes the K3 lattice L (cid:39) H ( S L , Z ) (cid:39) H ( S R , Z ). Furthermore,the inclusion maps ρ L/R : S L/R (cid:44) → X L/R induce the maps ρ ∗ L/R : H ( X L/R , Z ) → L ,which define the kernels k L/R := ker ρ ∗ L/R , the images N L/R := Im ρ ∗ L/R , and thetranscendental lattices T L/R = N ⊥ L/R = (cid:8) l ∈ L (cid:12)(cid:12) (cid:104) l, N L/R (cid:105) = 0 (cid:9) . (4.2)Note that — by the assumptions imposed on the cohomological properties of thebuildings blocks — the images N L/R are primitive sublattices of the K3 lattice L .We further assume that the sum N L + N R embeds primitively into the K3 lattice L . As a consequence the quotient L/ ( N L + N R ) is torsion-free, and — due to theassumed torsion-freeness of H ( Z L/R , Z ) — all the integral cohomology groups (4.1)are torsion-free as well. This can readily be seen as follows: The trivial normal bundle of S in X defines a tubularneighborhood T (cid:15) ( S ) ⊂ X . By construction T ∗ (cid:15) ( S ) := T (cid:15) ( S ) \ S is homeomorphic to ∆ cyl × S , whichhas the topology of a three-dimensional Calabi–Yau cylinder X ∞ to be viewed as the asymptoticregion of the asymptotically cylindrical Calabi–Yau threefold X , as discussed in Section 3.1. N = 1 supersymmetric spectrum of the M-theory compactification according to Table 2.1. Let us now in particular focus on theneutral chiral moduli multiplets Φ i associated to the three-form cohomology H ( Y, Z ).The derivation of the cohomology in ref. [24] is essentially based upon the Mayer–Vietoris sequence applied to the union (3.14) of the overlapping non-compact seven-manifolds Y L/R defined in eq. (3.13) in terms of the asymptotically cylindrical Calabi–Yau threefolds X L/R such that H ( Y, Z ) = ker (cid:18) H ( Y L , Z ) ⊕ H ( Y R , Z ) ( ι ∗ L , − ι ∗ R ) −−−−−→ H ( T × S, Z ) (cid:19) ⊕ coker (cid:18) H ( Y L , Z ) ⊕ H ( Y R , Z ) ( ι ∗ L , − ι ∗ R ) −−−−−→ H ( T × S, Z ) (cid:19) , (4.3)where Y R ∩ Y L deformation retracts to T × S and the maps are induced from thestandard inclusion maps ι L/R : T × S (cid:44) → Y L/R . The three-form cohomolology in (4.1)is distributed among these two summands in the following way [24]ker (cid:0) H ( Y L , Z ) ⊕ H ( Y R , Z ) → H ( T × S, Z ) (cid:1) = H ( Z L , Z ) ⊕ H ( Z R , Z ) ⊕ k L ⊕ k R ⊕ N L ∩ T R ⊕ N R ∩ T L , coker (cid:0) H ( Y L , Z ) ⊕ H ( Y R , Z ) → H ( T × S, Z ) (cid:1) = Z [ S ] ⊕ L/ ( N L + N R ) . (4.4)Expanding the torsion-free G -structure ϕ of Y in terms of the above assembled three-form cohomology basis as in eq. (2.14), the coefficients of the individual cohomologyelements capture (in the Kovalev limit) particular geometric moduli of the twistedconnected sum and their summands.First of all, the kernel contributions in (4.4) describe the moduli of the asymp-totically cylindrical Calabi–Yau manifolds X L/R . In particular, the coefficients of H ( Z L/R ) and k L/R realize the complex structure moduli and the K¨ahler moduli ofthe asymptotically cylindrical Calabi–Yau manifolds X L/R , respectively. Furthermore, N L ∩ T R captures mutual K¨ahler moduli of X L and complex structure moduli of X R ,which are interlinked in this way due to the non-trivial gluing with the hyper K¨ahlerrotation (3.9) — exchanging X L and X R . The intersection N R ∩ T L enjoys an analoginterpretation.Second of all, we analyze the cokernel contribution in eq. (4.4). To get a bettergeometric picture for these moduli, we first observe that — due to the K3 fibrations Z L/R → P — the Calabi–Yau threefolds X L/R are K3 fibrations over a disk D L/R .As a result Y L/R become K T L/R ≡ S L/R × D L/R , namely S L/R −−−→ Y L/R (cid:121) π T L/R . (4.5)24he gluing diffeomorphism (3.10) in the twisted connected sum identifies the boundaryof the disk D L with the circle S R and the circle S L with the boundary of the disk D R such that the two solid tori T L/R are glued together to a three-sphere S . Thus, theresulting G -manifold Y is a topological K3 fibration S −−−→ Y (cid:121) π S . (4.6)The cohomology three-forms of the cokernel (4.4) describe moduli of the asymptoticboundary of ∂Y L (cid:39) ∂Y R (cid:39) T × S . Their dual homology three-cycles restrict torelative three-cycles in the summands Y L and Y R , and hence the associated moduliare sensitive to the overlapping gluing regions Y L/R ( T ) \ K L/R , cf. Figure 3.1. Inparticular, the three-form generator [ S ] is Poincar´e dual to a K3 fiber S , and henceits dual homology three-cycle is the S base of the fibration (4.6). As a consequencethe modulus associated to [ S ] measures the volume of the S base. Similarly, theremaining cokernel moduli measure volumes of three-cycles that project under themap π : Y → S to paths in the S base connecting the disjoint compact subsets π ( K L ) and π ( K R ) of S . Note that 2 T + 1 is the distance between these two compactsubsets in terms of the parameter T introduced in Section 3.3. Therefore, it nowfollows that — in the Kovalev limit — all cokernel moduli depend linearly on theparameter T , and geometrically T enjoys the interpretation of a squashing parameterfor the S base of the K3 fibration (4.6).The split into two types of moduli fields in eq. (4.4) motivates us to introduce twouniversal geometric moduli v and b . For any G -manifold there is the universal volumemodulus v that is associated to the singlet H ( Y, Z ) of the three-form cohomology.It simply rescales the torsion-free G -structure ϕ . In the twisted connected sum weadditionally identify the squashing modulus b of the S base in the fibration (4.6).Note that b → + ∞ describes the Kovalev limit discussed in Section 3.3. Accordingto eq. (2.14), the torsion-free G -structure ϕ depends on these two moduli as ϕ ( v, b, ˜ S ) = v (cid:34)(cid:32) ρ ker0 + (cid:88) ˆ ı ˜ S ˆ ı ρ kerˆ ı (cid:33) + b (cid:32) [ S ] + (cid:88) ˜ ı ˜ S ˜ ı ρ coker˜ ı (cid:33)(cid:35) . (4.7)Here [ S ] is the harmonic three-form that is Poincar´e dual to the K3 fiber S . Fur-thermore, ( ρ ker0 , ρ kerˆ ı ) and ρ coker˜ ı form a basis of harmonic three-forms arising from thekernel contributions and the cokernel part L/ ( N L + N R ) in (4.4), respectively. ˜ S ˆ ı and˜ S ˜ ı are the respective associated geometric real moduli fields. The topological K3 fibration in the context of twisted connected sums has also been discussedin ref. [26]. Note that the kernel contribution (4.4) is at least one-dimensional, such that we can alwayschoose a basis element ρ ker [24]. G -structure ϕ ( v, b, ˜ S ) gives rise to twouniversal N = 1 neutral chiral moduli multiplets ν and κ given byRe( ν ) = v , Re( κ ) = vb . (4.8)In particular, we refer to the chiral multiplet κ as the Kovalevton since it describesin the limit Re( κ ) → + ∞ — while keeping Re( ν ) constant — the Kovalev limitdiscussed in Section 3.3.The remaining real moduli fields are not universal and relate to the non-universalneutral chiral multiplets asRe( φ ˆ ı ) = v ˜ S ˆ ı , Re( φ ˜ ı ) = vb ˜ S ˜ ı . (4.9)They depend on the topological details of the building blocks ( Z L/R , S L/R ) and thechoice of gluing diffeomorphism (3.10).Finally, the two-form cohomology H ( Y, Z ) for (smooth) G -manifolds yields four-dimensional massless abelian N = 1 vector multiplets, cf. Table 2.1. In Kovalev’stwisted connected sum we get two types of N = 1 vector multiplets according toeq. (4.1), which we discuss in the sequel.Firstly, the kernel contributions k L and k R associate to zero modes of the twosummands Y L and Y R . Thus, we can view Y L/R as the local geometries governing thesegauge theory degrees of freedom. As the individual summands Y L/R = S L/R × X L/R have SU (3) holonomy in the Kovalev limit, we expect that the two gauge theorysectors of the kernels k L and k R exhibit N = 2 supersymmetry. Indeed — in additionto the abelian N = 1 vector multiplets — the kernels k L/R of the local geometries Y L/R also contribute to the three-form cohomology H ( Y, Z ) resulting in N = 1 neutralchiral multiplets. Thus, the abelian N = 1 vector and the neutral N = 1 chiralmultiplets associated to the k L/R readily combine into four-dimensional N = 2 vectormultiplets.Secondly, in the Kovalev limit the abelian vector multiplets obtained from theintersection N L ∩ N R can be attributed to the local geometry of the asymptotic regions Y L ( T ) ∩ Y R ( T ) (cid:39) T × S × (0 , SU (2) holonomy. Thus, we expect thatthese vector multiplets give rise to a four-dimensional abelian N = 4 gauge theorysector, which can be seen as follows. To any two-form ω (2) in N L ∩ N R of the K3surface S , we attribute the three-forms ω (2) ∧ h ( t ) dθ L , ω (2) ∧ h ( t ) dθ R , ω (2) ∧ h ( t ) dt , (4.10)in terms of the coordinates θ L/R of S L × S R (cid:39) T and the smooth bump function h ( t )in the coordinate t of the interval (0 , These three-forms yield geometrically non-trivial cohomology elements of compact support in H c ( T × S × (0 , , Z ), which give The bump function h ( t ) is given by a smooth non-negative function h ( t ) : (0 , → R withcompact support, which is normalized such that (cid:82) h ( t ) dt = 1. S to three complex scalar moduli fields, whichfurnish three neutral four-dimensional N = 1 chiral multiplets. It is these three N = 1chiral multiplets that combine with the N = 1 vector multiplet of ω (2) to one N = 4vector multiplet. Note that the three-forms (4.10) canonically extend to Kovalev’s G -manifold Y . However, they become trivial in cohomology because N L ∩ N R is notan element of H ( Y, Z ) according to eq. (4.1). Nevertheless, we can Fourier expandany of these three-forms into eigenforms with respect to the three-form Laplacian ∆ ofthe G -manifold Y . By a simple scaling argument we find that the eigenvalues of thesethree-form Fourier modes scale with T − , i.e., they are inversely proportional to theparameter T realizing the Kovalev limit. Therefore, we argue that the normalizablezero modes associated to the three-forms (4.10) acquire a mass term m (cid:39) O ( T − ),which vanishes in the Kovalev limit. Furthermore, we expect that the scalar fieldsassociated to the hyper K¨ahler metric deformations are generically obstructed at firstorder by a mass term that also vanishes in the Kovalev limit. As a consequence, wededuce that the massless four-dimensional abelian N = 4 vector multiplets of theasymptotic region decomposes into a massless four-dimensional abelian N = 1 vectormultiplet and three massive four-dimensional N = 1 chiral multiplets with massesof order O ( T − / ). Thus, we expect that the four-dimensional N = 4 gauge theorysector is only realized in the strict Kovalev limit T → + ∞ .The discussed local abelian gauge theory sectors are summarized in Table 4.1. Inparticular, we find that in the Kovalev limit — at least in the absence of backgroundfour-form fluxes and for smooth G -manifolds Y — the spectrum of all abelian gaugetheory sectors exhibit extended supersymmetry. The observed extended supersym-metries of the local geometries appearing in Kovalev’s twisted connected sum becomerelevant in Sections 5 and 6 because they impose strong constraints on the non-Abeliangauge theory sectors with charged matter fields. The aim of this subsection is to describe the universal properties of the four-dimensionallow-energy effective action in terms of the universal chiral multiplets ν and κ . Wefirst establish that — while keeping the ratio Re( ν ) / Re( κ ) constant — the chiralmultiplet ν directly relates to the (dimensionless) volume modulus R of Section 3.3 Alternatively, we can consider the five-dimensional theory obtained from M-theory on T × S with SU (2) holonomy. Then the two-form cohomology element ω (2) is accompanied by the twothree-form cohomology elements ω (2) ∧ dθ L/R . Combined with the mentioned three hyper K¨ahlermetric deformations these cohomology elements provide the zero modes of five scalar fields, which— together with the vector field and the superpartners — assemble into a five-dimensional N = 2vector multiplet for each harmonic two-form ω (2) . Upon dimensional reduction to four dimensions,we arrive at four-dimensional N = 4 vector multiplets. N = 1 multiplets U (1) vector multiplets(Kovalev limit) U (1) vectors chirals multiplicity supersym. Y L = S L × X L dim k L dim k L dim k L N = 2 SU (3) holonomy Y R = S R × X R dim k R dim k R dim k R N = 2 SU (3) holonomy T × S × (0 , 1) dim N L ∩ N R · dim N L ∩ N R dim N L ∩ N R N = 4 SU (2) holonomyTable 4.1: Shown are the abelian gauge theory sectors of the local geometries appear-ing in twisted connected sum G -manifolds in the Kovalev limit T → + ∞ . The leftcolumn specifies the Ricci-flat local geometries with their holonomies. The middlecolumn lists the N = 1 chiral multiplets that assemble in the right column to vectormultiplets of extended supersymmetry.as Re( ν ) = R . (4.11)This relation comes about because the Re( ν ) measures (dimensionless) volumes ofthree-cycles while R measures (dimensionless) length scales in the G -manifold Y .Apart from the overall volume dependence, the Kovalevton κ measures the squashedvolume of the S base. Therefore, from expression (3.25) of the volume V Y ( ˜ S, R, T )we arrive at the relation Re( κ ) = (2 π ) R (2 T + α ( ˜ S )) , (4.12)where ˜ S denotes collectively the remaining geometric moduli fields ˜ S ˜ ı and ˜ S ˆ ı .Thus — using eqs. (2.31) and (3.25) — we find that the universal structure of thefour-dimensional low-energy effective N = 1 supergravity action is governed by theK¨ahler potential K ( ν, ¯ ν, κ , ¯ κ ) = − ν + ¯ ν ) − κ + ¯ κ ) − (cid:16) V ˜ gS ( ˜ S ) (cid:17) . (4.13)Note that this K¨ahler potential is only a valid approximation both in the large volumeregime and in the Kovalev regime, where quantum corrections and metric correctionsof the asymptotically cylindrical Calab–Yau threefolds are suppressed. The semi-classical large volume limit arises when both Re( ν ) and Re( κ ) are taken sufficientlylarge, and when Re( κ ) is (parametrically) larger than Re( ν ) — cf. the discussionat the end of Section 3.3 — while the corrections to the G -metric in the twistedconnected sum are suppressed. 28et us discuss some basic properties of the derived K¨ahler potential. First of all,the structure of the K¨ahler potential is reminiscent of the Kovalev limit, in whichthe volume of the G -manifold is dominated by the cylindrical region S × T × I interms of the interval I of size 2 T + 1. That is to say that, in this limit, the individualsummands in eq. (4.13) reflect the volume of the K3 surface S , the squashed volumeof the S base dominated by T × I , and the moduli dependence of the K3 fiber S on the non-universal moduli ˜ S . As long as we treat the non-universal modulifields ˜ S as constants, the K¨ahler geometry for the universal K¨ahler moduli ν and κ factorizes into two (complex) one-dimensional parts with a block diagonal K¨ahlermetric. However, this block structure in the K¨ahler metric vanishes as soon as wetreat the non-universal moduli fields ˜ S dynamically, because relation (4.9) impliesthat the real geometric moduli ˜ S also depend non-trivially on the chiral fields ν and κ as ˜ S ˆ ı = φ ˆ ı + ¯ φ ˆ ı ν + ¯ ν , ˜ S ˜ ı = φ ˜ ı + ¯ φ ˜ ı κ + ¯ κ . (4.14)We now briefly discuss the K¨ahler potential (4.13) with the non-universal modulifields ˜ S treated as constants. We first observe that g i ¯ ∂ i K∂ ¯ K − ≥ , i ∈ { ν, κ } , ¯ ∈ { ¯ ν, ¯ κ } , (4.15)in terms of the inverse K¨ahler metric g i ¯ . This implies that the no-scale inequality g i ¯ ∂ i K∂ ¯ K − ≥ N = 1 supergravity theory is positive semi-definite forany non-vanishing superpotential [55]. As a result the analyzed K¨ahler potential (ofthe two chiral fields ν and κ only) does not admit a negative cosmological constantand hence no (supersymmetric) anti-de-Sitter vacua.Finally, we record the K¨ahler potential including the leading order correction tothe Kovalev limit, which according to eq. (3.25) takes the form K = − log (cid:20)(cid:16) V ˜ gS ( ˜ S ) (cid:17) ( ν + ¯ ν ) ( κ + ¯ κ ) + A ( ˜ S, ν + ¯ ν, κ + ¯ κ ) e − λ κ +¯ κ ( ν +¯ ν )1 / (cid:21) , (4.16)where the coefficient of the exponentially suppressed correction is expected to generi-cally depend on both universal and non-universal geometric moduli fields. A detailedanalysis of this class of K¨ahler potential may exhibit interesting phenomenologicalproperties, which, is, however, beyond the scope of this work. In this section we discuss the explicit construction of twisted connected sum G -manifolds by the method of orthogonal gluing [24]. This construction offers a system-atic way to fulfill the matching condition (3.9) for a pair of asymptotically cylindrical29alabi–Yau threefolds X L/R that are obtained from building blocks ( Z L/R , S L/R ) as-sociated to semi-Fano threefolds P L/R . We focus on building blocks ( Z L/R , S L/R ) ofpolarized K3 surfaces S L/R with Picard lattices of low rank and generate a list ofnew examples in order to get an impression of the multitude of possibilities to realizetwisted connected sum G -manifolds in terms of orthogonal gluing.Our motivation for studying the method of orthogonal gluing is also to pan out thepossibilities to obtain gauge theory sectors in twisted connected sum G -manifolds.In Section 4 we have established that in the Kovalev limit this spectrum of vectormultiplets assembles itself into N = 2 and N = 4 sectors as summarized in Table 4.1.While we postpone the analysis of the N = 2 sectors to the next section, the focus inthis section is on the N = 4 gauge theory sectors.In the context of the orthogonal gluing construction, a certain intersection lattice R determines the N = 4 gauge theory sector. In particular, the rank of this intersectionlattice becomes the rank of the gauge group. We further argue that in the N = 4gauge theory sector the method of orthogonal gluing does not admit enhancementsto non-Abelian gauge groups. As a result, we therefore arrive in the Kovalev limit atfour-dimensional Abelian N = 4 gauge theory sectors with gauge group U (1) rk R . Following ref. [27], we obtain from toric weak Fano threefolds P a rich class of build-ing blocks ( Z, S ), which in turn give rise to asymptotically cylindrical Calabi–Yauthreefolds as discussed in Section 4.1. A projective smooth threefold P is weak Fanoif its anti-canonical divisor − K P is nef and big, which means that the intersectionsobey − K P C ≥ C in P and ( − K P ) > Assuming fur-ther that two global sections s and s of the anti-canonical divisor − K P intersecttransversely in a smooth reduced curve C = { s = 0 } ∩ { s = 0 } ⊂ P and that S = { α s + α s = 0 } ⊂ P is a smooth K3 surface for some choice of [ α : α ] ∈ P ,a building block ( Z, S ) is obtained from the blow up π C : Z → P along C , i.e., Z = Bl C P = (cid:8) ( x, z ) ∈ P × P (cid:12)(cid:12) z s + z s = 0 (cid:9) , (5.1)together with the proper transform S of the smooth anti-canonical divisor S on P [27].Note that, for ease of notation, we use the same symbol S for both the K3 surface inthe toric weak Fano threefold P and its proper transform in the blow-up Z . Then theK3 fibration π : Z → P becomes, cf. Section 4.1, π : Z → P , ( x, z ) (cid:55)→ z , (5.2)and S = π − ([ α , α ]) is the K3 surface of the building block ( Z, S ). Moreover, thethree-form Betti number b ( Z ) of the blown-up threefold Z becomes b ( Z ) = b ( P ) + 2 g ( C ) = b ( P ) + ( − K P ) + 2 . (5.3) A smooth projective variety is Fano if its anti-canonical divisor is ample, i.e., − K P C > C in P . g ( C ) denotes the genus of the curve C and the last equality follows from theadjunction formula.For twisted connected sum G -manifolds, however, it is essential to find a pairof asymptotically cylindrical Calabi–Yau threefolds that fulfill the matching condi-tion (3.9). A common strategy is to first focus only on the moduli spaces of the po-larized K S L/R , ignoring their origin from the building blocks ( Z L/R , S L/R )[24,27,56]. Once a matching pair of polarized K S L/R is found, it is necessaryto check if these particular K S L/R arise as zero sections of anti-canonicaldivisors in Z L/R . Beauville’s theorem guarantees that indeed any general K K S L/R to ensure theexistence of a pair of matching building blocks ( Z L/R , S L/R ) in the moduli spaces ofthese building blocks.However, for building blocks ( Z L/R , S L/R ) obtained from weak Fano threefolds theprocedure of simply matching their polarized K S L/R may not be sufficient.That is to say, in this more general setting the entire moduli space of the polarized K Z, S ) is constructed from toricweak Fano threefolds P Σ described in terms of a three-dimensional toric fan Σ. We de-scribe the toric fan Σ of a toric weak Fano threefold P Σ in terms of a three-dimensionalreflexive lattice polytope ∆ spanned by the one-dimensional cones of Σ, together witha triangulation, which encodes the higher-dimensional cones of the fan Σ. By theclassification of Kreuzer and Skarke [60, 61] there are 4 319 three-dimensional reflex-ive polytopes, which often admit several triangulations, i.e., typically of the orderof ten to a few hundred triangulations. In this work we focus on the class of toricsemi-Fano threefolds P Σ in order to follow the outlined recipe to construct explicitmatching pairs for twisted connected sum G -manifolds. In the toric setting thesemi-Fano threefolds P Σ are characterized by those reflexive polytopes ∆ that do nothave any interior points inside co-dimension one faces [27]. Note that there are 899three-dimensional reflexive polytopes of the semi-Fano type [27].The toric approach to semi-Fano threefolds provides a powerful combinatorial ma-chinery to explicitly carry out computations. For instance, a general global section s ∆ See, for instance, refs. [58, 59] for an introduction to toric geometry. 31f the anti-canonical line bundle − K P Σ is readily described by s ∆ = (cid:88) ν i ∈ ∆ s i (cid:89) ν ∗ k ∈ ∆ ∗ x (cid:104) ν i ,ν ∗ k (cid:105) +1 k , (5.4)with the points ν i and ν ∗ k of the lattice polytope ∆ and its dual lattice polytope ∆ ∗ , theduality lattice pairing (cid:104)· , ·(cid:105) , the toric homogenous coordinates x k , and the coordinates s i on the space of global anti-canonical sections. Furthermore, for the discussed three-dimensional semi-Fano lattice polytopes, a choice of generic sections s and s yieldsa smooth reduced curve C = { s = 0 } ∩ { s = 0 } ⊂ P Σ . This curve has a smoothK3 surface S = { α s + α s = 0 } ⊂ P Σ such that the blow-up (5.1) together withthe proper transform S yield indeed a well-defined building block ( Z, S ), which werefer to as the toric semi-Fano building block ( Z, S ) in the following. Such buildingblocks exhibit the required technical properties that the anti-canonical class − K Z isprimitive, that H ( Z, Z ) is torsion-free (because H ( P Σ , Z ) is torsion-free), and that H ( S, Z ) embeds primitively into H ( X, Z ) with X = Z \ S , cf. ref. [27]. G -manifolds To construct explicit examples of twisted connected sum G -manifolds, Corti et al.introduce the method of orthogonal gluing in ref. [24], which is a particular recipe tofulfill the matching condition for suitable pairs of building blocks ( Z L/R , S L/R ).In this work we apply the orthogonal gluing method mainly to Fano building blocksand toric semi-Fano building blocks to algorithmically find novel twisted connectedsum G -manifolds. In order to specify a particular semi-Fano threefold, in the fol-lowing we use the Mori–Mukai classification for Fano threefold varieties [62] and theKasprzyk classification for reflexive polytopes with terminal singularities for certaintoric semi-Fano threefolds [63,64]. We label the corresponding semi-Fano threefolds bytheir respective reference numbers MM ρ or/and K ρ of the Picard lattice of theFano threefold.Applying the method of orthogonal gluing to a pair of building blocks ( Z L/R , S L/R )of two semi-Fano threefolds P L/R , we use the following three-step algorithm [24]: • Construction of the orthogonal pushout lattice W : To achieve the matching con-dition (3.9) first for a pair of polarized K3 surfaces S L/R via orthogonal gluing,choose a negative definite lattice R embedded primitively into both Picard lat-tices N L/R of the polarized K3 surfaces S L/R . Then the lattice W is constructedas W = N L + N R , R = N L ∩ N R , (5.5)such that N ⊥ L ⊂ N R , N ⊥ R ⊂ N L , (5.6)32ith the orthogonal lattices N ⊥ L/R defined in eq. (4.2). The lattice W is calledthe orthogonal pushout of N L/R with respect to R and is also denoted by [24] W = N L ⊥ R N R . (5.7)Note that the pushout lattice W is unique but in general need not exist becausethe non-degenerate lattice pairing (cid:104)· , ·(cid:105) W : W × W → Z induced from the pairings (cid:104)· , ·(cid:105) N L/R : N L/R × N L/R → Z is not necessarily well-defined. That is to say, theinduced pairing (cid:104) e L + e R , f L + f R (cid:105) W = (cid:104) e L , f L (cid:105) N L + (cid:104) e R , f R (cid:105) N R (5.8)must be integral for any pair of lattice points ( e L + e R , f L + f R ) in W , which can berepresented — not necessarily uniquely — by e L/R , f L/R ∈ N L/R . Furthermore,we require that the intersections W L/R = N L/R ∩ N ⊥ R/L with the (generic) K¨ahlercones K ( P L/R ) of (the deformation families of) P L/R are non-empty, i.e., K ( P L/R ) ∩ W L/R (cid:54) = ∅ . (5.9) • Primitive embedding of pushout lattice W into K3 lattice L : The matchingcondition (3.9) for a suitable pair of polarized K3 surfaces S L/R in their mod-uli spaces is achieved if we embed primitively the pushout lattice W into theK3 lattice L . The existence of such an embedding can often be deduced fromresults by Nikulin [65]. In particular, such a primitive embedding is guaranteedto exist if the following rank condition is fulfilled [24]rk N L + rk N R ≤ . (5.10) • Lift matching condition of K3 surfaces S L/R to building blocks ( Z L/R , S L/R ) : Finally, we must ensure that the matching condition (3.9) for the polarizedK3 surfaces S L/R can be achieved within the moduli space of the building blocks( Z L/R , S L/R ). Corti et al. show in Proposition 6.18 of ref. [24] that the imposedassumptions on the orthogonal pushout W — namely that ( Z L/R , S L/R ) arebuilding blocks of semi-Fano threefolds, that the lattice R is negative definite,that the intersections (5.9) are non-empty, and that W embeds primitively intothe K3 lattice L — are sufficient to ensure that the matching conditions of thepolarized K3 surfaces S L/R can indeed be lifted to the moduli spaces of thebuilding blocks ( Z L/R , S L/R ).Let us now determine the cohomology groups (4.1) of the G -manifolds Y obtainedfrom orthogonal gluing. First we observe that rk N L ∩ T R and rk N R ∩ T L equal rk W L and rk W R , respectively, while N L + N R becomes the orthogonal pushout W withrk W = rk W L + rk W R + rk R . Therefore, we readily deduce for the Betti numbers33 ( Y ) = dim H ( Y ) and b ( Y ) = dim H ( Y ) for the G -manifolds Y or the orthogonalgluing type b ( Y ) = rk R + dim k L + dim k R ,b ( Y ) = b ( Z L ) + b ( Z R ) + dim k L + dim k R − rk R + 23 . (5.11)Here b ( Z L/R ) are the three-form Betti numbers of the threefolds Z L/R and dim k L/R are the dimensions of the kernels k L/R defined below eq. (4.1).Recall that in the Kovalev limit the kernels k L/R describe the N = 2 gauge theorysectors, respectively, whereas the rank of the intersection lattice R in the orthogonalpushout W coincides with the rank of the gauge group of the N = 4 gauge theorysector, cf. Table 4.1. A particular simple choice of orthogonal gluing is achieved ifthe intersection lattice R has rank zero, i.e., N L ∩ N R = { } . This special case oforthogonal gluing is referred to as perpendicular gluing with its trivial orthogonalpushout W denoted by [24] W = N L ⊥ N R . (5.12)As a consequence, the N = 4 gauge theory sector in the Kovalev limit is absent if thetwisted connected G -manifold is obtained via perpendicular gluing. G -manifolds from orthogonal gluing In this section we study concrete examples of twisted connected sum G -manifoldsobtained via orthogonal gluing. In particular, we focus on examples with non-trivialintersection lattices R in the orthogonal pushout W .As explained in Section 4.1, in the Kovalev limit such examples yield N = 4 gaugetheory sectors with the Abelian gauge group U (1) rk R . Enhancement to a non-Abeliangauge group G of rank r would occur if the intersection lattice R = N L ∩ N R had asublattice G ( − 1) or rank r with the pairing given by minus the Cartan matrix of theLie algebra of G [66]. Then we could blow-down the mutual r rational curves of bothpolarized K3 surfaces S L/R because — by the definition of the intersection lattice R — G ( − 1) resides in the intersection of both Picard lattices N L/R . In this way wewould arrive at singular polarized K3 surfaces S L/R resulting in the enhanced N = 4gauge theory sector with non-Abelian gauge group G × U (1) rk R − r . However, usingthe method of orthogonal gluing, such a gauge theory enhancement is not possiblebecause the orthogonal complement to R in the polarized K3 surfaces S L/R is requiredto contain an ample class, which — due to the ampleness — would always have anon-zero intersection with any rational curve. It would nevertheless be interesting tosee if non-Abelian gauge groups are possible in the N = 4 gauge theory sector bygeneralizing the orthogonal gluing construction. Orthogonal gluing of rank two semi-Fano threefolds: Let us consider orthog-onal gluings among building blocks of semi-Fano threefolds with Picard number two.34he only non-trivial orthogonal gluings among such building blocks — that is to sayapart from perpendicular gluings — have an intersection lattice R of rank one. Inref. [56] Crowley and Nordstr¨om classify, for rank two Fano threefold building blocks,all possible non-trivial orthogonal gluings to twisted connected sum G -manifolds —except for one missing pair. As a warm-up we want to extend this classificationby including all building blocks arising from toric semi-Fano threefolds with Picardnumber two.It turns out that there is actually a unique toric semi-Fano threefold P Σ of Picardnumber two that is not Fano and is given by the projective bundle [27] P ( O ⊕ O ( − ⊕ O ( − → P . (5.13)Its toric realization of P Σ arises from the reflexive lattice polytope ∆ and its dualreflexive polytope ∆ ∗ spanned by the lattice points ν , . . . , ν and the dual latticepoints ν ∗ , . . . , ν ∗ given by:∆ : ν = ( − , − , 0) ∆ ∗ : ν ∗ = ( − , − , ν = ( 1 , , ν ∗ = ( − , , − ν = ( 0 , , ν ∗ = ( − , , ν = ( 1 , , ν ∗ = ( 2 , − , − ν = ( 0 , , − ν ∗ = ( 2 , − , 1) (5.14)The reflexive lattice polytope ∆ appears as number K32 in the Kasprzyk classifi-cation [63, 64]. It admits two simplicial triangulations both realizing the projectivebundle (5.13). The toric variety P Σ associated to the fan Σ of one of these trian-gulations gives rise to the Mori cone spanned by the curves C B (cid:39) P and the curve C F (cid:39) P ⊂ P F in a projective fiber P F , such that these curves have the followingintersection numbers with the toric divisors D i associated to the vertices ν i : D D D D D C F : 1 1 1 0 0 C B : 0 − − D ∼ D , D ∼ D , D ∼ D − D and the K¨ahler cone K ( P Σ ) spanned by K ( P Σ ) = (cid:104)(cid:104) D , D (cid:105)(cid:105) . (5.16) In Table 2 and Table 4 of ref. [56] the authors list all rank two Fano building blocks and theresulting G -manifolds, respectively. However, the classification of G -manifolds in Table 4 missesthe orthogonal gluing between the building blocks MM5 and MM25 of Table 2. Therefore, Theorem5.10 of ref. [56] should enumerate nineteen instead of eighteen pairs of twisted connected sum G -manifolds. For the other triangulation the Mori cone takes the form: D D D D D C F : 1 0 0 1 1 C B : 0 1 1 − − − K P Σ = (cid:80) i D i ∼ D and − K P Σ = 54. The intersectionmatrix κ P Σ of the generators D and D with the anti-canonical class − K P Σ reads κ P Σ = (cid:18) (cid:19) , (5.17)and has discriminant ∆ κ = − 9. The intersection matrix κ P Σ furnishes the intersectionpairing of the Picard lattice of the anti-canonical K3 surface in P Σ .According to the described algorithm in Section 5.2, for a pair of Picard lattices( N L , N R ) of rank two to yield a non-trivial orthogonal pushout W , the rank onesublattices W L/R must be generated by ample classes in K ( P L/R ) with the intersectionlattice R orthogonal to both W L/R . Thus, we need to construct two ample classes A L/R together with orthogonal lattice vectors e L/R in N L/R with e L = e R , which generatethe rank one intersection lattice R . Then — as Crowley and Nordstr¨om show inref. [56] — the induced lattice pairing (cid:104)· , ·(cid:105) W is a well-defined integral lattice pairing,if and only if ∆ κL ∆ κR A L A R = k , for k ∈ Z , (5.18)in terms of the discriminants ∆ κL/R and the ample classes A L/R . Moreover, in orderto fulfill this matching condition with a rank two Fano threefold P R/L , Crowley andNordstr¨om deduce an upper bound for the rank two semi-Fano threefold P L/R [56] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∆ κL/R A L/R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . (5.19)To find a matching rank two Fano building block for non-trivial orthogonal gluingwith the unique rank two toric semi-Fano threefold (5.13), we first choose an ampleclass A , which according to eq. (5.16) is given by A = nD + mD with A = 6 n ( n + m ).Thus, in order to conform with the inequality (5.19) for ∆ κ = − 9, the only possibleample class is A = D + D with A = 12. For this class the orthogonal complement R is generated by e = − D + 3 D with e = − 12. Table 5.1 summarizes the data ofthis toric semi-Fano threefold together with the corresponding data for the buildingblocks of rank two Fano threefolds with compatible rank one intersection latticesgenerated by vectors of length square − 12. The latter entries are taken from theCrowley–Nordstr¨om classification [56].For the entries in Table 5.1, condition (5.18) tells us the two possible gluings withrank one intersection lattices, namely W K32MM25 = N K32 ⊥ e N MM25 : b ( Y K32MM25 ) = 1 , b ( Y K32MM25 ) = 114 ,W MM5 MM25 = N MM5 ⊥ e N MM25 : b ( Y MM5 MM25 ) = 1 , b ( Y MM5 MM25 ) = 84 . (5.20) We would like to thank Johannes Nordstr¨om for pointing out a numerical error in the third Bettinumbers in eq. (5.20) in an earlier version of this work. − K κ ∆ κ A e A e b ( Z )K32 (semi-Fano) 54 (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − 12 56MM5 (Fano) 12 (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − 12 26MM25 (Fano) 32 (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − 12 36Table 5.1: The rows of the table list the data of the unique rank two ρ = 2 toric semi-Fano threefolds together with all rank two Fano threefolds that admit an intersectionlattice R generated by a vector of length square − 12. The columns show the referencenumbers MM ρ in the Mori–Mukai [62] classification or K − K , the intersectionmatrix κ of the Picard lattice of the anti-canonical K3 surface, the discriminant ∆ κ of the intersection matrix κ , the chosen ample class A in the basis of the intersectionmatrix κ , the orthogonal complement e to the ample class A , the length squares ofthe classes A and e , and the three-form Betti number b ( Z ) of the associated buildingblock Z .Here Y ······ denotes the twisted connected G -manifolds obtained from the orthogonalpushout W ······ with their Betti numbers computed by eq. (5.11). The first G -manifoldrealizes the only possible combination with the rank two toric semi-Fano threefold K32,whereas the second G -manifold realizes the non-trivial orthogonal gluing among therank two Fano threefolds that have been overseen in ref. [56]. Orthogonal gluing of higher rank semi-Fano threefolds: As our next illus-trating examples, we consider orthogonal gluings along a rank one intersection latticewith the rank three Fano threefold P L = P × P × P , which has the reference numbersMM27 and K62 in the Mori–Mukai and Kasprzyk classifications, respectively [62–64].Let H i , i = 1 , , 3, be the hyperplane classes of the respective P -factors of this Fanothreefold, which generate the three-dimensional K¨ahler cone K ( P Σ ) = (cid:104)(cid:104) H , H , H (cid:105)(cid:105) . (5.21)The ample anti-canonical class becomes − K P L = 2 H +2 H +2 H , and the intersectionmatrix κ P L of the K¨ahler cone generators with the anti-canonical divisor − K P L reads κ P L = . (5.22)Now, we focus on orthogonal gluing with the rank one intersection lattice generatedby a vector e of length square e = − 4. Note that this length square realizes the37aximal negative value, as the pairing κ P L is even and vectors with length square e = − A , which is in violation with the orthogonal gluing assumption (5.9).The intersection pairing (5.22) corresponds to the ternary quadratic form q ( x, y, z ) =4( xy + yz + zx ), which allows us to parametrize, with the help of ref. [67], — up totrivial permutations of the K¨ahler cone generators H , H , H — all vectors e with e = − e = ( d − k ) H + ( d − k ) H + kH , (5.23)where the integers k, d , d obey k − d d = 1 , ≤ d < k ≤ d . (5.24)In order to fulfill condition (5.9), we need to check that the orthogonal complementcontains an ample class A = a H + a H + a H given in terms of positive integers a , a , a , i.e., 0 = A.e = a d + a d + a ( d + d − k ) . (5.25)As the sum of the first two terms are always positive, this orthogonality condition canonly be met if d + d < k ⇔ ( d + d ) < k ⇔ ( d − d ) < , (5.26)where we used the relations (5.24), which furthermore implies that d = d + 1 andhence k = 1 , d = 0 , d = 1, corresponding to the vector e = − H + H . (5.27)For this vector e , the ample class A = H + H + H is indeed orthogonal.Therefore, for a left building block ( Z L , S L ) of the Fano threefold P × P × P with a rank one intersection lattice R generated by a vector e with e = − 4, — upto trivial relabelling of the K¨ahler cone generators — the vector (5.27) is the onlypossibility. Its orthogonal complement W L is then generated by W L = Z w + Z w with w = H + H , w = H , (5.28)such that the Picard lattice N L in terms of ( w , w , e ) reads N L = Z w + Z w + Z e + 12 Z ( w + e ) . (5.29)So as to orthogonally glue this left Picard lattice N L along e with a Picard lat-tice N R of a right rank two Fano building block ( Z R , S R ), we find in the Crowley–Nordstr¨om classification [56] that the rank two Fano threefolds with Mori–Mukaireference numbers MM6 , MM12 , MM21 , and MM32 give rise to compatible inter-section lattices. For convenience, these particular building blocks together with some38o. − K κ ∆ κ A e A e b ( Z )MM6 (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − (cid:18) (cid:19) − (cid:18) (cid:19) (cid:18) − (cid:19) − R generated by a vector of length square − 4. The columns showthe reference number in the Mori–Mukai classification, the triple intersection of theanti-canonical divisor − K , the intersection matrix κ of the Picard lattice of the anti-canonical K3 surface, the discriminant ∆ κ of the intersection matrix κ , the chosenample class A in the basis of the intersection matrix κ , the orthogonal complement e to the ample class A , the length squares of the classes A and e , and the three-formBetti number b ( Z ) of the associated building block Z .geometric data are summarized in Table 5.2, and we readily see that for all theseexamples the rank two Picard lattices N R are generated in the orthogonal basis ( A, e )by N R = Z A + Z e + 12 Z ( A + e ) , (5.30)with e = − A as listed in Table 5.2.Next we can construct the orthogonal pushout W = N L ⊥ e N R , which in the basis( w , w , e, A ) takes the form W = Z w + Z w + Z e + Z A + 12 Z ( w + e ) + 12 Z ( A + e ) . (5.31)This orthogonal pushout is well-defined since the potentially non-integral intersectionpairing (cid:10) ( w + e ) , ( A + e ) (cid:11) W = − ( w + e ) , w , ( A + e ) , e ), the intersection pairing κ W of the pushout W becomes κ W = − − 22 0 0 0 − A − − − − − , det κ W = 4 A , (5.32)where — according to Table 5.2 — the entry A − W of the rank three Fano threefold P × P × P and the rank two Fano threefolds listed in Table 5.2,the twisted connected sum G -manifolds Y MM27 ··· W MM27 MM6 = N MM27 ⊥ e N MM6 : b ( Y MM27 MM6 ) = 1 , b ( Y MM27 MM6 ) = 104 ,W MM27 MM12 = N MM27 ⊥ e N MM12 : b ( Y MM27 MM12 ) = 1 , b ( Y MM27 MM12 ) = 100 ,W MM27 MM21 = N MM27 ⊥ e N MM21 : b ( Y MM27 MM21 ) = 1 , b ( Y MM27 MM21 ) = 102 ,W MM27 MM32 = N MM27 ⊥ e N MM32 : b ( Y MM27 MM32 ) = 1 , b ( Y MM27 MM32 ) = 122 . (5.33)Analogously, we can construct twisted connected sum G -manifolds via orthogonalgluing along rank one intersection lattices for semi-Fano threefolds with higher rankPicard lattices. In Table 5.3 we collect all (resolved) toric terminal Fano threefoldsof Picard rank three and four that allow for a rank one intersection lattice generatedby a vector e of length square e = − 4. The geometries of these threefolds areagain specified by their reference number MM ρ and/or K G -manifolds Y ······ obtained from orthogonal gluing along the rank one intersectionlattice R all have the two-form Betti number b ( Y ······ ) = 1 and their three-form Bettinumbers b ( Y ······ ) are listed in Table 5.4. These Betti numbers are easily calculatedwith relations (5.11). 40 o. rk N − K κ e e b ( Z )MM27 , K62 (Fano) 3 48 − − , K68 (Fano) 3 44 − − − , K105 (Fano) 3 52 − − − − − , K218 (Fano) 4 46 − − − , K266 (Fano) 4 42 − − − − − − − − − − − − − − − − − − − − − − Table 5.3: The rows of the table list the data of all rank three and four (resolved)toric terminal Fano threefolds for an intersection lattice R generated by a vector oflength square − 4. The columns show the reference number in the Mori–Mukai [62]and/or Kasprzyk [64] classification, the rank of the Picard lattice —note that in thesemi-Fano cases this rank as reported in [64] is smaller since it refers to the singularvariety— the triple intersection of the anti-canonical divisor − K , the intersectionmatrix κ of the Picard lattice of the anti-canonical K3 surface, the generator e ofthe lattice R and its length square, and the three-form Betti number b ( Z ) of theassociated building block Z . 41 ( Y ······ ) MM27 MM25 MM31 K124 MM12 MM10 K221 K232 K233 K247 K257MM27 122 118 126 122 120 116 112 114 112 118 120MM25 118 114 122 118 116 112 108 110 108 114 116MM31 126 122 130 126 124 120 116 118 116 122 124K124 122 118 126 122 120 116 112 114 112 118 120MM12 120 116 124 120 118 114 110 112 110 116 118MM10 116 112 120 116 114 110 106 108 106 112 114K221 112 108 116 112 110 106 102 104 102 108 110K232 114 110 118 114 112 108 104 106 104 110 112K233 112 108 116 112 110 106 102 104 102 108 110K247 118 114 122 118 116 112 108 110 108 114 116K257 120 116 124 120 118 114 110 112 110 116 118 Table 5.4: This table shows the three-form Betti numbers b ( Y ······ ) of the twistedconnected sum G -manifolds Y ······ arising from the orthogonal pushout N ··· ⊥ e N ··· along the rank one intersection lattice with e = − b ( Y ······ ) = 1. The referencenumbers MM ρ or K Orthogonal gluing along rank two intersection lattice: A systematic analysisof orthogonal pushouts for higher rank intersection lattices R is beyond the scopeof this work. Instead, we present a particular example with a rank two intersectionlattice R with two orthogonal generators e and e both of length square − 4. Certainlywe do not expect that orthogonality and the maximal negative value for the lengthsquares are necessary conditions to find a higher rank example. However, imposingthese two conditions certainly simplifies the construction of a matching pair.Our example is based upon gluing a pair of building blocks ( Z L/R , S L/R ) bothobtained from the rank five Fano threefold P L/R = P × dP , where dP denotesthe del Pezzo surface of degree six, which is the blow-up of P along three non-collinear points p , p , p . This rank five Fano threefold has the Mori–Mukai referencenumber MM3 and — as it is toric — the Kasprzky reference number K324.First, we collect some basic properties of the del Pezzo surface dP . Let E , E , E be the three exceptional divisors from the blow-ups at the points p , p , p , andlet H be the proper transform of the hyperplane class of P . These divisors span thePicard lattice of dP and their intersection numbers read (cid:104) E i , E j (cid:105) dP = − δ ij , (cid:104) H, H (cid:105) dP = 1 , (cid:104) H, E i (cid:105) dP = 0 . (5.34)The ample anti-canonical divisor reads − K dP = 3 H − E − E − E .Let us further define the two divisors e = E + E + E − H , e = E − E , (5.35)42hich are both differences of rational curves on dP . The most important point isthat the defined divisors e , e of length square − − K dP in the K¨ahler cone K ( dP ), i.e., (cid:104) e , e (cid:105) dP = (cid:104) e , K dP (cid:105) dP = (cid:104) e , K dP (cid:105) dP = 0 , (cid:104) e , e (cid:105) dP = (cid:104) e , e (cid:105) dP = − . (5.36)Now we return to the rank five Fano threefold P × dP . With the hyperplanedivisor h of P and the described divisors of dP , the anti-canonical divisor becomes − K P × dP = 2 h − K dP = 2 h + 3 H − E − E − E , (5.37)Furthermore, the Picard lattice N of the polarized K3 surface S on P × dP isgenerated by the divisors h, H, E , E , E together with the intersection pairing (cid:104) h, h (cid:105) N = 0 , (cid:104) h, D (cid:105) N = −(cid:104) K dP , D (cid:105) dP , (cid:104) D, F (cid:105) N = 2 (cid:104) D, F (cid:105) dP , (5.38)where D and F are some divisors on dP .For the orthogonal pushout we generate the rank two lattice R with the twodel Pezzo divisors e and e as R = Z e + Z e , (cid:104) e i , e j (cid:105) N = − δ ij , (5.39)where eqs. (5.36) and (5.38) determine the intersection pairing on R . Moreover, theorthogonal complement W of R becomes W = Z w + Z w + Z w with w = h − K dP , w = H − E , w = h , (5.40)where, in particular, the ample generator w is in the K¨ahler cone K ( P × dP ). As aresult for the rank five Picard lattice N of the polarized K3 surface S in P × dP wearrive with ( w , w , w , e , e ) at N = ( Z w + Z w + Z w )+( Z e + Z e )+ 12 ( Z ( w + e ) + Z ( w + w + e )) . (5.41)Now taking the decomposition (5.41) of the Picard lattice for both the left and theright Picard lattice, i.e., N L = N R = N , we consider the orthogonal pushout W = N L ⊥ R N R , which in the basis ( w L , w L , w L , w R , w R , w R , e , e ) takes the form W = (cid:0) Z w L + Z w L + Z w L (cid:1) + (cid:0) Z w R + Z w R + Z w R (cid:1) + ( Z e + Z e ) + 12 (cid:0) Z ( w L + e ) + Z ( w R + e ) (cid:1) + 12 (cid:0) Z ( w L + w L + e ) + Z ( w R + w R + e ) (cid:1) . (5.42)43his orthogonal pushout is well-defined because the potentially non-integral intersec-tions (cid:104) ( w L + e ) , ( w R + e ) (cid:105) = (cid:104) ( w L + w L + e ) , ( w R + w R + e ) (cid:105) W = − (cid:104) ( w L + e ) , ( w R + w R + e ) (cid:105) W = (cid:104) ( w R + e ) , ( w L + w L + e ) (cid:105) W = 0 are integral.As a result we obtain, from this orthogonal pushout along the rank two lattice R ,the twisted connected G -manifold Y MM3 MM3 with the Betti numbers W MM3 MM3 = N MM3 ⊥ R N MM3 : b ( Y MM3 MM3 ) = 2 , b ( Y MM3 MM3 ) = 97 . (5.43)Here we use that b ( Z L/R ) = (cid:104) K P × dP , K P × dP (cid:105) N + 2 = 6 (cid:104) K dP , K dP (cid:105) dP + 2 = 38because b ( P × dP ) = 0, cf. ref. [62]. N = 2 gauge sectors on twisted connected sums Compared to the Abelian N = 4 gauge theory sectors studied in the previous sec-tion, the structure of the N = 2 gauge theory sectors turns out to be much richer.The building blocks ( Z L/R , S L/R ) of the twisted connected sum G -manifolds admitenhancement to N = 2 non-Abelian gauge theory sectors with an interesting branchstructure that is geometrically accessible in terms of extremal transitions in the asymp-totically cylindrical Calabi–Yau threefolds X L/R .In order to see what kind of features we can expect by degenerating the buildingblocks ( Z L/R , S L/R ), we recall from ref. [33] that there is a simple hierarchy of realcodimension four, six and seven singularities in G -manifolds, which respectively leadto non-Abelian gauge groups, non-trivial matter representations, and chirality of thecharged N = 1 matter spectrum. While our setup admits non-Abelian gauge groupswith non-trivial matter representations, we should not expect singularities inducingchirality as the trivial S fibration in the non-compact seven-manifolds Y L/R preventsthe appearance of codimension seven singularities.The picture proposed in ref. [33] uses the heterotic/M-theory duality [66], theStrominger–Yau–Zaslow fibration of Calabi–Yau manifolds [68], and the fact that G -manifolds can be locally constructed as degenerating S fibrations over Calabi-Yau threefolds [36, 69], where the S can be identified in a hyper K¨ahler quotientconstruction starting in eight dimensions [37, 33]. Namely, consider the heteroticstring compactification on the Calabi–Yau threefold W . We further assume that thethreefold W admits a geometric mirror threefold such that it has a Strominger–Yau–Zaslow Lagrangian T fibration over a (real) three-dimensional Lagrangian cycle Q . In the best known examples — such as hypersurfaces in toric varieties — it has thetopology of a three-sphere. In the limit where the volume of the base Q is large Further suggestions on the role of mirror symmetry in the context of G -manifolds have beenproposed in refs. [70, 71]. It is possible to obtain a Lens space for the Lagrangian base Q , for instance by dividing out afreely acting finite group on a suitable Calabi–Yau threefold, see e.g., ref. [72]. T , the essential idea is now toadiabatically extend the duality between the heterotic string on T and M-theoryon K Q . The M-theory geometries defined in this way realizethe same fibration structure (4.6) as appearing in the twisted connected sum G -manifolds. Whenever the heterotic string has a non-Abelian ADE type gauge group G , the dual K3 fibers develop the corresponding ADE singularity extending over theentire real three-dimensional base Q . The proposed construction can be viewed as an N = 1 version of the N = 2 heterotic/type II duality between the heterotic stringon K × T and type IIA string on the dual K3 fibered Calabi–Yau threefolds, asproposed in refs. [73, 74]. In the context of twisted connected sum G -manifolds apossibility to arrive at non-Abelian ADE type gauge theories has been contemplatedin ref. [35].While the K3 fibration described in ref. [33] complies with the K3 fibration (4.6)of the twisted connected sum construction, we should stress that the non-Abeliangauge theory enhancement encountered in this work arises from singularities along athree-cycle S × C , where the curve C of genus g resides in K3 fibers along a circle S in the base Q . Thus, compared to ref. [33] the non-Abelian gauge group stillemerges from a real codimension four singularity, however, along different types ofthree-cycles. In the Kovalev limit, the three-cycle S × C resides in one of the seven-manifold Y L/R = X L/R × S L/R such that the curve C realizes an ADE singularityin one of the asymptotically cylindrical Calabi–Yau threefolds X L/R . Therefore, thenon-Abelian gauge theory enhancement discussed here directly relates to non-Abeliangauge groups from curves of ADE singularities in Calabi–Yau threefolds in the contextof type IIA strings [40, 41]. Specifically, in this setting an ADE singularity along acurve C yields a four-dimensional N = 2 gauge theory with the associated gauge group G together with g hypermultiplets in the adjoint representation. More general matterrepresentations occur at points along C where the ADE singularity further enhances,i.e., along real codimension six singularities. For instance, at the intersection point oftwo curves C and C (cid:48) of ADE singularities we encounter matter in the bi-fundamentalrepresentation of the two associated gauge groups G and G (cid:48) [43]. In the following wefind that the described N = 2 gauge theory spectra can indeed be realized within the N = 2 gauge theory sectors of the building blocks ( Z L/R , S L/R ). Remarkably, eventhe phase structure of the four-dimensional N = 2 gauge theory sectors — connectingtopologically distinct Calabi–Yau threefolds via extremal transitions — carries overto the four-dimensional N = 1 M-theory compactifications on twisted connected sum G -manifolds, where now the gauge theory branches relate topologically distinct G -manifolds.The picture presented in refs. [32, 66] finally explains chirality of non-Abelianmatter as a local effect that occurs in codimension seven. Since the twisted connectedsum breaks supersymmetry from N = 2 to N = 1 non-locally via the twisted gluingrecipe, the last step — namely to construct chiral charged matter — is more subtle45o achieve. We discuss some ideas in the conclusions. N = 2 Abelian gauge theory sectors Let us now focus on twisted connected sum G -manifolds with non-trivial N = 2gauge theory sectors in the Kovalev limit. According to Table 4.1 this amounts toconstructing building blocks with ( Z L/R , S L/R ) with non-trivial kernels k L/R as definedbelow eq. (4.1). This can be achieved with the proposal by Kovalev and Lee [75], gen-eralizing the construction of asymptotically cylindrical Calabi–Yau threefolds outlinedin Section 5.1. In a particular example, the possibility to realize N = 2 Abelian gaugetheory enhancement appeared in ref. [25].For the semi-Fano threefold P we pick again two global sections s and s of theanti-canonical divisor − K P . However, instead of choosing a generic section s , weassume that the global section s factors into a product s = s , · · · s ,n , (6.1)such that s ,i are global sections of line bundles L i with − K P = L ⊗ . . . ⊗ L n . As aconsequence, the curve C sing = { s = 0 } ∩ { s = 0 } becomes reducible and decomposesinto C sing = n (cid:88) i =1 C i , C i = { s ,i = 0 } ∩ { s = 0 } , (6.2)where we assume that the individual curves C i are smooth and reduced. FollowingKovalev and Lee [75], we construct the building block ( Z (cid:93) , S ) associated to P by thesequence of blow-ups π {C ,..., C n } : Z (cid:93) → P along the individual curves C i according to Z (cid:93) = Bl {C ,..., C n } P = Bl C n Bl C n − · · · Bl C P . (6.3)Since the curves C i and the semi-Fano threefold P are smooth, the blow-up Z (cid:93) issmooth as well. As before, the K3 surface S arises as the proper transform of asmooth anti-canonical divisor S (cid:93) = { α s + α s = 0 } ⊂ P for some [ α : α ] ∈ P .By blowing up a semi-Fano threefold P , the resulting dimension of the kernel k —defined below eq. (4.1) — is then given by [27]dim k = n − . (6.4)Furthermore, the three-form Betti number b ( Z (cid:93) ) of the blown-up threefold Z (cid:93) be-comes b ( Z (cid:93) ) = b ( P ) + 2 n (cid:88) i =1 g ( C i ) , (6.5)in terms of the three-form Betti number b ( P ) of the semi-Fano threefold P and thegenera g ( C i ) of the smooth curve components C i . As all these curves C i lie in the K346ultiplicity N = 2 multiplets N = 1 multiplets U (1) n − charges multiplet U (1) n − charges multiplet n − , , . . . , 0) vector (0 , . . . , 0) vector(0 , . . . , 0) chiral χ ij (0 , . . . , +1 i , . . . , +1 j , . . . , hyper (0 , . . . , +1 i , . . . , +1 j , . . . , chiral ≤ i < j < n (0 , . . . , − i , . . . , − j , . . . , chiral χ in (0 , . . . , +1 i , . . . , 0) hyper (0 , . . . , +1 i , . . . , 0) chiral1 ≤ i < n (0 , . . . , − i , . . . , 0) chiralTable 6.1: The table shows the spectrum of the Abelian N = 2 gauge theory sectorarising from the conifold singularities in the building block ( Z sing , S ). Listed are thefour-dimensional N = 2 multiplets and their decomposition into the four-dimensional N = 1 multiplets together with their multiplicities χ ij . The subscripts of the entriesof the U (1) charges indicate their position in the charged vector.fiber S , the genus g ( C i ) is readily computed by the adjunction formula g ( C i ) = 12 C i . C i + 1 , (6.6)with the self-intersections C i . C i in S .Corti et al. show in refs. [24, 27] that, with such a building block ( Z (cid:93) , S (cid:93) ), theorthogonal gluing recipe of Section 5.2 can still be carried out in the same way. Inparticular, the lift from a matching pair of K3 surfaces S L/R to their asymptoticallycylindrical Calabi–Yau threefolds X L/R still exists if such generalized building blocksare involved in the orthogonal gluing procedure.Thus, we observe that, from a single semi-Fano threefold P , often several build-ing blocks can be constructed depending on the properties of the curve C = { s =0 } ∩ { s = 0 } . Namely, for smooth irreducible curves C (cid:91) , we obtain a smooth buildingblock ( Z (cid:91) , S (cid:91) ) with vanishing kernel k , whereas for reducible curves C sing with smoothcomponents C i we arrive, with the sequence of blow-ups (6.3), at a smooth build-ing block ( Z (cid:93) , S (cid:93) ) with non-vanishing kernel k . The former building block does notcontribute with any vector multiplets to the N = 2 gauge theory sector, while thelatter building block contributes with Abelian vector multiplets to the N = 2 gaugetheory sector. In the sequel we argue that these different possibilities realize distinctbranches of the N = 2 gauge theory sectors.To arrive at this gauge theory interpretation, let us consider a semi-Fano threefold P with a curve C sing of the reducible type (6.2) with the factorized global anti-canonicalsection (6.1). Performing a blow-up along this reducible curve yields the fibration47 : Z sing → P with Z sing = Bl C sing P = (cid:8) ( x, z ) ∈ P × P (cid:12)(cid:12) z s , · · · s ,n + z s = 0 (cid:9) . (6.7)In the vicinity of the fiber π − ([1 , Z sing becomes singular because inthe patch of the affine coordinate t = z z we get s , · · · s ,n + ts = 0 . (6.8)Thus — assuming transverse intersections among the smooth curves C i — there areconifold singularities at the discrete intersection loci I ij = { t = 0 }∩{ s = 0 }∩{ s ,i =0 } ∩ { s ,j = 0 } for 1 ≤ i ≤ j ≤ n with χ ij = |I ij | intersection points. These numbersare given by χ ij = C i . C j , (6.9)in terms of the intersection numbers of the reduced curves C i and C j within the K3surface S . This singularity structure prevails in the asymptotically cylindrical Calabi–Yau threefold X sing = Z sing \ S since the asymptotic fiber S = π − ([ α , α ]) (for α (cid:54) = 0)is disjoint from the singular fiber π − ([1 , π − ([1 , ⊂ X sing , we interpret the dimen-sional reduction of M-theory on the local seven-dimensional singular space S × X sing as the dimensional reduction of type IIA string theory on the asymptotically cylin-drical Calabi–Yau threefold X sing , where the S factor corresponds to the M-theorycircle of type IIA string theory. In this IIA picture, refs. [39, 38] establish that theconifold singularities (6.8) yield an Abelian N = 2 gauge theory with charged mattermultiplets. Namely, to each curve C i we assign an Abelian group factor U (1) i suchthat the total Abelian gauge group of rank n − U (1) n − (cid:39) U (1) × . . . × U (1) n U (1) Diag , (6.10)where U (1) Diag is the diagonal subgroup of U (1) × . . . × U (1) n . Thus, in the low-energyeffective theory, we obtain ( n − 1) four-dimensional N = 2 U (1) vector multiplets,which decomposes into ( n − 1) four-dimensional N = 1 U (1) vector multiplets and( n − 1) four-dimensional N = 1 neutral chiral multiplets. Furthermore, to eachintersection point in I ij one assigns a four-dimensional N = 2 hypermultiplet of charge(+1 , +1) with respect to the U (1) i × U (1) j group factor. Then each of these N = 2hypermultiplet of charge (+1 , +1) decomposes into two four-dimensional N = 1 chiralmultiplets of charge (+1 , +1) and ( − , − Starting from the Fano threefold P , it has also been proposed in ref. [25] that the singularbuilding block ( Z sing , S ) with conifold singularities realizes an Abelian gauge theory with chargedmatter. T × X sing to three space-time dimensions. Then the conifold points in X sing become genus one curves of conifold singularities. This analysis has been carriedout in ref. [76], and one arrives at three-dimensional N = 4 gauge theory sectors,which agree with the four-dimensional N = 2 spectrum in Table 6.1 upon further di-mensional reduction on a circle S . This further justifies that the local dimensionalreduction of type IIA theory on X sing correctly describes the gauge theory of M-theoryon S × X sing without requiring that the S factor realizes the M-theory circle for thedual type IIA description.The described four-dimensional N = 2 Abelian gauge theory now predicts a Higgsbranch H (cid:91) and a Coulomb branch C (cid:93) . On the one hand, generic non-vanishing expec-tation values of the charged hypermultiplets break the U (1) n − gauge theory entirelyand parametrize the Higgs branch H (cid:91) of the gauge theory. As a consequence ( n − N = 2 hypermultiplets play the role of N = 2 Goldstone multiplets thatcombine with the ( n − 1) short massless N = 2 vector multiplets to ( n − 1) long mas-sive N = 2 vector multiplets. As a result — according to the spectrum in Table 6.1— we arrive at the Higgs branch H (cid:91) of complex dimension h (cid:91) [38] h (cid:91) = dim C H (cid:91) = 2 (cid:32) (cid:88) ≤ i 1) + δ (2) R , b ( Y (cid:93) ) = b ( P ) + (cid:32) n (cid:88) i =1 C i . C i (cid:33) + 3 n + 22 + δ (3) R , (6.16)with the contributions from the right building block ( Z R , S R ) δ (2) R = dim k R + rk R , δ (3) R = b ( Z R ) + dim k R − rk R . (6.17)The relations allow us to further define what we call the reduced Betti numbers b (cid:91)(cid:96) and b (cid:93)(cid:96) , (cid:96) = 1 , 2, given by b (cid:91)(cid:96) = b (cid:96) ( Y (cid:91) ) − δ ( (cid:96) ) R , b (cid:93)(cid:96) = b (cid:96) ( Y (cid:93) ) − δ ( (cid:96) ) R , (cid:96) = 1 , . (6.18)Using the equivalence C (cid:91) ∼ C + . . . + C n on the semi-Fano threefold P and thedefinition (6.9) of the multiplicities χ ij , we finally arrive at b ( Y (cid:91) ) = b ( Y (cid:93) ) − ( n − ,b ( Y (cid:91) ) = b ( Y (cid:93) ) + 2 (cid:32) (cid:88) ≤ i 1) of the gaugegroup. Furthermore, the difference in the four-dimensional N = 1 chiral multipletsagrees with the change in dimension of the moduli space of these gauge theory phases,i.e., b ( Y (cid:91) ) − b ( Y (cid:93) ) = b (cid:91) − b (cid:93) = h (cid:91) − c (cid:93) . (6.20)In a similar fashion, it is straightforward to establish the correspondence betweenthe gauge theory and the geometry for mixed Higgs–Coulomb branches, where thegauge group U (1) n − is broken to a subgroup U (1) k − with 1 < k < n . The geometriesof such mixed phases are obtained by partially resolving and partially deforming theconifold singularities in the asymptotically cylindrical Calabi–Yau threefold X sing . Weillustrate the analysis of mixed Higgs–Coulomb branches with an explicit example inSection 6.3. N = 2 non-Abelian gauge theory sectors Let us now turn to the enhancement to non-Abelian N = 2 gauge theory sectorsin the context of twisted connected sum G -manifolds, indicated as a possibility inref. [35]. Let us assume that the anti-canonical line bundle − K P of the semi-Fanothreefold P factors as − K P = ˜ L ⊗ k ⊗ . . . ⊗ ˜ L ⊗ k s s with n = k + . . . + k s , (6.21)where ˜ L i are line bundles with global sections ˜ s ,i . Then the global section s of − K P can further degenerate to s = ˜ s k , · · · ˜ s k s ,s and the singular building block (6.7) reads Z sing = (cid:8) ( x, z ) ∈ P × P (cid:12)(cid:12) z ˜ s k , · · · ˜ s k s ,s + z s = 0 (cid:9) , (6.22)with the singular equation in the affine coordinate t = z z given by˜ s k , · · · ˜ s k s ,s + ts = 0 . (6.23)As before, we assume that all curves ˜ C i = { ˜ s ,i = 0 } ∩ { s = 0 } are smooth. In thevicinity of the singular fiber π − ([1 , ⊂ Z sing , the singular building block ( Z sing , S )develops A k i − -singularities along those curves ˜ C i with k i > S × X sing in terms of its dual type IIA picture on the asymptoticallycylindrical Calabi-Yau threefold X sing . Refs. [40, 41] establish that type IIA string51ultiplicity N = 2 multiplets N = 1 multiplets G reps. multiplet G reps. multiplet s − U (1) vector U (1) vector chiral i = 1 , . . . , s adj SU ( k i ) SU ( k i ) vector adj SU ( k i ) SU ( k i ) vector adj SU ( k i ) chiral g ( ˜ C i ) adj SU ( k i ) hyper adj SU ( k i ) chiral ≤ i ≤ s adj SU ( k i ) chiral˜ χ ij ( k i , k j ) (+1 i , +1 j ) hyper ( k i , k j ) (+1 i , +1 j ) chiral ≤ i < j < s ( ¯k i , ¯k j ) ( − i , − j ) chiral˜ χ is ( k i , k s ) (+1 i ) hyper ( k i , k s ) (+1 i ) chiral1 ≤ i < s ( ¯k i , ¯k s ) ( − i ) chiralTable 6.2: The table shows the spectrum of the N = 2 gauge theory sector with gaugegroup G = SU ( k ) × . . . × SU ( k s ) × U (1) s − as arising from the non-Abelian buildingblocks ( Z sing , S ). It lists both the four-dimensional N = 2 and the four-dimensional N = 1 multiplet structure. The adjoint matter is determined by the genus g ( ˜ C i ) ofthe curves ˜ C i , whereas the bi-fundamental matter is determined by their intersectionnumbers ˜ χ ij within the K3 surface S .theory on Calabi–Yau threefolds with a genus g curve of A k − singularities devel-ops a N = 2 SU ( k ) gauge theory with g four-dimensional N = 2 hypermultipletsin the adjoint representation of SU ( N ). Furthermore, according to ref. [43], eachintersection point of two such curves of A k − and A k − singularities contributes afour-dimensional N = 2 hypermultiplet in the bi-fundamental representation ( k , k )of SU ( k ) × SU ( k ).Therefore — putting all these ingredients together and including the U (1) gaugetheory factors of the previously discussed Abelian gauge theory sectors — we proposethe following non-Abelian gauge theory description for M-theory on the local singularseven space S × X sing . Firstly, the singularities along the curves ˜ C i determine thegauge group G = SU ( k ) × . . . × SU ( k s ) × U (1) s − (cid:39) U ( k ) × . . . × U ( k s ) U (1) Diag , (6.24)where any SU (1) factors must be dropped out and U (1) Diag is the diagonal subgroup of U ( k ) × . . . × U ( k s ). Secondly, for any i with k i > 0, there are g ( ˜ C i ) four-dimensional52 = 2 hypermultiplets in the adjoint representation of SU ( k i ). Thirdly, we have˜ χ ij four-dimensional N = 2 hypermultiplets in the bi-fundamental representation( k i , k j ) (+1 i , +1 j ) of the gauge group factors SU ( k i ) × SU ( k j ), where the subscriptsindicate the U (1)-charges with respect to the diagonal U (1) i and U (1) j subgroups ofthe respective unitary groups U ( k i ) and U ( k j ) in the relation (6.24). The multiplicities˜ χ ij are again determined by the intersection numbers of the curves ˜ C i and ˜ C j in theK3 fiber S . The resulting gauge theory spectrum is summarized in Table 6.2.From the described spectrum and the results of ref. [41], we are now ready toanalyze the branches of the N = 2 gauge theory sectors. First, we determine thecomplex dimension h (cid:91) of the Higgs branch h (cid:91) = dim C H (cid:91) = 2 (cid:32) s (cid:88) i =1 ( g ( ˜ C i ) − k i − (cid:33) +2 (cid:32) (cid:88) ≤ i 1) complex degrees of freedom of the four-dimensional N = 2 hypermultiplets in the corresponding adjoint representations ofthe SU ( k i ) gauge group factors — reduced by one adjoint N = 2 Goldstone hyper-multiplet rendering the four-dimensional N = 2 SU ( k i ) vector multiplet massive. Thesecond term realizes the complex degrees of freedom of the four-dimensional N = 2matter hypermultiplets in the bi-fundamental representations of the associated specialunitary gauge groups and charged with respect to the appropriate U (1) factors. Thelast term subtracts from the second term the N = 2 Goldstone hypermultiplets forhiggsing the ( s − 1) four-dimensional N = 2 U (1) vector multiplets.Next, we derive the complex dimension of the Coulomb branch C (cid:91) , in which themaximal Abelian subgroup U (1) n − remains unbroken. It is parametrized by theexpectation value of all four-dimensional N = 2 hypermultiplet components that areneutral with respect to this unbroken maximal Abelian subgroup. Therefore, thecomplex dimension c (cid:93) of the Coulomb branch becomes c (cid:93) = dim C C (cid:93) = 2 (cid:32) s (cid:88) i =1 g ( ˜ C i )( k i − (cid:33) + ( n − . (6.26)The first term counts the traceless neutral diagonal degrees of freedom of the four-dimensional N = 2 matter hypermultiplets in the adjoint representation, while thesecond term adds the contributions of the complex scalar fields in the four-dimensionalunbroken Abelian N = 2 vector multiplets.The next task is to compute the Betti numbers of the twisted connected sum G -manifolds Y (cid:91) and Y (cid:93) , which geometrically realize the Higgs and Coulomb branchby orthogonal gluing of the building blocks ( Z (cid:91) , S (cid:91) ) and ( Z (cid:93) , S (cid:93) ) to a common rightbuilding block ( Z R , S R ). We construct the building block ( Z (cid:91) , S (cid:91) ) by blowing-up thesemi-Fano threefold P along the smooth irreducible curve C (cid:91) , which — as in theHiggs branch of the Abelian gauge theories — arises from a generic deformation of53he section s of the anti-canonical line bundle − K P . Then, relations (6.15) determineagain the two-form and three-form Betti numbers of the G -manifold Y (cid:91) . The smoothCoulomb branch building block ( Z (cid:93) , S (cid:93) ) results from the sequence of n = k + . . . + k s blow-ups Z (cid:93) = Bl { ˜ C k ,..., ˜ C kss } P , (6.27)where each individual curve ˜ C i is resolved k i times such that dim k (cid:93) = n − 1. Therefore,using eqs. (5.11), (6.5) and (6.6), we arrive at the Betti numbers for the smooth G -manifold Y (cid:93) b ( Y (cid:93) ) = ( n − 1) + δ (2) R , b ( Y (cid:93) ) = b ( P ) + (cid:32) s (cid:88) i =1 k i ˜ C i . ˜ C i (cid:33) + 3 n + 22 + δ (3) R , (6.28)with the definitions (6.17). Using the equivalence relation C (cid:91) ∼ k ˜ C + . . . + k s ˜ C s , wecalculate the change of Betti numbers b ( Y (cid:91) ) = b ( Y (cid:93) ) − ( n − ,b ( Y (cid:91) ) = b ( Y (cid:93) ) + (cid:32) s (cid:88) i =1 ˜ χ ii k i ( k i − (cid:33) + 2 (cid:32) (cid:88) ≤ i 1) (6.29)in terms of the intersection numbers ˜ χ ij = ˜ C i . ˜ C j on the K3 surface S .As for the Abelian gauge theory sectors, the computed change of Betti numbersis also in accord with the phase structure of the proposed non-Abelian gauge theorydescription. Namely, the change of the two-form Betti number conforms with thedifference of the four-dimensional N = 1 vector multiplets in the Higgs and Coulombbranches, given by the rank of the non-Abelian gauge group (6.24). The differenceof four-dimensional N = 1 chiral multiplets is accurately predicted by the complexdimensions of the Higgs and Coulomb branches. That is to say that, with eqs. (6.6),(6.25) and (6.26), we find for the discussed non-Abelian gauge theories b ( Y (cid:91) ) − b ( Y (cid:93) ) = b (cid:91) − b (cid:93) = dim C H (cid:91) − dim C C (cid:93) . (6.30)As for the previously discussed Abelian gauge theories, the established correspon-dence between G -manifolds and non-Abelian Higgs and Coulomb branches carriesover to mixed Higgs–Coulomb branches as well, which we illustrate with an explicitexample in Section 6.3. The fact that the performed analysis of the non-Abeliangauge theory sectors closely parallels the study of the Abelian gauge theories doesnot come as a surprise, because the Abelian gauge group (6.10) arises from partiallyhiggsing the non-Abelian gauge group (6.24) to its maximal Abelian subgroup. As aresult, the topological data of the G -manifolds for the Higgs, Coulomb and mixedHiggs–Coulomb phases resulting from a given semi-Fano threefold P are the same forboth the discussed Abelian and non-Abelian gauge theory sectors.54 .3 Examples of G -manifolds with N = 2 gauge theories Following the general discussion of N = 2 gauge theory sectors in Section 6.1 andSection 6.2, we now illustrate the emergence of N = 2 gauge theory sectors in twistedconnected sum G -manifolds with a few explicit examples: SU (4) gauge theory with adjoint matter from the Fano threefold P : Con-sider the Fano threefold P with the anti-canonical divisor − K P = 4 H in terms of thehyperplane class H . Let ˜ s , and s be a (generic) global section of H and − K P , re-spectively. Then we obtain, with eq. (6.22), the resolved building block Z sing ⊂ P × P as the hypersurface equation ˜ s , + ts = 0 , (6.31)with the affine coordinate t of the factor P . This equation exhibits an A singularityalong the curve ˜ C = { ˜ s , = 0 }∩{ s = 0 }∩{ t = 0 } , which yields a N = 2 gauge theorysector with gauge group SU (4). Note that for this particular example the deformedphases of the non-enhanced N = 2 Abelian gauge theory sector are discussed as wellin ref. [25].We first note that the curves C ( k ) = ( − K P ) ∩ ( kH ) have the following intersectionnumbers on the K3 surface S and — according to eq. (6.6) — genera C ( k ) . C ( l ) = 4 kl , g ( C ( k ) ) = 12 C ( k ) . C ( k ) + 1 = 2 k + 1 . (6.32)Due to the equivalence ˜ C ∼ C ( k ) , we arrive at g ( ˜ C ) = 3 four-dimensional N = 2hypermultiplets in the adjoint representation of SU (4). This spectrum predicts witheqs. (6.25) and (6.24) the dimensions of the Higgs and Coulomb branchesdim C H (cid:91) = 60 , dim C C (cid:93) = 21 , dim C H (cid:91) − dim C C (cid:93) = 39 . (6.33)As proposed in Section 6.2, by sequentially blowing-up P four times along the curve ˜ C ,we arrive at the building block ( Z (cid:93) , S (cid:93) ) withdim k (cid:93) = 3 , b ( Z (cid:93) ) = 4 · g ( ˜ C ) = 24 . (6.34)Deforming the hypersurface equation (6.31) to s + ts = 0 with a generic section of − K P , we resolve along the reduced smooth curve C (cid:91) ⊂ P with C (cid:91) ∼ C (4) in order todetermine the building block ( Z (cid:91) , S (cid:91) ) of the Higgs branch H (cid:91) withdim k (cid:91) = 0 , b ( Z (cid:91) ) = 2 g ( C (cid:91) ) = 66 . (6.35)Finally, orthogonally gluing the building blocks ( Z (cid:91) , S (cid:91) ) and ( Z (cid:93) , S (cid:93) ) to a suitableright building block ( Z R , S R ), we obtain, with eq. (5.11), the twisted connected sum G -manifolds Y (cid:91) and Y (cid:93) with the reduced Betti numbers b (cid:91) = 0 , b (cid:91) = 89 ,b (cid:93) = 3 , b (cid:93) = 50 , (6.36)55 factors Gauge Group N = 2 Hypermultiplet spectrum h (cid:91) c (cid:93) b (cid:91) b (cid:93) k (cid:93) SU (4) 3 × adj 60 21 89 50 31 · SU (3) × U (1) 3 × adj ; 4 × +1 54 15 89 50 31 · SU (2) × U (1) 3 × ( adj , ); 3 × ( , adj ); 4 × ( , ) +1 54 15 89 50 31 · · SU (2) × U (1) × adj ; 4 × (+1 , +1) ; 4 × (+1 , ; 4 × (0 , +1) 48 9 89 50 31 · · · U (1) × (+1 , +1 , × (+1 , , +1); 4 × (0 , +1 , +1); 42 3 89 50 34 × (+1 , , × (0 , +1 , × (0 , , +1)2 · SU (2) × U (1) 3 × adj ; 8 × +1 42 8 89 55 22 · · U (1) × (+1 , +1); 8 × (+1 , × (0 , +1) 36 2 89 55 22 SU (2) 9 × adj 48 19 89 60 12 · U (1) 16 × (+1) 30 1 89 60 13 · U (1) 12 × (+1) 22 1 89 68 1 Table 6.3: Depicted in this table are the gauge theory branches of the SU (4) gauge the-ory of the building blocks associated to the rank one Fano threefold P . The columnslist the factorization of the anti-canonical section s with degrees and multiplicities,the gauge group of the gauge theory branch, the matter spectrum of N = 2 hypermul-tiplets with their non-Abelian representations together with the Abelian U (1) charges,the complex dimensions h (cid:91) and c (cid:93) of the Higgs and Coulomb branches, the reducedthree-form Betti numbers b (cid:91) and b (cid:93) of the twisted connected sum G -manifolds Y (cid:91) and Y (cid:93) , and the kernel k (cid:93) of the Coulomb phase building block ( Z (cid:93) , S (cid:93) ).which we defined in eq. (6.18). We observe that the differences b (cid:93) − b (cid:91) = 3 and b (cid:91) − b (cid:93) =39 agree with the rank of the gauge group and the change in the dimensionality of theHiggs and Coulomb branches, respectively, which is in accord with the anticipatedgauge theory description established in Section 6.2.By partially deforming the first term ˜ s , in the hypersurface equation (6.31),we can realize hypersurface singularities describing various Abelian and non-Abeliansubgroups of SU (4). Such partial deformations geometrically realize mixed Higgs–Coulomb branches of the SU (4) gauge theory. We collect the geometry and phasestructure of these mixed Higgs–Coulomb branches in Table 6.3, where the entriesof this table are determined by eqs. (6.4), (6.5), (6.25), (6.26), and (6.32). Notethat — depending on the breaking pattern of SU (4) arising from partially higgsing— the dimensions of Higgs and Coulomb branches vary because only the chargedmatter spectrum of the unbroken gauge group plays a role for the Higgs and Coulombbranches in this gauge theory sector. For all entries in Table 6.3 we find that b (cid:91) − b (cid:93) = h (cid:91) − c (cid:93) , dim k (cid:93) = rk G . (6.37)This agreement confirms nicely the correspondence between gauge theory branches56 factors Gauge Group N = 2 Hypermultiplet spectrum h (cid:91) c (cid:93) b (cid:91) b (cid:93) k (cid:93) (1 , (0 , SU (2) × SU (2) × U (1) 2 × ( adj , ); 2 × ( , adj ); 4 × ( , ) +1 42 11 50 19 3(1 , (0 , , SU (2) × U (1) × adj ; 4 × (1 , ; 4 × (0 , ; 2 × (1 , 38 7 50 19 3(1 , , U (1) × (1 , , × (1 , , × (0 , , · (0 , , 1) 4 × (1 , , × (0 , , × (0 , , , , SU (2) × U (1) 2 × adj ; 8 × +1 36 6 50 20 2(2 , , , U (1) × (1 , × (0 , × (1 , 1) 32 2 50 20 2(1 , SU (2) 7 × adj 36 15 50 29 1(1 , , U (1) 12 × (+1) 22 1 50 29 1(2 , , U (1) 16 × (+1) 30 1 50 21 1(2 , , U (1) 10 × (+1) 18 1 50 33 1 Table 6.4: The table shows the branches of the SU (2) × SU (2) × U (1) gauge theoryassociated to the Fano threefold W with Mori–Mukai label MM48 [62]. Listed are thefactors of the anti-canonical section s with bi-degrees and multiplicities, the unbrokengauge subgroup, the N = 2 matter hypermultiplets, the complex dimensions h (cid:91) and c (cid:93) of the Higgs and Coulomb branches, the reduced three-form Betti numbers b (cid:91) and b (cid:93) of the twisted connected G -manifolds Y (cid:91) and Y (cid:93) , and the kernel k (cid:93) of the Coulombphase.and phases of twisted connected sum G -manifolds. SU (2) × SU (2) × U (1) gauge theory from the Fano threefold MM48 : The ranktwo Fano threefold W with reference number MM48 is a hypersurface of bidegree(1 , 1) in P × P with b ( W ) = 0 [62]. Let H and H be the hyperplane classes of P × P . Then, by adjunction, the anti-canonical divisor of W reads − K W = 2 H + 2 H .Furthermore, the self-intersection numbers of the curves C ( k,l ) = ( − K W ) ∩ ( kH + lH )in the anti-canonical divisor − K W and hence their genera are C ( k ,l ) . C ( k ,l ) = 2( k k + l l +2 k l +2 l k ) , g ( C ( k,l ) ) = k + l +4 kl +1 . (6.38)With generic global sections ˜ s , , ˜ s , and s of H , H and − K W , the equation forthe singular building block Z sing ⊂ W × P becomes, with the affine coordinate t of P , ˜ s , ˜ s , + ts = 0 . (6.39)Thus, we find A singularities along the two curves ˜ C i = { ˜ s ,i = 0 } ∩ { s = 0 } ∩ { t =0 } with i = 1 , 2. Thus — following the general discussion in Section 6.2 — wefind a SU (2) × SU (2) × U (1) gauge theory both with adjoint matter and with bi-fundamental matter from the intersection points ˜ C ∩ ˜ C . Due to the equivalences57 C ∼ C (1 , and ˜ C ∼ C (0 , and relations (6.38), we arrive at the four-dimensional N = 2 hypermultiplet matter spectrum2 × ( adj , ) ; 2 × ( , adj ) ; 4 × ( , ) +1 . (6.40)The resulting correspondence between the gauge theory branches and the phase struc-ture of the twisted connected sum G -manifold is summarized in Table 6.4, where theentries are computed with the formulas (6.4), (6.5), (6.25), (6.26), and (6.38). Further examples from toric semi-Fano threefolds: Our last class of examplesconcerns N = 2 gauge theory sectors from toric semi-Fano threefolds P Σ , wherethe fan Σ is obtained from a triangulation of a three-dimensional reflexive latticepolytope ∆. In this toric setup, the anti-canonical divisor reads − K P Σ = D + . . . + D n , (6.41)where the toric divisors D i correspond to the one-dimensional cones of Σ, that is to sayto the rays of the lattice polytope ∆. For smooth toric varieties P Σ , the toric divisors D i are smooth and intersect transversely [59]. As the anti-canonical divisor − K P isbase point free, we can apply Bertini’s theorem — see, e.g., ref. [77] — to argue thatwe can find a smooth global section s of the anti-canonical divisor − K P Σ and furthergeneric global sections s ,i of D i such that the curves C i = { s ,i = 0 } ∩ { s = 0 } aresmooth and mutually intersect transversely. Hence, the toric semi-Fano threefold P Σ realizes indeed a U (1) n − gauge theory sector. The four-dimensional matter spectrumis then given by Table 6.1, where the multiplicities χ ij are the toric triple intersectionnumbers χ ij = − K P Σ .D i .D j . (6.42)As proposed in Section 6.1, we construct the building blocks ( Z (cid:93) , S (cid:93) ) of the Coulombbranch C (cid:93) by the sequential blow-ups (6.3) along the curves C i , while we determinethe building block ( Z (cid:91) , S (cid:91) ) of the Higgs branch H (cid:91) by blowing a smooth curve C (cid:91) = { s = 0 } ∩ { s = 0 } obtained by deforming the singular section s , · · · s ,n to a genericanti-canonical section s . Then we arrive at the twisted connected sum G -manifolds Y (cid:93) and Y (cid:91) by orthogonally gluing these gauge theory building blocks with a rightbuilding block ( Z R , S R ) in the usual way. Note that, due to linear equivalences among the toric divisors D i , the Abeliangauge theory can enhance to non-Abelian gauge groups as well. Namely, assume thatthe anti-canonical bundle − K P Σ is linearly equivalent to − K P Σ ∼ k ˜ D + . . . + k s ˜ D s , (6.43) For toric semi-Fano threefolds P Σ , some of the performed blow-ups discussed here and in thefollowing can also be described with toric geometry techniques [26]. However, such a toric descriptionis not advantageous to extract the relevant geometric data for us. D α ∼ (cid:80) i a αi D i with global sections ˜ s ,α . Furthermore, werequire that the curves ˜ C α are smooth and mutually intersect transversely. Then,following Section 6.2, we arrive at the N = 2 gauge theory sector with gauge group G = SU ( k ) × . . . × SU ( k s ) × U (1) s − . (6.44)Note that rank of the gauge group ˜ n = k + . . . + k s − n of toric divisors. Instead, it depends on the precise nature of thelinear equivalences among the toric divisors D i , i = 1 , . . . , n , and the divisors ˜ D α , α = 1 , . . . , s .Let us exemplify the study of four-dimensional N = 2 gauge theory sectors withthe rank two toric semi-Fano threefold P Σ of reference number K32 described in somedetail in Section 5.3. Using for this example the linear equivalences among the toricdivisors D , . . . , D — cf. below eq. (5.14) — we find for the anti-canonical divisor − K P Σ = D + . . . + D ∼ D ∼ D + 3 D . (6.45)With these linearly equivalent representations for − K P Σ , we arrive, for instance, at thegauge groups U (1) of rank four, SU (3) of rank three, or SU (3) × SU (3) × U (1) of rankfive. Note that the phases of the lower rank gauge groups U (1) and SU (3) enjoy againthe interpretation as mixed Higgs–Coulomb branches of the SU (3) × SU (3) × U (1)gauge theory of rank five which, by applying eq. (6.42) and eq. (6.6), yields thespectrum 1 × ( adj , ) ; 1 × ( , adj ) ; 3 × ( , ) +1 . (6.46)In Table 6.5 we list the gauge theory sectors of a few toric semi-Fano threefolds P Σ .This table does not display all mixed Higgs–Coulomb branches. Here, we focus on theresulting twisted connected sum G -manifolds Y (cid:91) and Y (cid:93) associated to the Higgs H (cid:91) and Coulomb branches C (cid:93) of the maximally enhanced gauge group of maximal rank,as obtained by the factorization of the anti-canonical bundle − K P Σ .59 o. ρ Gauge Group N = 2 Hypermultiplet spectrum h (cid:91) c (cid:93) b (cid:91) b (cid:93) k (cid:93) K24, 2 SU (3) × SU (2) 2 × ( adj , ); ( , adj ); 3 × ( , ) +1 50 14 79 43 4MM34 × U (1)K32 2 SU (3) × U (1) ( adj , ); ( , adj ); 3 × ( , ) +1 52 13 79 40 5K35, 2 SU (5) × SU (2) 2 × ( adj , ); ( , ) +1 60 22 87 49 6MM36 × U (1)K36, 2 SU (4) × SU (2) 2 × ( adj , ); 2 × ( , ) +1 54 17 81 44 5MM35 × U (1)K37, 2 SU (4) × SU (3) ( adj , ); 3 × ( , ) +1 54 12 79 37 6MM33 × U (1)K62, 3 SU (2) × U (1) ( adj , ); ( , adj , ); ( , adj ); 2 × ( , ) (1 , 44 11 73 40 5MM27 × ( , , ) (1 , ; 2 × ( , ) (0 , K68, 3 SU (3) × SU (2) ( adj , ); 3 × ( , ) (1 , ; 2 × ( , ) (1 , ; ( , ) (0 , 42 9 69 36 6MM25 × U (1) K105, 3 SU (3) × SU (2) ( adj , ); ( , adj , ); 2 × ( , ) (1 , ; ( , , ) (1 , ; 50 15 77 42 7MM31 × U (1) ( , , ) (0 , K124 3 SU (4) × SU (2) ( adj , , × ( , , ) (1 , ; 2 × ( , , ) (1 , 48 13 73 38 7 × U (1) K218, 4 SU (4) × SU (3) ( adj , ); ( , , ) (1 , , ; ( , , , ) (1 , , ; 46 16 71 41 10MM12 × SU (2) × U (1) ( , , ) (1 , , ; ( , , , ) (0 , , ; ( , , , ) (0 , , K266, 4 SU (3) × SU (2) ( , adj , ); ( , , ) (1 , , ; 2 × ( , , , ) (1 , , ; 42 10 67 35 8MM10 × U (1) × ( , , ) (1 , , ; ( , , ) (0 , , ; ( , , , ) (0 , , K221 4 SU (3) × SU (2) × ( , , ) (1 , , ; 3 × ( , , ) (1 , , ; ( , ) (1 , , 40 7 63 30 7 × U (1) × ( , , ) (0 , , K232 4 SU (4) × SU (2) × ( , , ) (1 , , ; 2 × ( , , , ) (1 , , ; 42 9 65 32 9 × U (1) × ( , , ) (1 , , K233 4 SU (3) × SU (2) × ( , , ) (1 , ; 3 × ( , , ) (1 , 40 6 63 29 6 × U (1) K247 4 SU (4) × SU (3) × ( , , ) (1 , , ; 2 × ( , , , ) (1 , , ; 46 11 69 34 11 × SU (2) × U (1) ( , , , ) (0 , , ; ( , , ) (0 , , K257 4 SU (5) × SU (3) × ( , , ) (1 , , ; 2 × ( , , , ) (1 , , ; 48 12 71 35 12 × SU (2) × U (1) ( , , ) (1 , , Table 6.5: The table exhibits the N = 2 gauge theory sectors for some smooth toricsemi-Fano threefolds P Σ of Picard rank two and higher. The columns display thenumber of the threefold P Σ in the Mori–Mukai [62] and/or Kasprzyk [64] classification,its Picard rank ρ , the maximally enhanced gauge group of maximal rank by factorizingthe anti-canonical bundle, the N = 2 matter hypermultiplets, the complex dimensions h (cid:91) and c (cid:93) of the Higgs and Coulomb branches, the reduced three-form Betti numbers b (cid:91) and b (cid:93) , and the kernel k (cid:93) of the Coulomb branch.60 .4 Transitions among twisted connected sum G -manifolds The proposed correspondence between phases of twisted connected sum G -manifoldsand gauge theory branches of the described N = 2 gauge theory sectors is essentiallybased upon the correspondence between extremal transitions in the asymptoticallycylindrically Calabi–Yau threefolds X L/R and the Higgs–Coulomb phase structure ofthe associated N = 2 gauge theories. In the original type IIA string theory settingthe N = 2 matter spectrum arises from solitons of massless D2-branes wrapping thevanishing cycles of the singular Calabi–Yau threefolds at the transition point [39, 38],which become membranes in the discussed context of M-theory. However, while inthe type IIA setting these D2-branes furnish BPS states of the N = 2 algebra, thecorresponding interpretation of membrane states becomes more subtle in the con-text of M-theory on twisted connected sum G -manifolds because the correspondingmembrane states do not admit a BPS interpretation due to minimal four-dimensional N = 1 supersymmetry. Therefore, a natural question now is whether the describedM-theory transitions are actually dynamically realized.As discussed in Section 2, the semi-classical moduli space M C of M-theory on G -manifolds has the geometric moduli space M of Ricci-flat G -manifolds as a realsubspace. From the low-energy effective N = 1 supergravity point of view, this is aconsequence of the semi-classical shift symmetries with respect to the real parts of thechiral fields (2.30). However, due to arguments about the absence of global continuoussymmetries in consistent theories of gravity, see e.g., ref. [78], these shift symmetriesshould be broken non-perturbatively such that the flat directions of the chiral modulifields are lifted. In the context of M-theory on G -manifolds, membrane instantonson suitable three-cycles generate non-perturbative superpotential terms that breakthese continuous shift symmetries [22]. As these non-perturbative corrections are ex-ponentially suppressed in the volume of the wrapped three-cycles, the flat directions— as described by the semi-classical moduli space M C — are expected to be onlyrealized in the large volume limit of the G -compactification. Hence, M-theory transi-tions among G -manifolds should only occur in the absence of such non-perturbativeeffects, as for instance in the case of the large volume limit. If we now take both the large volume limit and the Kovalev limit simultaneously,gravity decouples, and we arrive at a genuine four-dimensional N = 2 gauge theorysector with eight unbroken supercharges. Then the lower energy dynamics is indeeddescribed as in refs. [39,38,40,41], and the gauge theory phases connect asymptoticallycylindrical Calabi–Yau threefolds via extremal transitions. Thus, we claim that, inthe large volume and in the large Kovalev limit, the transitions among the N = 2gauge theory sectors geometrically realize the anticipated transitions among twistedconnected sum G -manifolds. In the presence of small non-perturbative obstructions we can still have quantum-mechanicaltransitions among four-dimensional vacua. Then the transition probability is governed by the tun-neling rate through the barrier of the non-perturbative scalar potential. 61f we maintain the large volume limit but allow for finite Kovalevton, the situationbecomes more subtle. While the massless spectrum is still N = 2, we expect that theappearance of further interaction terms breaks N = 2 supersymmetry to N = 1. Thenthe N = 2 gauge theory sector is partially broken to a N = 1 gauge theory, whosesupersymmetry breaking couplings are governed by the scale of the Kovalevton. Inthis N = 1 language, the transition between non-Abelian N = 2 Higgs and Coulombbranches essentially describes an enhancement to an Abelian gauge symmetry withinthe N = 1 Higgs branches. Namely, in the N = 1 language, the N = 2 Coulombphase corresponds to the partially higgsing of the non-Abelian group to its maximalAbelian subgroup. Thus, at low energies, the proposed (non-Abelian) N = 2 Higgs–Coulomb phase transition describes the Higgs mechanism of a weakly-coupled Abelian N = 1 gauge theory. These observations provide for some evidence that, in the largevolume limit, the anticipated phase structure among the described twisted connectedsum G -manifolds is still realized — even for finite Kovalevton.Geometrically, we therefore propose that in the M-theory moduli space M C thepresented transitions among twisted connected sum G -manifolds are indeed unob-structed. That is to say, we conjecture that the construction of orthogonally glu-ing commutes with extremal transitions in the asymptotically cylindrical Calabi–Yauthreefolds X L/R . Furthermore, our proposal implies that the moduli space M ofRicci-flat G -metrics of the twisted connected sum type should exhibit a stratificationstructure as predicted by the phase structure of the analyzed N = 2 gauge theoriessectors. In the context of Abelian gauge theory sectors our proposal conforms with asimilar conjecture put forward in ref. [25]. In this work we have studied the four-dimensional low-energy effective N = 1 super-gravity action arising from M-theory compactified on G -manifolds of the Kovalev’stwisted connected sum type. By suitably gluing a pair of non-compact asymptot-ically cylindrical Calabi–Yau threefolds times a circle Y L/R = X L/R × S in theirasymptotic regions [23], this construction realizes a large class of examples for com-pact G -manifolds [24–26], which are yet of a specific type realizing only a particular(non-trivial) homotopy invariant of G -manifolds [29].From the cohomology of such G -manifolds, we established that this class of M-theory compactifications yields two neutral universal N = 1 chiral moduli fields asso-ciated to the complexified overall volume modulus ν and the gluing modulus — calledthe Kovalevton κ — respectively. The latter parametrizes the Kovalev limit taken byRe( κ ) → ∞ .The proper interpretation of the different contributions in (4.4) to the cohomologyof the G -manifold Y implies that there is a decomposition of the fields of the N = 1effective supergravity theory on Y into N = 1 neutral chiral moduli multiplets, into62wo N = 2 gauge theory sectors coming from the two asymptotic regions Y L/R , andinto one N = 4 gauge sector that comes from the trivial K S inthe gluing region T × S × (0 , N = 1 supergravity action resulting from a compactification ona smooth twisted connected sum G -manifold Y . Moreover, the obtained two scalesdo also control the behavior of M -theory corrections. Let us now list mathemati-cal, physical and eventually even phenomenological prospects of this decomposition,specific to M -theory compactifications on twisted connected sum G -manifolds.A first consequence is that we can identify Abelian and non-Abelian gauge the-ory enhancements with various matter content from singularities in the asymptoticcylindrical Calabi–Yau threefolds X L/R in codimension four and six that occur in thetwisted connected sum Y away from the gluing region. These lead to transitions inthe threefolds X L/R , whose deformations and resolutions can be described by methodsof algebraic geometry familiar in the context of N = 2 theories. The significant pointestablished in Section 6 is that these transitions commute with the Kovalev limit andthe gluing construction. Namely, they connect G -manifolds whose change in the co-homology groups corresponds exactly to the change in the spectrum of N = 1 vectorand chiral superfields as predicted by the transitions. Concretely, starting with theequations that describe the blow-up of the anti-canonical divisor in semi-Fano three-folds and analyzing all their possible degenerations lead to a great variety of gaugegroups and matter spectra as well as to many novel examples of twisted connectedsum G -manifolds corresponding to the different branches of these gauge theories.This suggests that, in a suitably compactified moduli space of the Ricci-flat G -metrics, there are many new types of singular loci through which it is possible toreach topological inequivalent G -manifolds. This question is a priori independentof the possible N = 1 non-perturbative membrane instanton corrections that couldlift the flat directions in the N = 1 scalar potential (which are protected in the pure N = 2 limit). Therefore, by taking the large volume limit and the Kovalev limit, thesedirections certainly remain flat. However, we argued in Section 6.4 that even for finiteexpectation values of the Kovalevton κ , the transitions should remain physical inthe effective N = 1 theory, as long as non-perturbative membrane instantons remainsuppressed.Another interesting physical consequence of the decomposition and the Kovalevlimit is that the more advanced N = 2 techniques — like calculating the exact gaugecoupling and the exact BPS masses from the periods of the holomorphic three-form —are approached in this limit and serve as a zeroth order approximation with inversepower laws or exponential corrections in the Kovalevton κ and the volume modu-63us ν , similarly as the calculations carried out in refs. [36, 69] in the context of local G -manifolds. Those corrections leading to holomorphic terms in the four-dimensional N = 1 effective theory are expected to be accessible by techniques similar to the onesused to calculate four-dimensional N = 1 F-terms in flux and/or brane compactifi-cations of type II theories. Note, however, that these computations would require adetailed study of the relative Calabi–Yau three-form periods on the two non-compactCalabi–Yau threefolds — for instance by using variation of mixed Hodge structuretechniques along the lines of refs. [79–85] — and a moduli-dependent analysis of thematching conditions (3.9). An attractive feature of the twisted connected sum compactification is that in thetwo individual N = 2 gauge theory sectors from X L/R we have algebraic methodsto geometrically engineer gauge groups, spectra and interactions. Already the fewexamples presented in Table 6.5 yield small rank gauge groups such as the standardmodel group and possible grand unification scenarios. The matter contents couldin principle be broken into phenomenologically more suitable massless N = 1 chiralmatter multiplets. In fact, the computed N = 2 spectra can be broken to N = 1 mul-tiplets by various non-local effects. As discussed for finite Kovalevton κ the twistedgluing itself and non-perturbative effects — such as membrane instantons — intro-duce genuine N = 1 interaction terms. Adding a flux-induced superpotential (2.35)offers yet another attractive mechanism to break the N = 2 spectra into N = 1multiplets [8], potentially introducing chirality as well. Due to the absence of tadpoleconstraints for four-form fluxes on G -manifolds, the local scenario for fluxes in type IIstring theories proposed in ref. [86] is readily realized on the level of the non-compactasymptotically cylindrical Calabi–Yau summands X L/R . We expect that non-trivialbackground four-form fluxes provides for a much more intricate and genuine N = 1gauge theory branch structure, similarly as in refs. [76,87]. All these effects come withdifferent scales — partially exponentially suppressed — which exhibit potentially at-tractive hierarchies. A systematic analysis of phenomenologically attractive N = 1 oreven N = 0 interaction terms in the context of M-theory on twisted connected sum G -manifolds is beyond the scope of this work and will be addressed elsewhere.While in type II Calabi–Yau threefold compactifcations we arrive at four-dimen-sional N = 2 effective supergravity theories with two massless gravitinos realizingextended supersymmetry, breaking the N = 2 gravity multiplet down to the N = 1is rather non-trivial, see for instance the discussion in ref. [88]. In our case, however,the obtained four-dimensional supergravity theory has already minimal supersym-metry. It is only the gauge theory sectors in the Kovalev limit that approximatelyexhibit extended global supersymmetries. Therefore, introducing background fluxesto break supersymmetry in the gauge theory sectors is much simpler than in thetype II Calabi–Yau threefold compactifications. In particular, turning on background A similar analysis of moduli dependent matching conditions is required for building blocks ( Z, S )arising from general weak Fano threefold for which Beauville’s theorem [57] is not applicable. X L and X R in Figure 3.1 iscontrolled by the real part of the Kovalevton κ . Together with the local constructionof the spectra on X L/R described in Section 6, this offers the possibility to considera hidden and a visible sector and to employ the mechanism of mediation of super-symmetry breaking only in the gravitational sector with a controllable scale set byKovalevton κ . Or alternatively, as there is an anomaly inflow mechanism in the localtheories [32, 33], one could use the anomaly mediation of supersymmetry breaking asproposed in [89].Finally, we comment on the possible relation of the twisted connected sum con-struction to other non-perturbative descriptions of N = 1 theories. In lower dimen-sions the algebraic-geometrical approach towards the Hoˇrava–Witten setup describesa duality to F-theory [90]. Namely, certain Calabi–Yau compactifications of the het-erotic string are dual to F-theory on elliptically fibered Calabi–Yau fourfolds in aparticular stable degeneration limit [91]. To obtain four-dimensional N = 1 su-pergravity theory this heterotic–F-theory correspondence is realized on the level ofelliptically-fibered Calabi–Yau fourfolds. It is intriguing to observe that such Calabi–Yau fourfolds in the stable degeneration limit are obtained by gluing a pair of suitablychosen Fano fourfolds along their mutual anti-canonical Calabi–Yau threefold divi-sor [92,93]. This construction of Calabi–Yau fourfolds in the stable degeneration limitshows a certain resemblance — yet in one real dimension higher — to twisted con-nected sum G -manifolds in the Kovalev limit. It would be interesting to see if sucha speculation could be made precise, namely establishing a duality between M-theoryon G -manifolds in the Kovalev limit and F-theory on elliptically-fibered Calabi–Yaufourfolds in a certain degeneration limit. Acknowledgements We would like to thank Dominic Joyce, Dave Morrison, Stefan Schreieder, and EricZaslow for discussions and correspondences. Thaisa C. da C. Guio would like tothank financial support from CNPq (Brazil) under grant number 205626/2014-9 andthe Bonn-Cologne Graduate School of Physics and Astronomy (BCGS). Hung-Yu Yehwould like to thank the IMPRS scholarship and great research environment in the MaxPlanck Institute for Mathematics during the period of the PhD study. A Kaluza-Klein reduction of fermionic terms In this appendix we present the Kaluza–Klein reduction of the fermionic terms ob-tained from the compactification of the eleven-dimensional supergravity action (2.22)65n a seven-dimensional G -manifold. We analyze the four-dimensional fermionic spec-trum, and explicitly derive the zero modes of the eleven-dimensional Rarita–Schwingerfield compactified on the G -manifold. We further determine the holomorphic su-perpotential induced by internal four-form fluxes from certain fermionic interactionterms. Compared to the purely bosonic interactions with quadratic dependences onthe superpotential, the fermionic interactions are linear in the superpotential, see forinstance ref. [51]. Thus, analyzing the fermionic interactions, as opposed to the purelybosonic ones, is more tractable for determining the superpotential. A.1 Definitions and useful relations Following the definitions and conventions of ref. [20], we first spell out the properties ofthe representation of the used eleven-, seven- and four-dimensional gamma matrices.The eleven-dimensional gamma matrices are represented by 32-dimensional matrices,which satisfy the usual Clifford algebra { ˆΓ M , ˆΓ N } = 2 g MN , (A.1)with the eleven-dimensional Lorentzian metric g MN . Furthermore, in the chosen 32-dimensional Majorana representation, the gamma matrices obey [20]ˆΓ · · · ˆΓ = I , (A.2)in terms of the 32-dimesional identity matrix I . With the compactification ansatz M , = M , × Y , the eleven-dimensional gamma matrices split into two sets ofcommuting gamma matrices, i.e.,ˆΓ M = (ˆΓ µ , ˆΓ m ) , ˆΓ µ = γ µ ⊗ I , ˆΓ m = γ ⊗ γ m , (A.3)where I is the seven-dimensional identity matrix, γ µ , µ = 0 , , , 3, are the four-dimensional imaginary gamma matrices, γ m , m = 4 , . . . , 10, are purely imaginaryseven-dimensional gamma matrices satisfying γ · · · γ = i . Furthermore, we define γ = ( i/ (cid:15) µνρσ γ µ γ ν γ ρ γ σ as the four-dimensional chirality matrix, which is purelyimaginary and satisfies γ = 1.The four- and seven-dimensional gamma matrices satisfy the Clifford algebra intheir corresponding dimensions { γ µ , γ ν } = 2 η µν , { γ m , γ n } = 2 g mn . (A.4)Here we use the Minkowski metric η µν with signature ( − , +1 , +1 , +1) and g mn de-notes the Riemannian metric of the seven-dimensional compactification space.We define the anti-symmetrized product of eleven-dimensional gamma matrices asˆΓ M ··· M n = ˆΓ [ M · · · ˆΓ M n ] , (A.5)66nd we use the same notation for the anti-symmetrized products of four- and seven-dimensional gamma matrices, i.e., γ µ ··· µ n = γ [ µ · · · γ µ n ] and γ m ··· m n = γ [ m · · · γ m n ] .For the decomposition of ˆΓ MNP into lower-dimensional gamma matrices we arrive atthe useful relationˆΓ MNP = ( γ µνρ ⊗ I ) + ( γ µν − ⊗ γ p ) + ( γ µ − ρ ⊗ γ n ) + ( γ − νρ ⊗ γ m )+ 13 ( γ ρ ⊗ γ mn + γ ν ⊗ γ pm + γ µ ⊗ γ np )+ γ ⊗ γ mnp , (A.6)where the index ‘ − ’ refers to the four-dimensional chirality matrix γ .For the forthcoming zero mode analysis we record here a few useful identities. Firstof all, we record a few useful identities among products of anti-symmetrized gammamatrices, namely γ mnp γ q = γ mnpq + 3 g q [ m γ np ] ,γ mnp γ qr = γ mnpqr + 3 (cid:0) g q [ m γ np ] r − g r [ m γ np ] q (cid:1) + 6 g q [ m γ n g p ] r . (A.7)Furthermore, the G -structure ϕ fulfills the contraction relations ϕ mnp ϕ npq = 6 δ qn , ϕ mnp ϕ pqr = Φ mnqr + δ qm δ rn − δ rm δ qn , (A.8)with the Hodge dual form Φ = ∗ ϕ , and the Fierz identity γ mn η = − iϕ mnp γ p η , (A.9)in terms of the covariantly constant spinor η . Finally, the Levi–Civita connection ∇ ,the exterior derivative d and its adjoint d † fulfill the relations( dA ) n ...n p +1 = ( p + 1) ∇ [ n A n ...n p +1 ] , ( d † A ) n ...n p − = −∇ m A mn ...n p − , (A.10)for any p -form A = p ! A n ...n p dy n ∧ . . . ∧ dy n p . A.2 G -representations and the Rarita–Schwinger G -bundle In this section we present further details for the decomposition of the global section(2.18) of the Rarita–Schwinger bundle T ∗ Y ⊗ SY on G -manifolds, as introduced inSection 2.First of all, the differential p -forms on a manifold with G -structure fall into ir-reducible representations with respect to the structure group G . Specifically, for aseven-manifold with G -structure the spaces of differential p -forms Λ p decompose ac-cording to ref. [7], namelyΛ = Λ , Λ = Λ , Λ = Λ ⊕ Λ , Λ = Λ ⊕ Λ ⊕ Λ , Λ = Λ , Λ = Λ , Λ = Λ ⊕ Λ , Λ = Λ ⊕ Λ ⊕ Λ , (A.11)67here the summands Λ p q are p -forms transforming in the q -dimensional irreduciblerepresentations of the structure group G . As indicated in the arrangement of theform space Λ p q in eq. (A.11), the Hodge star ∗ provides for an isometry betweenΛ p q and Λ − p q . The differential p -form spaces Λ p are isomorphic to each other for p = 1 , . . . , 6. Moreover, Λ and Λ are generated by ϕ and ∗ ϕ , respectively. Fora compact G -manifold Y equipped with a torsion-free G -structure, the de Rhamcohomologies H p ( Y, R ) have a similar decomposition into H p q ( Y, R ) with harmonicrepresentatives, see, e.g., ref. [7], H ( Y, R ) = H ( Y, R ) ⊕ H ( Y, R ) ,H ( Y, R ) = H ( Y, R ) ⊕ H ( Y, R ) ⊕ H ( Y, R ) ,H ( Y, R ) = H ( Y, R ) ⊕ H ( Y, R ) ⊕ H ( Y, R ) ,H ( Y, R ) = H ( Y, R ) ⊕ H ( Y, R ) . (A.12)Notice that H ( Y, R ) = (cid:104)(cid:104) [ ϕ ] (cid:105)(cid:105) and H ( Y, R ) = (cid:104)(cid:104) [ ∗ ϕ ] (cid:105)(cid:105) . Moreover, H p q ( Y, R ) ∼ = H − p q ( Y, R ), which implies for the Betti numbers b p q ( Y ) = b − p q ( Y ) and b ( Y ) = b ( Y ) = 1. If the holonomy group is G and not a subgroup thereof, we furtherhave H p = { } for p = 1 , . . . , G -manifolds, which — dueto the covariantly constant spinor η — becomes reducible, namely T ∗ Y ⊗ SY ∼ = T ∗ Y ⊗ ( T ∗ Y ⊕ R ). This allows us to make the following identification T ∗ Y ⊗ SY ∼ = T ∗ Y ⊗ ( T ∗ Y ⊕ R )= ( T ∗ Y ⊗ T ∗ Y ) ⊕ T ∗ Y = Sym ( T ∗ Y ) ⊕ Λ T ∗ Y ⊕ T ∗ Y , (A.13)where Sym ( T ∗ Y ) is the space of symmetric two-tensors on Y and Λ T ∗ Y is the spaceof two-forms. Furthermore, it is shown in ref. [94] that Sym ( T ∗ Y ) ∼ = Λ ⊕ Λ . Sincethe spaces Λ and Λ are isomorphic to the cotangent bundle Λ , we arrive at T ∗ Y ⊗ SY ∼ = Λ ⊕ Λ ⊕ Λ ⊕ Λ . (A.14)This decomposition of the Rarita–Schwinger G -bundle justifies the ansatz for theglobal Rarita–Schwinger section (2.18) in Section 2. A.3 The massless four-dimensional fermionic spectrum We are now ready to determine the four-dimensional fermionic terms in the dimen-sional reduction of eleven-dimensional supergravity action on G -manifolds. We focuson the four-dimensional fermionic kinetic and mass terms.Let us perform the dimensional reduction of the Rarita–Schwinger kinetic term forthe gravitino ˆΨ, which is given by the third term of the first line in (2.22). Inserting68he expansion (2.16) for the gravitino ˆΨ and relation (A.6) we obtain − κ (cid:90) ∗ i ¯ˆΨ M ˆΓ MNP ˆ ∇ N ˆΨ P = − i κ (cid:90) M , ∗ ¯ ψ µ γ µνρ ∇ ν ψ ∗ ρ (cid:90) Y ∗ ¯ ζζ − i κ (cid:90) M , ∗ ¯ ψ µ γ µ − ρ ψ ∗ ρ (cid:90) Y ∗ ¯ ζγ n ∇ n ζ − i κ (cid:90) M , ∗ 13 ¯ χγ ν ∇ ν χ ∗ (cid:90) Y ∗ ¯ ζ (1) m γ pm ζ (1) p − i κ (cid:90) M , ∗ ¯ χγχ ∗ (cid:90) Y ∗ ¯ ζ (1) m γ mnp ∇ n ζ (1) p + c.c. . (A.15)The resulting terms comprise the kinetic and mass terms for both the four-dimensionalgravitinos ψ µ — the first and second line on the right hand side of eq. (A.15), respec-tively — and the four-dimensional fermions χ — the third and fourth line on theright hand side of eq. (A.15), respectively. It also gives rise to mixed terms between ψ and χ . However, since such mixed terms are not present in standard four-dimensionalsupergravity theories, they have been neglected in our analysis. Now, we turn to the discussion of the massless four-dimensional fermionic spec-trum, which is obtained from the zero modes of the Dirac operator /D = γ n ∇ n andthe Rarita–Schwinger operator /D RS = γ mnp ∇ n , i.e., /Dζ = 0 and /D RS ζ (1) = 0.With the ansatz (2.17) for the section ζ of the spin bundle SY , we arrive at thezero modes equation /Dζ = ( ∇ n a m ) γ n γ m η + ( ∂ n b ) γ n η = ∇ [ n a m ] γ nm η + ( ∇ n a n ) η + ( ∂ n b ) γ n η = 0 , (A.16)which — due to the linear independence of η , γ n η , and γ nm η — yields for the coefficientone-form a ( y ) = a n ( y ) dy n and the function b ( y ) together with eqs. (A.10) da ( y ) = 0 , d † a ( y ) = 0 , db ( y ) = 0 . (A.17)The first two equations imply that a ( y ) must be a harmonic one-form, whereas thelast equation determines the function b ( y ) to be constant. As there are no harmonicone-forms on the G -manifold Y , the covariantly constant spinor η furnishes the onlyzero mode in the spin bundle SY . This zero mode gives rise to the massless four-dimensional gravitino field ψ µ and its conjugate ψ ∗ µ of the four-dimensional massless N = 1 gravity multiplet listed in Table 2.1. Actually, one should perform a redefinition of Ψ µ with Ψ µ → Ψ (cid:48) µ = Ψ µ + ˆΓ µ ˆΓ m Ψ m in order forsuch terms to cancel out. However, we do not consider this field redefinition as such a shift does notaffect the gravitino mass [20, 95]. /D RS on the ansatz (2.18)for ζ (1) and using eqs. (A.7), (A.8) and (A.9), we arrive after a straightforward butsomewhat tedious calculation at /D RS ζ (1) = ( ∇ [ n b m ] ) γ mnp dy p ⊗ η − (cid:0) ∇ n a nm (cid:1) dy m ⊗ η + 32 (cid:0) ∇ [ n a pq ] (cid:1) γ mnpq dy m ⊗ η − i (cid:0) ∇ n a npq (cid:1) dy p ⊗ γ q η + i (cid:0) ∇ [ m a npq ] (cid:1) γ mnpqr dy r ⊗ η − ∂ p (tr g a mn ) ) dy q ⊗ γ pq η , (A.18)in terms of the singlet tr g a mn ) = a nm g nm and the three-form a [ nmp ] a mnp ] = g rs a r [ m ϕ np ] s , a nm ) = 34 a npq ] ϕ pqr g rm − g nm a pqr ] ϕ pqr . (A.19)Let us now analyze the zero modes of the Rarita–Schwinger section ζ (1) fromeq. (A.18). The one-form b ( y ) = b n ( y ) dy n does not contribute any zero modes, be-cause with eq. (A.10) such a zero mode must be a closed one-form db ( y ) = 0. Fur-thermore, due to b ( Y ) = 0 it also must be exact b ( y ) = df ( y ). However, an exactone-form df ( y ) furnishes no physical degrees of freedom as it can be removed by agauge transformation of the Rarita–Schwinger section, i.e., ζ (1) → ζ (1) − ∇ ( f ( y ) ⊗ η ).For the remaining tensors we find that, with the help of eqs. (A.10), the zero modesof the Rarita–Schwinger operator /D RS are given by da ( y ) = 0 , d † a ( y ) = 0 ,da ( y ) = 0 , d † a ( y ) = 0 , (A.20)in terms of the two-form a ( y ) = a nm ( y ) dy n ∧ dy m and the three-form a ( y ) = a nmp ( y ) dy n ∧ dy m ∧ dy p . Thus, the zero modes are in one-to-one correspondence withharmonic two-forms a ( y ) and three-forms a ( y ) on the G -manifolds Y , where theharmonic property of a ( y ) implies that the symmetric tensor a nm ) must be solutionsto the Lichnerowicz Laplacian as well, cf. eq. (2.10). Altogether, we can thereforededuce from the cohomology of the G -manifold Y the fermionic zero modes listed inTable 2.1. A.4 The flux-induced holomorphic superpotential Let us now determine the holomorphic superpotential generated by a cohomologicallynon-trivial four-form background flux G on the G -manifold Y , which is locally givenby d ˆ C . The superpotential can be read off from the four-dimensional gravitino mass70erm (2.34). Such a term arises from the dimensional reduction of the fourth term inthe eleven-dimensional action (2.22). That is to say, we find − κ (cid:90) ∗ ¯ˆΨ M ˆΓ MNP QRS ˆΨ N ( d ˆ C ) [ P QRS ] ⊃ − κ (cid:90) ∗ ¯ˆΨ µ ˆΓ µνpqrs ˆΨ ν ( d ˆ C ) [ pqrs ] = − κ (cid:90) ∗ ( ¯ ψ µ + ¯ ψ ∗ µ ) ¯ ζγ µν γ pqrs ( ψ ν + ψ ∗ ν ) ζ ( d ˆ C ) [ pqrs ] . (A.21)Since there are no harmonic one-forms on the G -manifold Y , we can identify thespinorial section ζ with the unique covariant constant spinor η on the G -manifold Y ,cf. Section A.3. Furthermore, we notice that the covariantly constant three-form ϕ and its Hodge dual four-form Φ = ∗ ϕ are bilinear in η , namely ϕ mnp = i ¯ ηγ mnp η andΦ [ mnpq ] = ( ∗ ϕ ) [ mnpq ] = − ¯ ηγ mnpq η such that − κ (cid:90) ∗ ¯ˆΨ M ˆΓ MNP QRS ˆΨ N ( d ˆ C ) [ P QRS ] ⊃ κ (cid:90) ∗ ¯ ψ µ γ µν ψ ∗ ν Φ [ pqrs ] ( d ˆ C ) [ pqrs ] + c.c. (A.22)To arrive at the four-dimensional N = 1 supergravity action in the conventionalEinstein frame, we employ the Weyl rescalings g µν → g µν λ ( S i ) , γ µ → (cid:112) λ ( S i ) γ µ , ψ µ → ψ µ ( λ ( S i )) / . (A.23)Using the dimensionless volume factor defined in eq. (2.25) and κ = V Y κ in termsof the reference volume V Y = V Y ( S i ) defined in Section 2, we obtain − κ (cid:90) ∗ ¯ˆΨ M ˆΓ MNP QRS ˆΨ N ( d ˆ C ) [ P QRS ] ⊃ λ / κ (cid:90) Y G ∧ ϕ (cid:90) M , ∗ ¯ ψ µ γ µν ψ ∗ ν + c.c. 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