On N=1 4d Effective Couplings for F-theory and Heterotic Vacua
aa r X i v : . [ h e p - t h ] M a r CERN-PH-TH/2009-250LMU-ASC 59/09SU-ITP-09/52 On N = 1 4d Effective Couplings forF-theory and Heterotic Vacua Hans Jockers a , Peter Mayr b , Johannes Walcher ca Department of Physics, Stanford UniversityStanford, CA 94305-4060, USA b Arnold Sommerfeld Center for Theoretical PhysicsLMU, Theresienstr. 37, D-80333 Munich, Germany c PH-TH Division, CERNCH-1211 Geneva 23, Switzerland
Abstract
We show that certain superpotential and K¨ahler potential couplings of N = 1 super-symmetric compactifications with branes or bundles can be computed from Hodge theoryand mirror symmetry. This applies to F-theory on a Calabi–Yau four-fold and three-foldcompactifications of type II and heterotic strings with branes. The heterotic case includesa class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten.Mirror symmetry of the four-fold computes non-perturbative corrections to mirror sym-metry on the three-folds, including D-instanton corrections. We also propose a physicalinterpretation for the observation by Warner that relates the deformation spaces of certainmatrix factorizations and the periods of non-compact 4-folds that are ALE fibrations. December 2009 ontents1. Introduction2. Hodge theoretic data and N = 1 superpotentials
3. Quantum corrected superpotentials in F-theory from mirror symmetry of 4-folds N = 1 Duality chain3.3. The decoupling limit as a stable degeneration3.4. Open-closed duality as a limit of F-theory/heterotic duality3.5. Instanton corrections and mirror symmetry in F-theory
4. Heterotic superpotential from F-theory/heterotic duality W F ( X B )4.2. The Chern-Simons contribution to W F ( X B )4.3. Type II / heterotic map
5. Type II/heterotic duality in two space-time dimensions X A and X B T × Z B
6. A heterotic bundle on the mirror of the quintic X B S corrections: perturbative contributions6.4. D-instanton corrections and Gromov–Witten invariants on the 4-fold
7. Heterotic five-branes and non-trivial Jacobians SU (1): Heterotic five-branes7.2. Non-trivial Jacobians: SU (2) bundle on a degree 9 hypersurface
8. ADE Singularities, Kazama-Suzuki models and matrix factorizations9. ConclusionsAppendix A: Some toric data for the examples
A.1. The quintic in P (1 , , , , SU (2) bundle of the degree 9 hypersurface in P (1 , , , , . Introduction Let Z B be a Calabi–Yau (CY) three-fold and E a holomorphic bundle or sheaf onit. In a certain decoupling limit, where one neglects the backreaction of the full stringtheory to the degrees of freedom of the bundle, E can describe either a (sub-)bundle of aheterotic string compactification on Z B , a heterotic 5-brane or a B -type brane in a typeII compactification on Z B . In the latter case we will also be interested in the geometry( Z A , L ) associated to ( Z B , E ) by open string mirror symmetry, which consists of an A -type brane L on the mirror three-fold Z A of Z B . The contribution of the bundle to thespace-time superpotential of a string compactification on Z B is, in a certain approximation,given by the holomorphic Chern-Simons functional for both the heterotic bundle [1] andthe B -type brane [2] W CS = Z Z B Ω ∧ tr( 12 A ∧ ¯ ∂A + 13 A ∧ A ∧ A ) . (1 . Z B and A is the (0 ,
1) part of the connectionon E . There is another superpotential proportional to the periods of Ω, which, again in acertain approximation, is of the form W G = Z Z B Ω ∧ G = ( N Σ + S ˆ N Σ ) Z γ Σ Ω , γ Σ ∈ H ( Z B , Z ) . (1 . Z B , W G is the superpotential induced by NS and RR3-form fluxes [3], and S the complex dilaton. In heterotic compactifications, W G will berelated below to the superpotential of a compactification on non-K¨ahler manifolds with H -flux [4]. Depending on the type of string theory and its compactification, the combinedsuperpotential W = W CS + W G , (1 . relative cohomology group H ( Z B , D ), which captures the brane/bundle data in additionto the geometry of Z B . This was shown previously in the context of B -type branes in[8,9,10,11] and we generalize this relation here to heterotic 5-branes and general bundles,including the bundles on elliptically fibered 3-fold Z B constructed by Friedman, Morganand Witten in [12] (see also ref. [13]). The ’classical’ Hodge theory on the 3-fold gives anexplicit evaluation of the 3-fold integrals in (1.1),(1.2) and a preferred choice of physicalcoordinates, which leads to the prediction of world-sheet corrections from sphere and discinstantons of the appropriately defined mirror theories.The second step involves Hodge theory and mirror symmetry on a mirror pair ofdual CY 4-folds. 4-folds enter the stage in two seemingly different ways, in remarkableparallel with the two appearances of (1.1) in heterotic and type II compactifications on Z B . Firstly, through the duality of heterotic strings on elliptically fibered CY 3-fold Z B toF-theory on a CY 4-fold X B [14,15]. This duality motivated the systematic constructionof “heterotic” bundles on elliptically fibered Z B in refs. [12,13]. Secondly, 4-folds appearin the computation of brane superpotentials of type II strings via an “open-closed stringduality”, which associates a non-compact 4-fold geometry X ncB to a B -type brane on a 3-fold Z B [16,10,17]. In this approach, the superpotential (1.1) of the brane compactificationon ( Z B , E ) is computed from the periods of the holomorphic (4 ,
0) form on the dual 4-fold X ncB . Moreover, mirror symmetry of 4-folds relates the sphere instanton corrected periodson the mirror 4-fold X ncA of X ncB to the disc instanton corrected superpotential of thecompactification with A -type brane L on the mirror manifold Z A of Z B . This surprisingrelation between mirror symmetry of the 4-folds X ncA and X ncB and open string mirrorsymmetry of the brane geometries ( Z B , E ) and ( Z A , L ) has been tested in various differentcontexts, see e.g. [18,11,19,20].As we will argue below, these two 4-fold strands are in fact connected by a certainphysical and geometrical limit, that relates open-closed duality to heterotic/F-theory du-ality. In this limit part of the bundle degrees of freedoms decouple (in a physical sense) A related explanation of type II open-closed duality based on T-duality of 5-branes [21] hasbeen recently given in ref. [17]. X B reduces to the’classical’ type II/heterotic superpotential (1.3) on the 3-fold Z B , as has been observedpreviously in [11].The result obtained from an F-theory/type IIA compactification on the dual 4-folddiffers from the 3-fold result away from the decoupling limit. We assert that these devi-ations represent physical corrections to the dual type II/heterotic compactification fromperturbative and instanton effects and describe how Hodge theory and mirror symmetryon the 4-fold provides a powerful computational tool to determine these perturbative andnon-perturbative contributions. Depending on the point of view, the corrections computedby mirror symmetry of 4-folds describe world-sheet, D-brane or space-time instanton effectsin the dual type II and heterotic compactifications.Finally we discuss the type II/heterotic duality in the context of non-compact 4-foldsthat arise as two-dimensional ALE fibrations. For a particular choice of background fluxesthese models admit a description in terms of certain Kazama-Suzuki coset models [24,25],whose deformation spaces coincide with the deformation spaces of matrix factorizations of N = 2 minimal models [26]. We give a physical interpretation of this relation via typeII/heterotic duality and we propose that this correspondence holds even more generally.The organization of this note is as follows. In sect. 2 we discuss the applicationof Hogde theory to the evaluation of the Chern-Simons functional (1.1) with a focus onbundles on elliptic CY 3-fold constructed by Friedman, Morgan and Witten [12]. For aperturbative bundle with structure group SU ( N ) the superpotential captures obstructionsto the deformation of the spectral cover Σ imposed by a certain choice of line bundle. Wediscuss also the case of a general structure group G and heterotic 5-branes. In sect. 3we describe the decoupling limit in the type II and heterotic compactifications and use itto relate open-closed string duality to F-theory/heterotic duality, giving an explicit mapbetween type II and heterotic compactifications. We discuss the relevant string dualitiesand the meaning of the quantum corrections in the dual theories. In sect. 4, we argue, thatthe F-theory superpotential on the 4-fold captures more generally the heterotic superpo-tential for a bundle compactification on a generalized Calabi–Yau manifold and describethe map from the F-theory superpotential to the superpotential for heterotic bundles and4eterotic 5-branes. In sect. 5 we extend the previous discussion to the K¨ahler potential andthe twisted superpotential by studying the effective supergravity for the two-dimensionalcompactification of type IIA on the 4-fold and heterotic strings on T × Z B . In sect. 6 westart to demonstrate our techniques for an example of an N = 1 supersymmetric bundlecompactification on the quintic. We discuss the perturbative heterotic theory, the generalstructure of the quantum corrections and give explicit results for the example. In sect. 7we consider other interesting examples, including heterotic 5-branes wrapping a curve inthe base of the heterotic CY manifold and bundles with non-trivial Jacobians. In sect. 8we connect via heterotic/type II duality the deformation spaces of certain matrix factoriza-tions to the deformation spaces of type II on non-compact 4-folds that are ALE fibrationswith fluxes. Sect. 9 contains our conclusions. In the appendix we present further technicaldetails on the computations for the toric hypersurface examples analyzed in the main text.
2. Hodge theoretic data and N = 1 superpotentials In the approach of refs.[8,9,11], the superpotential of B -type brane compactificationswith 5-brane charge on a Calabi–Yau Z B is computed from the mixed Hodge variation on acertain relative cohomology group H ( Z B , D ). The superpotential is a linear combinationof the period integrals of the relative (3,0) form Ω ∈ H , ( Z B , D ) W II ( Z B , D ) = X γ Σ ∈ H ( Z B ) N Σ Z γ Σ Ω (3 , + X γ Σ ∈ H ZB,D ) D ⊃ ∂γ Σ =0 ˆ N Σ Z γ Σ Ω (3 , . (2 . γ Σ ∈ H ( Z B ) and thesecond term an off-shell version of the brane superpotential [27,28,7] defined on 3-chains γ Σ with non-empty boundary. Note that the superpotential W II ( Z B , D ) associated withthe Hodge bundle does not include the NS part of the type II flux potential.The boundary ∂γ Σ is required to lie in a hypersurface D ⊂ Z B , ∂γ Σ ∈ H ( D ). Themoduli of the hypersurface D parametrize certain deformations of the brane configura-tion ( Z B , E ). Infinitesimally, the accessible deformations are described by elements in H , ( Z B , D ) and come in two varieties, φ a ∈ H , ( Z B ) , ˆ φ α ∈ H , ( D ) . (2 . H , ( Z B ) captures the deformations of the complex structure of the 3-fold Z B and H , ( D ) the deformations of the holomorphic hypersurface i : D ֒ → Z B .Mirror symmetry maps the B -type brane configuration ( Z B , E ) to an A -type braneconfiguration ( Z A , L ) on the mirror 3-fold Z A . The flat Gauss-Manin connection on H ( Z B , D ) determines the mirror map z ( t ) between the complex structure moduli z of( Z B , E ) and the K¨ahler moduli t of ( Z A , L ). Inserting the mirror map into (2.1) then givesthe disc instanton corrected superpotential of the A -type geometry near a suitable largevolume point of ( Z A , L ) [11].The relative cohomology problem and open string mirror symmetry is related to abso-lute cohomology and mirror symmetry of CY 4-folds by a certain open-closed string duality[16,10,17]. The constructions of these papers associate to a B -type brane compactification( Z B , E ) and its mirror ( Z A , L ) a pair of non-compact mirror 4-folds ( X ncA , X ncB ), such thatthe “flux” superpotential of [24] agrees with the combined “flux” and brane superpotential(2.1) of the three-fold compactification, W ( X ncB ) = X γ Σ ∈ H ( X ncB ) N Σ Z γ Σ Ω (4 , = W II ( Z B , D ) , (2 . N Σ , ˆ N Σ , N Σ . Open-closed string duality thus linksthe pure Hodge variation on H hor ( X ncB ) to the mixed Hodge variation on the relativecohomology space H ( Z B , D ) ≃ H ( Z B ) ⊕ H var ( D ). The relation between the pure Hodgespaces appearing in this relation is schematically H , ( Z B ) δ / / H , ( Z B ) δ / / H , ( Z B ) δ / / H , ( Z B ) H , ( X ncB ) δ / / α O O H , ( X ncB ) δ / / α O O β (cid:15) (cid:15) H , hor ( X ncB ) δ / / α O O β (cid:15) (cid:15) H , ( X ncB ) δ / / α O O β (cid:15) (cid:15) H , ( X ncB ) H , ( D ) δ / / H , var ( D ) δ / / H , ( D ) (2 . δ denotes universally a variation in the complex structure of the respective geometries,represented by the Gauss-Manin derivative and projecting onto pure pieces.The two maps α, β : H hor ( X ncB ) → H ( Z B , D ) identify an element of H hor ( X ncB )either with an element in H ( Z B ) of the closed string state space or an element in H ( D )6ssociated with the brane geometry i : D ֒ → Z B . These maps can be explicitly realizedon the level of 4-fold period integrals by integrating out certain directions of the 4-cyclesΓ Σ ∈ H ( X ncB ) [16,17]. The map α : H hor ( X ncB ) → H ( Z B ) can be represented as anintegration over a particular S in X ncB and shifts the Hodge degree by ( − , D as in [5], andleads to the map β : H hor ( X ncB ) → H ( D ) that shifts by ( − , − X ncB split into the closed and openstring deformations (2.2) as H , ( X ncB ) ≃ H , ( Z B ) ⊕ H , ( D ) . The above deformation problem is a priori unobstructed , but becomes obstructedby the superpotential (2.3) upon adding the appropriate “flux”. In the brane geometry( Z B , E ) this can be realized by a brane flux, adding a D5-charge ˜ γ ∈ H ( D ) [11,19,17]. Anon-trivial obstruction in the open string direction arises for the choice˜ γ ∈ H var ( D ) = coker (cid:0) H ( Z B ) i ∗ → H ( D ) (cid:1) . (2 . γ remains of type (1,1)leads to a superpotential for the closed string moduli as in refs. [29,30]. Note also that aclass ˜ γ in the image of i ∗ is always of type (1,1) and thus does not impose a restriction onthe moduli of D , as the variation δW II of eq.(2.1) is automatically zero for a holomorphicboundary ∂ Γ Σ . In the following we consider a similar Hodge theoretic approach to superpotentials of“heterotic” bundles on elliptically fibered Calabi–Yau manifolds constructed in [12,13].In the framework Friedmann, Morgan and Witten, an SU ( n ) bundle E on an ellipti-cally fibered CY 3-fold π Z B : Z B → B with section σ : B → Z B is described in termsof a spectral cover Σ, which is an n -fold cover π Σ : Σ → B , and certain twisting dataspecifying a line bundle on Σ. Fixing the projection of the second Chern class of E to thebase B , the latter comprise a continuous part related to the Jacobian of Σ and a discretepart from elements γ ∈ ker (cid:0) H , (Σ) π Σ ∗ −→ H , ( B ) (cid:1) (2 . X B , the elements of the Hodge spaces of the spectralcover are related to those on X B schematically as [12,13,31]:Σ X B H , H , H , H , H , H , The first line identifies the infinitesimal deformations of Σ with infinitesimal defor-mations of the 4-fold. The second relation relates the discrete data described by the class γ with 4-form flux in the F-theory compactification on X B . The last relation reflects theisomorphism of the Jacobian of Σ and the corresponding Jacobian in X B related to it byduality (see also [32]). Note that the heterotic/F-theory relation between H ( X B ) and H (Σ) is formally given by the same ( − , −
1) shift in Hodge degree as in the map β inthe open-closed duality relation (2.4). As argued below, this similarity is not accidental,but a reflection of the fact, that the heterotic and type II data can be related by the aforementioned decoupling limit.Again the deformations of the spectral cover Σ in H , (Σ) are unobstructed if γ is the“generic” (1 ,
1) class discussed in [12]. Consider instead a class γ that is of type (1 , z = 0 of the deformation space. Twisting by γ then should obstructthe deformations of Σ in the direction ˆ z = 0, which destroy the property γ ∈ H , (Σ).We propose that the heterotic superpotential describing this obstruction is capturedby the chain integral W het ( Z B , Σ , γ ) = Z Γ Ω , , (2 . ∈ H ( Z B , Σ) a 3-chain with non-zero boundary on Σ. The dual space H ( Z B , Σ) ≃ H ( Z B ) ⊕ H var (Σ) is the relative cohomology group defined by the spectral cover Σ with H var (Σ) the mid-dimensional horizontal Hodge cohomology of Σ. Moreover the boundary2-cycle C = ∂ Γ ⊂ Σ is the cycle Poincar´e dual to γ . The chain integral can then becomputed from the Hodge variation on the relative cohomology group, as has been usedin refs. [8,9,11] to compute brane superpotentials in type II strings. As a first check on therelevance of the mixed Hodge variation on H ( Z B , Σ) for the heterotic theory, note that However, the existence of this class is a consequence of insisting on a section for π Σ : Σ → B . H , (Σ) is indeed captured by the Hodge space H , ( Z B , Σ), as inthe type II case.In the type II context, the mixed Hodge variation gives more physical informationthan just the superpotential, specifically appropriate coordinates on the deformation space,which lead to the interpretation of the superpotential as a disc instanton sum in the mirror A model. The physical interpretation of the corrections in the heterotic theory will bediscussed below.The expression (2.7) of the heterotic string can be argued for by relating it to theholomorphic Chern-Simons functional (1.1), which is the holomorphic superpotential forthe bundle moduli in the heterotic string [1]. Before turning to the derivation for a genuineCY 3-fold of holonomy SU (3), it is instructive to reflect on the argument at the hand ofthe simpler N = 2 supersymmetric case of dual compactifications of F-theory on K × K T × K
3. The perturbative F -term superpotential associated witha heterotic flux on K3 in the i -th U (1) factor is [33,34] W N =2 het = A i Z C ω , , (2 . A i is the Wilson line on T , C the cycle Poincar´e dual to the flux and ω , theholomorphic (2 ,
0) form on the heterotic K3. In this simple case, the spectral cover is justpoints on the dual T times K3, and the chain integral (2.7) over the holomorphic (3 , dz ∧ ω , becomes W het = Z Γ Ω = Z p i dz Z C ω , = A i Z C ω , , (2 . A i ∼ A i +1 ∼ A i + τ appearing in (2.8) are defined by the Abel-Jacobi map on T . Furthermore, p i denotes the associated point in the Jacobian. In the N = 1 case, the points p i vary overthe base and the bounding 2-cycles are not of the simple form (0 , p i ) × C . An importantconsequence is that holomorphy of C gets linked to the deformations A i . There is also a simple generalization of this N = 2 superpotential to the case, wherethe heterotic vacuum contains heterotic 5-branes [36], and this is also true for the N = 1supersymmetric case studied below. The 5-brane superpotential is in fact the most straight-forward part starting from the results on type II brane superpotentials of refs. [8,9,11], asthe brane deformations of the type II brane map to the brane deformations of the het-erotic 5-brane in a simple way. The type II/heterotic map providing this identification andexplicit examples will be discussed later on. See ref. [35] for a similar discussion. .3. Holomorphic-Chern Simons functional for heterotic bundles The holomorphic Chern-Simons functional is (a projection of) the transgression ofthe Chern-Weil representation of the algebraic second Chern class for a supersymmetricvector bundle configuration. Thus, in order to establish for a supersymmetric heteroticbundle configuration that (1.1) agrees with eq. (2.7) on-shell, we need to show that theboundary 2-cycle C = ∂ Γ of the 3-chain Γ in eq. (2.7) is given by a curve representingthe algebraic second Chern class of the holomorphic heterotic vector bundle. The latter isencoded in the zero and pole structure of a global meromorphic section s E : Z → E of thesupersymmetric holomorphic heterotic bundle E [37]. This is described in ref. [38] for ageneral SU (2) bundle and in ref. [30] for a bundle associated with a matrix factorization.To apply this reasoning to the SU ( N ) bundles of [12], we need to construct an explicitrepresentative for the algebraic Chern class. As explained in ref. [12], the spectral coverΣ together with the class γ of eq. (2.6) defines the SU ( n ) bundle E over the ellipticallyfibered 3-fold π Z : Z → B by E = π ∗ R , R = P B ⊗ S , R → Σ × B Z .
Here π is the projection to the second factor of the fiberwise product Σ × B Z of the3-fold Z and of the spectral cover Σ over the common base B . P B is the restriction of thePoincar´e bundle of the product Z × B Z to Σ × B Z , while S →
Σ denotes the line bundleover the spectral cover Σ, which is given by S = N ⊗ L γ . The bundle N ensures that the first Chern class c ( E ) of the SU ( n ) bundle vanishes andits explicit form is thoroughly analyzed in ref. [12]. The holomorphic line bundle L γ with c ( L γ ) = γ governs the twisting associated to the class γ in (2.6), and it is responsible forthe discussed obstructions to the deformations of the spectral cover Σ. Note that, due tothe property (2.6), the line bundle L γ does not further modify the first Chern class c ( E )[12]. To avoid cluttering of notation, the heterotic manifold Z B is denoted simply by Z in thefollowing argument. For ease of notation its pull-back to Σ × B Z is also denoted by the same symbol S .
10n order to construct a section s E of the SU ( n )-bundle, we need to push-forward aglobal (meromorphic) section s R = s P · s S of the line bundle R , which in turn is theproduct of a section of the Poincar´e bundle P B and the line bundle S . The Poincar´ebundle is given by P B = O (∆ − Σ × σ ) ⊗ K B , where ∆ is (the restriction of) the diagonaldivisor in Z × B Z , K B is the canonical bundle of the base (pulled back to Σ × B Z ) and σ : B → Z the section of the elliptic fibration Z . Therefore the section s P = s K · s F canbe chosen to be the product of the section s K of the canonical bundle of the base B andthe section s F , which has a (simple) zero set along the diagonal divisor ∆ and a (simple)pole set along the divisor Σ × B σ . Finally, the zero set/pole set of the section s S is inducedfrom the (algebraic) first Chern class c ( S ) of the line bundle S over the spectral cover Σ.Here we are in particular interested in the contribution from the line bundle L γ , whoseglobal (meromorphic) section extended to the fiber-product space Σ × B Z is denoted by s γ . For an SU ( n )-bundle the projection map π is an n -fold branched cover of the 3-fold Z , and therefore in a open neighborhood U ⊂ B of the base the push-forward of the section s R yields s E = π ∗ s R = s K · ( s F · s S , s F · s S , . . . , s nF · s n S ) . (2 . s K originates from the canonical bundle over the base, it appears as anoverall pre-factor of the bundle section s E , while the entries s iF and s i S arise from the n sheets of the n -fold branched cover. The entries s iF restrict on the elliptic fiber to a sectionof ⊕ ni =1 ( O ( p i ) ⊗ O (0) − ) that have a simple zero at p i and a simple pole at 0. Here 0denotes the distinguished point corresponding to the section σ : B → Z and P i p i = 0 for SU ( n ). The n entries s i S arise again from the section s S on the n different sheets. Sincethe section s S is induced from a line bundle over the spectral cover, the zeros/poles of thesections s i S correspond to co-dimension one sub-spaces on the base.Now we are ready to determine the algebraic Chern classes of the SU ( n )-bundle E from the global section (2.10). By construction the first topological Chern class is trivial,which implies that also the first algebraic Chern class vanishes since the Abel-Jacobi map At branch points of the spectral cover (at least) two points p i and p j , i = j , coincide, andthe restriction of the bundle E to the elliptic fiber becomes a sum of n − n line bundles.
11s trivial for the simply-connected Calabi-Yau 3-folds discussed here. The second algebraicChern class is determined by the “transverse zero/pole sets” of the section s E , whichcorrespond to the co-dimension two cycles of the mutual zero/pole sets of distinct entries s iE and s jE , i = j .Since s iE = s iF · s i S , this computation exhibits c ( E ) as a sum of three contributions:The joint vanishing of s iF and s jF is empty since p i = p j generically. The joint vanishingof s i S and s j S is a sum of fibers, which we may neglect since, moving in a rational family,they do not contribute to the superpotential. Equivalently, we may use the relation ch ( E ) = c ( E ) − c ( E ) between the secondChern class and the second Chern character ch ( E ), which thanks to the vanishing of c reduces to ch ( E ) = − c ( E ), to compute c ( E ) from the transverse zero/pole sets of thelocal sections s kF and s k S of the same entry k . This will more directly lead to the desiredboundary 2-cycle C = ∂ Γ. (Again, we may neglect the self-intersections of s kF and s k S .)We focus now on the contribution c ( E γ ) to the second algebraic Chern class, whichis associated to the intersection of the zero/pole sets of the local sections s kγ and the localsections s kF for k = 1 , . . . , n . As argued the obtained divisor is rational equivalent tothe (negative) boundary 2-cycle C arising form the Poincar´e dual of the 2-form γ on thespectral cover Σ, and we obtain for the second algebraic Chern class c ( E ) = c ( E γ ) + c ( V ) , c ( E γ ) = − [ C ] , (2 . C ] the cycle class, which arises from embedding the two-cycle C ofthe spectral spectral cover Σ into the Calabi-Yau 3-fold Z . Due to the property (2.6) thecurve associated to c ( E γ ) is (up to a minus sign) rational equivalent to the boundaryof the same 3-chain Γ appearing in eq. (2.7). The other piece c ( V ), which is (locally)independent of the analyzed deformations of the spectral cover, is discussed in detail inref. [12]. In general it gives rise to a non-trivial second topological Chern class. In a globallyconsistent heterotic string compactification this contribution is compensated by the second An equivalent way to see this is to note that five-branes wrapped on the fiber on the ellip-tic threefold map under heterotic/F-theory duality to mobile D3-branes which clearly have nosuperpotential. Thus, by reproducing the 3-chain Γ from the second algebraic Chern class of theholomorphic SU ( n ) bundles, the holomorphic Chern-Simons functional is demonstratedto be agreement with the holomorphic superpotential (2.7). Analogously to the non-supersymmetric off-shell deformations of branes in type II compactifications [11,17],we propose that the correspondence between the superpotential (2.7) and the Chern-Simons functional even persists along deformations of the spectral cover, which yield non-supersymmetric SU ( n ) bundle configurations.To illustrate the presented construction, we briefly return to the N = 2 compacti-fication of the heterotic string on T × K
3. For this example the spectral cover of an SU ( n ) bundle is a disjoint union of n K3 surfaces ` ni =1 { p i } × K T × K γ fulfilling the property (2.6) can be thought of as a non-trivial (1,1)-form ω γ ,which appears in the component p i × K p j × K i = j , with opposite signs. Thenthe Poincar´e dual curve C of γ embedded into T × K p i , p j ) × C , where ( p i , p j ) denotes the 1-chain on the torus bounded by the points p i and p j . The resulting chain integral over dz ∧ ω , exhibits the same structure as thenaive equation (2.8). In the next section we will consider a dual F-theory compactification on a 4-fold andargue that mirror symmetry of the 4-fold computes interesting quantum corrections tothe Chern-Simons functional. Here we want to motivate the following ’classical’ relationbetween the 4-fold periods and the Chern-Simons functional (1.1) Z X B Ω , ∧ G A = Z Z Ω , ∧ tr( 12 A ∧ ¯ ∂A + 13 A ∧ A ∧ A ) + O ( S − , e πiS ) . (2 . X B is a CY 4-fold which will support the F-theory compactification dualto the heterotic compactification on the 3-fold Z and G A is a 4-form ’flux’ related tothe connection A of a bundle E → Z as described below. Moreover S is a distinguished In generalized Calabi-Yau compactifications of the heterotic string additional contributionsenter into the anomaly equation due to non-trivial background fluxes and the modified generalizedgeometry [4]. X B such that Im S → ∞ imposes a so-calledstable degeneration (s.d.) limit in the complex structure of X B . In this limit the 4-fold X degenerates into two components X Im S →∞ −→ X ♯ = X ∪ Z X , intersecting over the elliptically fibered heterotic 3-fold Z → B [15,12,13,39]. The two4-fold components X i are also fibered over the same base B and capture (part of) thebundle data of the two E factors of the heterotic string, respectively.The idea is now to view Z as a complex boundary within one of the components X i and to apply a theorem of [38], which relates the holomorphic Chern-Simons functionalon a 3-fold Z to an integral of the Pontryagin class of a connection A on an extension E → X ′ of the bundle E → Z defined over a Fano 4-fold X ′ : Z X ′ tr (cid:16) F , A ∧ F , A (cid:17) ∧ s − = CS ( Z, A ) . (2 . CS ( Z, A ) is short for the Chern-Simons functional on the r.h.s. of (2.12) without thefinite S corrections. Moreover s ∈ H ( K − X ′ ) is a section of the anti-canonical bundle of X ′ whose zero set defines the 3-fold Z as a ’boundary’ of X ′ .Now it is straightforward to show, that the components X i of the degenerate F-theory4-fold X ♯ are Fano in the sense required by the theorem and moreover that the heteroticCalabi-Yau 3-fold Z can be defined as the zero set of appropriate sections s i of the anti-canonical bundles K − X i , as required by the theorem. This will be discussed in more detailin sect. 4.2, where we explicitly discuss hypersurface representations for X ♯ to match theF-theory/heterotic deformation spaces.The above line of argument then leads to a relation of the form (2.12), provided oneidentifies the 4-form flux G A with the Pontryagin class of a gauge connection A on anextension E of the bundle over the component X . Up to terms of lower Hodge type, weshall have G A | X ∼ tr (cid:16) F , A ∧ F , A (cid:17) . (2 . G , but rather to view the flux superpotential as a potential onthe moduli space of the 4-fold X , which fixes the moduli to the critical locus. The idea is,that the periods R X Ω , on the l.h.s. of (2.12) have a well-defined meaning as the sectionof a bundle over the unobstructed complex structure moduli space M CS ( X ) of the 4-fold before turning on a the flux; in particular they define the K¨ahler metric on M CS ( X B ). Inthis way, viewing non-zero G as a ’perturbation’ on top of an unobstructed moduli space,the section W ( X B ) is considered as an off-shell potential for fields parametrizing M CS ( X ).Although it is not clear in general under which conditions it is valid to restrict the effectivefield theory to the fields parametrizing M CS ( X ) and to interprete W ( X B ) as the relevantlow energy potential for the light fields, this working definition for an off-shell deformationspace seems to make sense in many situations. The relation (2.12) suggests that it should be possible to give a sensible notion ofa distinguished, finite-dimensional ’off-shell moduli space’ for non-holomorphic bundlesand to treat the obstruction induced by the Chern-Simons superpotential as some sort of’perturbation’ to an unobstructed problem. This is also suggested by the recent successto compute off-shell superpotentials for brane compactifications from open string mirrorsymmetry. We plan to circle around these questions in the future.
3. Quantum corrected superpotentials in F-theory from mirror symmetry of4-folds
In this section we show, that the various Hodge theoretic computations of superpo-tentials in CY 3-fold and 4-fold compactifications discussed above are in some cases linkedtogether by a chain of dualities. The unifying framework is the type IIA compactificationon a pair ( X A , X B ) of compact mirror CY 4-folds and its F-theory limits. As will be arguedbelow, mirror symmetry of the 4-folds computes interesting quantum corrections, most no-tably D-instanton corrections to type II orientifolds and world-sheet corrections to heterotic(0,2) compactifications, which are hard to compute by other means at present. Anotherinteresting connection is that to the heterotic superpotential for generalized Calabi–Yaumanifold. The purpose of this section is to study the general framework, which involves asomewhat involved chain of dualities, while explicit examples are given in sects. 6, 7. There is a considerable literature on this subject. We suggest ref. [40] for a justification inthe context of type IIA flux compactifications on 3-folds, ref. [41] in the type IIB context, ref. [42]in non-geometric phases, and ref. [43] for a recent general discussion. .1. Four-fold superpotentials: a first look at the quantum corrections For orientation it is useful to keep in mind the concrete structure of the superpotentialon compact 4-folds that we want to study, as it links the different dual theories discussedbelow at the level of effective supergravity. The compact 4-fold X B for F-theory compacti-fication is obtained from the non-compact 4-fold X ncB of open-closed in eq.(2.3) by a simplecompactification [10,11,19], discussed in more detail later on. In a certain decoupling limitdefined in [11], the F-theory superpotential on X B reproduces the type II superpotential(2.1) plus further terms: W F ( X B ) = X γ Σ ∈ H ( X B ) N Σ Z γ Σ Ω (4 , S →∞ = X γ Σ ∈ H ( Z B ) ( N Σ + S M Σ ) Z γ Σ Ω (3 , + X γ Σ ∈ H ZB,D ) ∂γ Σ =0 ˆ N Σ Z γ Σ Ω (3 , + . . . . (3 . S of the compactification X B of X ncB . This modulus is identified in [11]with the decoupling limit Im S ∼ /g s → ∞ . (3 . ∼ S M Λ in the 4-fold superpotential W F ( X B ) correspond toNS fluxes in the type II string on Z B , which were missing in (2.1). In addition there aresubleading corrections for finite S , denoted by the dots in (3.1), which include an infinitesum of exponentials with the characteristic weight e − /g s of D-instantons. Before studyingthese corrections in detail, it is instructive to consider the dualities involved in the picture,which leads to a somewhat surprising reinterpretation of the open-closed duality of [16,10]. This has been observed already earlier in a related context in ref. [44], see also the discussionin sect. 5 below. .2. N = 1 Duality chain
The relevant duality chain for understanding the quantum corrections in (3.1), andthe relation to open-closed duality, relates the following N = 1 supersymmetric compact-ifications: type II OF T × Z B ∼ F-theory K × Z B ∼ heterotic T × Z B ∼ type IIA X B /X A ∼ F-theory X B × T (3 . Z B is a CY 3-fold and ( X A , X B ) a mirror pair of 4-folds which is related to theheterotic compactification on Z B by type IIA/heterotic duality. Here and in the followingit is assumed that the 3-fold Z B and the 4-fold X B have suitable elliptic fibrations, inaddition to the K3 fibration of X B required by heterotic/type IIA duality [45]. Thisguarantees the existence of the F-theory dual in the last step. For an appropriate choiceof bundle one can take the large volume of the T factor to obtain the four-dimensionalduality between heterotic on Z B and F-theory on X B [15].The remaining section will center around the identification of the limit (3.2) in thevarious dual theories. Note that there are two different F-theory compactifications involvedin the duality chain (3.3), namely on the manifolds K × Z B and X B × T , respectively.The identification (3.2) is associated with the F-theory compactification on K × Z B , or thetype II orientifold on T × Z B , in the orientifold limit [46]. The decoupling limit describesalso a certain limit of the heterotic compactification on the same 3-fold Z B , which will beidentified as a large fiber limit of the elliptic fibration Z B below.In order to make contact with the brane configuration ( Z B , E ) discussed in sect. 2.1,we combine the orientifold limit of F-theory with a particular Fourier-Mukai transformation[47,48] type II OF ˇ T × ˇ Z B ∼ type II OF T × Z B ∼ F-theory K × Z B . The relevant Fourier-Mukai transformation is discussed in detail in ref. [48]. Heuristically,it implements T duality in both directions of the torus T to the dual torus ˇ T together witha fiberwise T duality in both directions of the elliptic fibers of the 3-fold Z B to the 3-fold ˇ Z B with dual elliptic fibers. This operation does not change the complex structure of the bulk In this note, for ease of notation and to emphasize the relation to four-dimensional theories, N = 1 compactifications to two space-time dimensions also refer to low energy effective theorieswith four supercharges. Z B , E ). These orientifold limits of F-theory, the type II and heterotic compactificationson Z B can be also connected as: type II OF ˇ T × ˇ Z B ∼ type I ˇ T × Z B ∼ heterotic T × Z B ≀ type II OF T × Z B (3 . S duality associates the type I to the heterotic string, T duality on ˇ T relates thetype I compactification to the type II orientifold on T × Z B , while the afore mentionedFourier Mukai transformation, which realizes fiberwise T duality, applied to the 3-fold Z B of the type I theory maps to the type II orientifold on ˇ T × ˇ Z B [46,47,48]. The meaning of the decoupling limit in the mirror pair ( X A , X B ) of 4-folds and thedual heterotic string on Z B ( × T ) can be understood with the help of the following twopropositions obtained in the study of F-theory/heterotic duality and mirror symmetry ontoric 4-folds in ref. [23]. It is shown there that ( C
1) If F-theory on the 4-fold X B is dual to a heterotic compactification on a 3-fold Z B then the mirror 4-fold X A is a fibration Z A → X A → P , where the generic fiber Z A is the 3-fold mirror of Z B .( C
2) In the above situation, the large base limit in the K¨ahler moduli of the fibration X A → P maps under mirror symmetry to a “stable degeneration” limit in the complexstructure moduli of the mirror X B .The first part applies, since the 4-fold duals constructed in the context of open-closedstring duality have precisely the fibration structure required by ( C X ncA , X ncB ) of open-closed dual 4-folds, dual to an A -brane geometry ( Z A , L ) and its For concreteness, we quote the result for F-theory on a 4-fold, although it applies moregenerally to n -folds, as will be also used below. B -brane geometry ( Z B , E ), is constructed in refs. [16,10] as a fibration over thecomplex plane, where the generic fiber is the CY 3-fold Z A : Z A / / X ncAπ ( L ) (cid:15) (cid:15) − foldmirror symmetry / / X ncB o o C (3 . π ( L ) for the fiber projection is a reminder of the fact that the data ofthe bundle L are encoded in the singularity of the central fiber as described in detail inrefs. [16,18,10,11]. The manifold X ncB may be defined as the 4-fold mirror of the fibration X ncA . Since the pair of compact 4-folds ( X A , X B ) is obtained by a simple compactificationof the base to a P [11,19], it follows that the F-theory 4-fold X B has a mirror X A , whichis a 3-fold fibration π : X A → P with generic fiber Z A . The multiple fibration structuresare summarized below: F-theory X B ∼ heterotic Z B closed X A ∼ open ( Z A ,L ) Elliptic Fib. T → X B ↓ B T → Z B ↓ B –K3 Fib. K → X B ↓ B –3-fold Fib. Z A → X A ↓ P Z A → X A ↓ P (3 . B and B denote the corresponding three- and two-dimensional base spaces, where B is common to the F-theory manifold and the heterotic dual. The crucial link is the3-fold fibration of X A , which is required by both, F-theory/heterotic and open-closedduality. ( C
1) then implies that F-theory on X B has an open-closed dual interpretationas a B -type brane on a 3-fold Z B and an A -type brane on the mirror Z A . The reverseconclusion, namely that an open-closed dual pair ( X A , X B ) also has an F-theory/heteroticinterpretation, requires the additional condition, that X B is elliptic and K3 fibered. Thisleaves the possibility, that open-closed duality holds for more general 4-fold geometriesthan F-theory/heterotic duality. For simplicity we impose in the following, that X B iselliptically and K3 fibered, which implies that ( C
1) holds also in the reverse direction.19art two of the proposition applies, since the decoupling limit Im S → ∞ in thecomplex structure of X B was defined in ref. [11] as the mirror of the large base volume inthe K¨ahler moduli of the fibration π : X A → P . The image of this limit under the mirrormap in the complex structure of X B is a local mirror limit in the sense of [22] and effectivelyimposes the stable degeneration (s.d.) limit of X B studied in refs. [15,12,39]. Under F-theory/heterotic duality, the s.d. limit maps to a large fiber limit of the heterotic stringcompactification on the elliptic fibration Z B and this is the sought for identification of limit(3.2) in the heterotic string. The meaning as a physical decoupling limit of a sector of theheterotic string can be understood from both, the world-sheet and the effective supergravitypoint of view, as will be discussed in sect. 5. Explicit examples for the relation betweenthe hypersurface geometries X B and Z B in the s.d. limit will be considered in sects. 6,7. The relation in (3.3) between the type II orientifold on Z B and type IIA on the 4-folds( X B , X A ) is similar as in the open-closed duality of refs. [16,10,17]. These papers claimto compute the type II superpotential for a B -type brane compactification on Z B with agiven 5-brane charge from the periods of a dual (non-compact) 4-fold X ncB . As explainedin refs. [11,19,17], this 5-brane charge can be generated by non-trivial fluxes on higherdimensional branes. The only difference to the type II orientifold on T × Z B appearingin (3.3) is the extra T compactification and the presence of 7-branes wrapping Z B , whichdoes not change the superpotential associated with the 5-brane charge.In the decoupling limit Im S → ∞ , which sends X B to the non-compact manifold X ncB , the “local” B -type brane with 5-brane charge decouples from the global orientifoldcompactification and we recover the type II result W II ( Z B ) in eq. (1.2). Note that in thislimit there are two different paths connecting the B -type orientifold to the non-compactopen-closed string dual X ncB . The first one goes via the open-closed string duality ofrefs. [16,10,17], while the second goes via F-theory/heterotic/type IIA duality of eq. (3.3). type II OFT × Z B F/het/IIAduality / / g s → (cid:15) (cid:15) type IIAX B Im S →∞ (cid:15) (cid:15) local B − brane ( Z B ,E ) open − closedduality / / type IIAX ncB (3 . In the type II string without branes/orientifold, ˆ N Σ = 0 and the subleading corrections tothe superpotential would be absent [3]. C in Z B in the ori-entifold to a heterotic 5-brane wrapping the same curve C in the heterotic dual Z B . Theheterotic 5-brane can be locally viewed as an M-theory 5-brane [49], which is in turn relatedto the type IIA 5-brane used in [17] to derive open-closed string duality from T-duality.The original observation of open-closed string duality of ref. [16] is that it maps the discinstanton generated superpotential of the brane geometry ( Z A , L ) (mirror to ( Z B , E )) tothe sphere instanton generated superpotential for the dual 4-fold X ncA (mirror to X ncB ). Attree-level, this map is term by term , that is it maps an individual Ooguri–Vafa invariant fora given class β ∈ H ( Z A , L ) to a Gromov-Witten invariant for a related class β ′ ∈ H ( X ncA ).This genus zero correspondence left the important question, whether there is a full stringduality, that extends this relation between the 3-fold and the 4-fold data beyond thesuperpotential. From the above diagram we see, that there is at least one true stringduality which reduces to open-closed string duality of refs. [16,10,17] at g s = 0 and extendsit to a true string duality: F-theory/heterotic duality! The above discussion has lead to the qualitative identification of the dual interpreta-tions of the expansion in (3.1) in terms of a weak coupling limit of the type II orientifold, alarge fiber volume of the heterotic string on the elliptic fibration Z B , a stable degenerationlimit of the F-theory 4-fold X B and a large base limit of the 3-fold fibration X A → P .We will now argue that the quantum corrections computed by 4-fold mirror symmetry canbe tentatively assigned to the two 4-fold superpotentials in refs. [24,50] as W ( X B ) = R X B Ω ∧ F hor ↔ D-1,D1/finite-fiber corrections in type II OF/Het f W ( X B ) = R X B e B + iJ ∧ F ver ↔ D3/space-time instantons in type II OF/Het (3 . W ( X B ) is the 4-fold superpotential of eq. (3.1), while f W ( X B ) is the twisted super-potential associated with the type IIA compactification on X B . The latter computes See the discussion in sect. 5 below. N = 2 supersymmetry: type II OF T × Z H ∼ F-theory ˜ Z V × Z H ∼ heterotic T × Z H ∼ type IIA/IIB X B /X A ∼ F-theory X B × T (3 . Z V , Z H are two K3 manifolds and ( X A , X B ) denotes a mirror pair of CY 3-folds;differently then in (3.3), mirror symmetry of the 3-folds exchanges the IIA compactificationon X B with a type IIB compactification on X A . As before, we assume that the 3-fold X B iselliptically fibered, such that one can decompactify the T of the heterotic string to obtainF-theory in six dimensions. Note that the N = 1 duality chain (3.3) can be heuristicallythought of as a chain of dualities obtained by “fibering” (3.9) over P , so that someobservations from the N = 2 supersymmetric case will carry over to N = 1.The two basic questions that we want to study in this simpler setup are the meaningof mirror symmetry in F-theory and the identification of quantum corrections computedby it. It will turn out that, under favorable conditions, the distinguished modulus S hasa mirror partner ρ and mirror symmetry of the CY manifolds X A and X B exchanges thetwo weak coupling expansions in Im S and Im ρ .The quantum corrections to the N = 2 supersymmetric duality chain (3.9) have arich structure studied previously in [36,51]. The F-theory superpotential for the K × K N = 2 supergravity theory from certaingaugings in the hypermultiplet sector, can be written as a bilinear in the period integralson the two K3 factors [52,33] W F,pert = X I, Λ (cid:0) Z Z H ω , ∧ µ I (cid:1) G I Λ (cid:0) Z ˜ Z V ω , ∧ ˜ µ Λ (cid:1) . (3 . G I Λ labels the 4-form flux in F-theory, decomposed on a basis { ˜ µ Λ } for H prim ( ˜ Z V )and { µ I } for H prim ( Z H ) as G = P I, Λ G I Λ µ I ∧ ˜ µ Λ .The periods on Z H depend on N = 2 hyper multiplets and are mapped under dualityto the type IIA/F-theory compactification on X B to the 3-fold periods, by a similar relationas (3.1): Z X B ω , ∧ γ I = Z Z H ω , ∧ µ I + O ( e πiS , S − ) . (3 . X B defined on the basis γ I ∈ H ( X B , Z ) compute finite S corrections to the periods on the 2-fold Z H of thedual type II compactification. As explained in the 4-fold case, ( C
2) says that these arecorrections to the s.d. limit in the complex structure of X B .Note that (3 .
10) is apparently symmetric in the periods of the two K3 factors. This issomewhat misleading, as the periods on ˜ Z V depend on N = 2 vector multiplets. It wasargued in [36], that there is also a similar relation as (3.11) for the second period vectoron ˜ Z V (3.11), Z X A ω , ∧ ˜ γ Λ = Z ˜ Z V ω , ∧ ˜ µ Λ + O ( e πiρ , ρ − ) , (3 . ρ is a distinguished vector multiplet related to the heterotic string coupling asdiscussed below. This relation describes corrections to the result (3.10) computed by theperiods of the mirror manifold X A . Here it is understood, that one uses mirror symmetryto map the periods of the holomorphic (3,0) form on H ( X A , Z ) defined on the basis˜ γ Λ ∈ H ( X A , Z ) to the periods of the K¨ahler form on a dual basis γ Λ ∈ ⊕ k H k ( X B , Z ), Z X A ω , ∧ ˜ γ Λ −→ Z X B k ! J k ∧ ˜ γ Λ . (3 . X B are the 3-fold equivalent of the integrals appearingin the twisted superpotential f W ( X B ) in (3.8). However, replacing the K3 periods in(3 .
10) by the quantum corrected expressions (3.11),(3.12), we get a superpotential thatis proportional to both, the periods of the manifold X B and of its mirror X A . It wasargued in [36], that this ’quadratic’ superpotential in the 3-fold periods is in agreementwith the S -duality of topological strings predicted in ref. [53]. Similar expressions havebeen obtained in refs. [54,55] from the study of type II compactification on generalizedCY manifolds.The similarity of the two expansions (3.11),(3.12) is no accident. By ( C S → ∞ is mirror to the large base limit of the fibration X A → P , which is a K3fibration by ( C
1) in the 3-fold case. By type IIA/heterotic duality, X B is also a K3 fibration X B → P and eq. (3.12) represents the large base limit Im ρ → ∞ of X B , where ρ is theK¨ahler volume of the base P . By heterotic/type IIA duality, the K¨ahler volume of thebase of X B is identified with the four-dimensional heterotic string coupling [56]. Adding See refs. [52] for a discussion of the effective supergravity theory for the orientifold limit of K × K S provided by ( C V A/B of the base P ’s of the fibrations X A/B → P : V B = λ − ,het = Im ρ , V A = V E het = Im S . (3 . V E het denotes the volume of the elliptic fiber of Z H in the heterotic compactification in(3.9). Clearly, mirror symmetry exchanges the two expansions (3.11) and (3.12) associatedwith a compactification on X A or on X B , respectively S (3 . mirror ←→ symmetry ρ (3 . . (3 . K × K
3, mirror symmetry represents theexchange of the two K3 factors [57,51], which gives rise to two dual heterotic T × K K × K T × S × K Z transformation on the moduli ofthe two heterotic duals: V E ′ het = λ − , λ ′ − = V E het . Comparing with the relation (3.14) between the four-dimensional heterotic coupling andthe volumes of the bases of the fibrations ( X A , X B ), one concludes that the result of [51] isin accord with the claim ( C
2) of [23] and its consequence (3.15) in this case. It is reassuringto observe that these conclusions, reached by rather different arguments in refs. [23,36] and[51], agree so nicely.As further argued in [36], the expansion (3.12) computed from mirror symmetry of the3-folds X B and X A computes D3 instanton corrections to the orientifold on K × T (orF-theory on K × K K
3, which is mappedunder the duality (3.4) to a 5-brane instanton of the heterotic brane wrapping T × K ρ is the K3 volume.Compactifying the N = 2 chain on a further P , the previous arguments leads to theassignements (3.8). In particular the identification of D3 instantons in [36] continues tohold with the appropriate replacement of K3 with 4-cycles in Z B . The above argumentbased on ( C
2) is in fact independent of the dimension and can be phrased more generallyas the following statement on mirror symmetry in F-theory. Let X B be an F-theory n -foldwith heterotic dual ( Z B , V B ), where V B denotes the gauge bundle. If the mirror X A of24 B is also elliptically and K3 fibered, we have the following relations between the F-theorycompactifications on ( X A , X B ) and heterotic compactifications on ( Z A , Z B ): F − theoryZ A → X B → P mirrorsymmetry / / (cid:15) (cid:15) F − theoryZ B → X A → P o o (cid:15) (cid:15) heterotic ( Z B ,V B ) het/hetmap / / ( C lllllllllllllll heterotic ( Z A ,V A ) o o i i RRRRRRRRRRRRRRR (3 . Z A → X A → P Z B → X B → P + + VVVVVVVVVVVVVVVVVVV stable deg s s hhhhhhhhhhhhhhhhhhh large base hhhhhhhhhhhhhhhhhhh large base k k VVVVVVVVVVVVVVVVVVV (3 . not dual but becomedual after further circle compactifications.The simplest example is F-theory on a K3 X B dual to heterotic on ( Z B = T , V G ),where V G denotes a flat gauge bundle on T with structure group G . The eight-dimensionalheterotic compactification has an unbroken gauge group H , where H is the centralizer of G in the ten-dimensional heterotic gauge group. In a further compactification on T one hasto choose a flat H bundle on the second T . Assuming that the bundles factorize, one canexchange the two T factors and thus H and G . In F-theory this exchange corresponds tomirror symmetry of K3 and this was used in [22,23] to construct local mirrors of bundleson T from local ADE singularities.The next simple example is the above N = 2 supersymmetric case, where X B isthe 3-fold in (3.9), with a heterotic dual compactified on K × T . Assuming a suitablefactorization of the heterotic bundle, the action of 3-fold mirror symmetry maps to theexchange of the two K3 factors ( ˜ Z V , Z H ) in the dual F-theory compactification in (3.16).In the heterotic string this symmetry relates two different K3 compactifications ( Z H , V )and ( ˜ Z V , V ′ ) which become dual after compactification on T × S [59,23]. One needs the T compactification to get two type IIA compactifications on the mirror pair( X A , X B ), which become T-dual after a further circle compactification.
25n the 4-fold case, the fibrations required by the above arguments are not granted,since ( C
1) now implies that the 4-fold X A is a 3-fold fibration X A → P (as opposed tothe K3 fibration in the 3-fold case). If X A is K3 fibered, the N = 1 chain can be viewed asa N = 2 chain fibered over P and the above arguments apply, leading to the assignment(3.8). In the other case, the large Im S expansion of W ( X B ) always exists, but there is nocorresponding large ρ expansion of the twisted superpotential f W ( X B ).
4. Heterotic superpotential from F-theory/heterotic duality
Having identified the limit S → i ∞ as a large fiber limit in the heterotic interpretation,the next elementary question is to identify the “flux quanta” of the 4-fold superpotential(3.1) in the context of the heterotic string. This task can be divided into identifyingthe origin of the terms ∼ N Σ , M Σ captured by the bulk periods and the terms ∼ ˆ N Σ proportional to chain integrals. W F ( X B )The back-reaction of the bulk background fluxes in the heterotic string requires thecompactification space to be a generalized Calabi-Yau space [4,60,61,62,63,64,65]. Usingdimensional reduction techniques of the heterotic string on such generalized Calabi-Yaugeometries ˜ Z B reveals that the flux-induced superpotential reads [66,67,64,65,68] W het = Z ˜ Z B ˜Ω ∧ (cid:16) H − i d ˜ J (cid:17) , (4 . H is the non-trivial NS 3-form flux and d ˜ J is often called the geometric flux of thegeneralized 3-fold ˜ Z B . The 3-forms ˜Ω and the 2-form ˜ J are the generalized counterparts ofthe holomorphic 3-form Ω and the (complexified) K¨ahler form J of the associated Calabi-Yau 3-fold Z B . In general the direct evaluation of the heterotic superpotential (4.1) of the3-fold ˜ Z B is rather complicated, therefore we argue here that under certain circumstancesthe heterotic fluxes can be computed from the periods of the original 3-fold Z B .It is instructive to examine first the fluxes of the heterotic string compactified on the N = 2 background T × K
3. For this particular geometry the analyzed fluxes induce In the context of generalized Calabi-Yau spaces ˜ J and ˜Ω are in general not closed with respectto the de Rahm differential d .
26 deformation to the non-K¨ahler geometry ˜ K , which is a non-trivial toroidal bundle π : T → ˜ K → K K K . Choosing a good opencovering U = { U α } of the K ϕ ( k ) αβ : U αβ → R , k = 1 , U αβ = U α ∩ U β . These transition functions patch together the angularcoordinates of the two circles S × S in the torodial fibers. Due to the periodicity of theangular variables the transition functions fulfill on triple overlaps U αβγ = U α ∩ U β ∩ U γ the condition ε ( k ) αβγ = 12 π (cid:16) ϕ ( k ) αβ − ϕ ( k ) αγ + ϕ ( k ) βγ (cid:17) ∈ Z , k = 1 , . The constructed functions ε ( k ) : U αβγ → Z specify 2-cocycles in the ˇCech cohomologygroup ˇ H ( K , Z ). The classes ε ( k ) correspond to the Euler classes e ( k ) of the two circularbundles in the integral de Rham cohomology H ( K , Z ). The non-K¨ahler manifold ˜ K is equipped with the hermitian form ˜ J and the holomor-phic 3-form ˜Ω [70,71] ˜ J = π ∗ J K − S i θ (1) ∧ θ (2) , ˜Ω = ω , ∧ ( θ (1) + iθ (2) ) . Here θ ( k ) , k = 1 , , are the two 1-forms of the toroidal fibers, while J K is the (complexified)K¨ahler form and ω , is the holomorphic 2-form of the K S is the (complexified)Volume modulus of the toroidal fiber. On-shell the value of the volume modulus S becomesstabilized at S = i [71], since the equations of motions impose the torsional constraint[4,60,61] H = ( ∂ − ¯ ∂ ) ˜ J . (4 . For details and background material on ˇCech cohomology and on the construction of theEuler classes we refer the interested reader, for instance, to ref. [72]. For simplicity, we ignore a warp factor in front of the K¨ahler form J K , as it is not relevantfor the analysis of the superpotential. Also note that in our conventions the imaginary part of ˜ J corresponds to the hermitian volume form. The stabilization of volume moduli in the context of heterotic string compactifications withfluxes is also discussed in refs. [62,67].
27s the two-forms dθ ( k ) restrict to the Euler classes e ( k ) on the K K encodes the background fluxes d ˜ J = − i S ( π ∗ e (1) ∧ θ (2) − π ∗ e (2) ∧ θ (1) ) , H = π ∗ e (1) ∧ θ (1) + π ∗ e (2) ∧ θ (2) , where the H -flux is determined by imposing the torsional constraint (4.2) for the on-shellvalue S = i of the fiber volume. Then evaluating the superpotential (4.1) with these fluxesyields W het = Z ˜ K ˜Ω ∧ ( H − i d ˜ J ) = Z C H dz ∧ ω , − iS Z C J dz ∧ ω , . (4 . K are integrated out, andin a second step the resulting period integrals of the K dz ∧ ω , on the original 3-fold T × K C H and C J , which are Poincar´e dual to the integral 3-forms e (1) ∧ dy − e (2) ∧ dx and e (1) ∧ dx + e (2) ∧ dy .Note that the structure of the derived superpotential is in agreement with the super-potential periods obtained in ref. [36].The idea is now to generalize the construction by “twisting” the fibers of the ellipticallyfibered 3-fold π : Z B → B with a section σ : B → Z B , such that we arrive at the generalizedCalabi-Yau 3-fold ˜ Z B . In order to eventually relate the periods of the two manifolds Z B and˜ Z B , we first translate the 3-form cohomology of the 3-fold Z B to appropriate cohomologygroups on the common base B . This is achieved with the Leray-Serre spectral sequence,which associates the cohomology of a fiber bundle to cohomology groups on the base.Let U = { U α } be a good open covering of the base B . Then the cohomology group H k ( Z B , Z ) is iteratively approximated by the Leray-Serre spectral sequence. The leadingorder E terms of the spectral sequence read [72] E p,q = ˇ H p ( B, H q ) ≃ H p ( B, H q ) . Here the (pre-)sheaf H q of the base B is defined by assigning to each open set U thegroup H q ( U ) = H q ( π − U, Z ), and the inclusion of open sets ι VU : V ֒ → U induces the28omomorphism ι VU ∗ : H q ( U ) → H q ( V ) via pullback of forms. Then the spectral sequenceabuts to H ( Z B , Z ), and we get H ( Z B , Z ) ≃ M n =0 E n, − n = M n =0 H n ( B, H − n ) . Due to the simple connectedness of the examined Calabi-Yau 3-fold Z B we arrive at thesimplified relation H ( Z B , Z ) ≃ E , = ˇ H ( B, H ) ≃ H ( B, H ) . (4 . H is not locally constant, because the dimension of the sheaf H differs at a singular fiber from the dimension of the sheaf H at a generic regular fiber.In terms of the open covering U a ˇCech cohomology element ε in ˇ H ( B, H ) is a mapthat assigns to each triple intersection set U αβγ an element in H ( U αβγ ) and fulfills thecocycle condition on quartic intersections U αβγδ ρ δ ◦ ε )( U αβγ ) − ( ρ γ ◦ ε )( U αβδ ) + ( ρ γ ◦ ε )( U αβδ ) − ( ρ α ◦ ε )( U βγδ ) . The map ρ δ , for instance, is the pull-back induced from the inclusion ι δ : U αβγ ֒ → U αβγδ .Then the cohomology element ε is called a 2-cocycle with coefficients in the (pre-)sheaf H , and it is non-trivial if it does not arise form a 1-cochain on double intersections U αβ .To proceed we assume that the generalized Calabi-Yau manifold ˜ Z B is also fibered˜ π : ˜ Z B → B over the same base B and that it arises from “twisting” the elliptic fibers ofthe 3-fold Z B . This “twist” is measured by the 1-cochain ϕ , which assigns to each doubleintersection U αβ an element in H ( U αβ ) ⊗ Z R and which captures the distortion of theangular variable of the 1-cycles in the elliptic fibers of the original 3-fold Z B .In general the 1-chain ϕ does not fulfill the cocycle condition due to the periodicityof the angular variables of the 1-cycles. Instead we find on triple intersections U αβγ ε : U αβγ π [( ρ γ ◦ ϕ )( U αβ ) − ( ρ β ◦ ϕ )( U αγ ) + ( ρ α ◦ ϕ )( U βγ )] ∈ H ( U αβγ ) , which defines a 2-cocycle in ˇ H ( B, H ) characterizing the “twist” of the 3-fold ˜ Z B . Strictly speaking the first relation is not an equality ‘ ≃ ’ but an inclusion ‘ ⊆ ’, because weignore the “higher order corrections” from the spectral sequence. This implies that some of theelements on the right hand side might actually be trivial in H ( Z B , Z ). e in H ( Z B , Z ), which corresponds to the ˇCech cohomologyelement ε in ˇ H ( B, H ), is explicitly constructed. Namely, there are 1-forms ξ α defined onthe open sets U α , which are exact on double overlaps U αβ [72]12 π dϕ ( U αβ ) = ρ β ( ξ α ) − ρ α ( ξ β ) . (4 . dξ α patch together to a global 2-form s e in H ( B, H ), which inturn can be identified with the 3-form e in H ( Z B , Z ) according to (4 . Z B , we need to get a handle onthe 2-form ˜ J in the superpotential (4.1). Due to the fibered structure of the 3-fold Z B theK¨ahler form J splits into two pieces J = π ∗ J B + J F , where J B = σ ∗ J , J F = J − π ∗ J B , and J F = S ω F in terms of the integral generator ω F and the (complexified) K¨ahler volume of the generic elliptic fiber. Then upon the “twist”to the 3-fold ˜ Z B the K¨ahler form J is transformed into the 2-form˜ J = ˜ π ∗ J B + ˜ J F = ˜ π ∗ J B + S ˜ ω F . The 2-form ˜ ω F is defined on each open-set ˜ π − U α by˜ ω F | ˜ π − U α = ω F | ˜ π − U α + ξ α , where we now view ξ α as a two form in the open set ˜ π − U α . Due to the “twist” the 2-forms˜ ω F , which are defined on open sets, patch together to a global 2-form on the 3-fold ˜ Z B .Furthermore, as a consequence of eq. (4.5) we observe that d ˜ J = S d ˜ ω F = S s e , (4 . s e in H ( B, H ).In order to evaluate the heterotic superpotential (4.1) we express the 3-forms of ˜Ω, H and d ˜ J of the “twisted” 3-fold ˜ Z B as elements s Ω , s H and s e of the sheaf cohomology H ( B, H ⊗ C ). Using eq. (4.4) we induce s Ω from the holomorphic 3-form Ω in H , ( Z B )and the NS flux s H from an integral 3-form in H ( Z B , Z ). Furthermore, we also inherit30he the pairing h· , ·i on H ( B, H ⊗ C ) from the 3-form pairing R Z B · ∧ · on the Calabi-Yau3-fold Z B . Then the superpotential (4.1) for the “twisted” manifold ˜ Z B becomes W het = h s Ω , s H i − i S h s Ω , s e i = Z C H Ω − iS Z C J Ω . (4 . s H and s e to their Poincar´edual 3-cycles C H and C J in the original Calabi-Yau manifold Z B .In the context of heterotic string compactifications on the 3-fold Z B the presentedarguments provide further evidence for the encountered structure of the closed-string pe-riods in eq. (3.1). In particular we find that the complex modulus S should be identifiedwith the complexified volume of the generic elliptic fiber.There is, however, a cautious remark overdue. We tacitly assumed that the manifold˜ Z B can be constructed by simply “twisting” the elliptic fibers of Z B . In general, how-ever, we expect that such a construction is obstructed and additional modifications arenecessary to arrive at a “true” generalized Calabi-Yau manifold. A detailed analysis ofsuch obstructions is beyond the scope of this note. However, we believe that the outlinedconstruction is still suitable to anticipate the (geometric) flux quanta, which are responsi-ble for the transition to the generalized Calabi-Yau manifold ˜ Z B to leading order. Fromthe duality perspective of the previous section we actually expect further corrections tothe superpotential (4.7). These corrections should be suppressed in the large fiber limitIm S → ∞ . It is in this limit, in which we expect the “twisting” construction to becomeaccurate. W F ( X B )The F-theory prediction from the last term in (3.1) is the equality, up to finite S corrections, of certain 4-fold period integrals on X B and the Chern-Simons superpotentialon Z B , for appropriate choice of G ∈ H ( X B ) and a connection on E → Z B . The generalrelation of this type has already been described in sect. 2.4 where we used that the 3-fold Z B may be viewed as a ’boundary’ within the F-theory 4-fold X B in the s.d. limit.Here we complete the argument and discuss the map of the deformation spaces by usinghypersurface representations for X ♯ and Z B . This will also lead to a direct identificationof the open-closed dual 4-fold geometries for type II branes and the local mirror geometriesfor (heterotic) bundles of [22,23]. 31o this end, we represent the s.d. limit X ♯B of the F-theory 4-fold X B as a reduciblefiber of a CY 5-fold W obtained by fibering X B over C as in [39,23]. Let µ be the localcoordinate on the base C which serves as a deformation space for the 4-fold fiber X B . Westart from the Weierstrass form p W = y + x + x X α,β s − α ˜ s α µ − β a α,β f α ( x k ) + X α,β s − α ˜ s α µ − β b α,β g α ( x k ) , where f α ( x k ) and g α ( x k ) are functions of the coordinates on the two-dimensional base B of the K3 fibration of the 4-fold X B . Moreover ( y, x ) and ( s, ˜ s ) can be thought of as(homogeneous) coordinates on the elliptic fiber and the base P of the K3 fiber, respec-tively. Finally a α,β , b α,β are some complex constants entering the complex structure of W . The fiber of W → C over a point p ∈ C represents a smooth F-theory 4-fold X B with a complex structure determined by the values of the constants a α,β , b α,β and of thecoordinate µ at p .Tuning the complex structure of W by choosing a α,β = 0 for α + β > b α,β = 0for α + β >
6, the central fiber of W at µ = 0 acquires a non-minimal singularity at y = x = s = 0, which can be blown up by y = ρ y, x = ρ x, s = ρs, µ = ρµ, to obtain the hypersurface p W ♯ = y + x + x X α,β s − α ˜ s α µ − β ρ − α − β f α ( x k ) + X α,β s − α ˜ s α µ − β ρ − α − β g α ( x k ) , (4 . X ♯ = X ∪ X with two components X i defined by ρ = 0 and µ = 0, respectively. The component ρ = 0 is described by p X = p + p + ,p = y + x + xf ( x k ) + g ( x k ) ,p + = x X α> s − α µ α f α ( x k ) + X α> s − α µ α g α ( x k ) , (4 . µ into the two poly-nomials p and p + for later use. The hypersurface X is a fibration X → B with fiber a The non-zero constants a α,β , b α,β are set to one in the following. S . The expressions in (4.9) are sections of line bundles, specificallythe anti-canonical bundle L = K − B , a line bundle M over B that enters the definitionof the fibration X B → B and a bundle N associated with a C ∗ symmetry acting on thehomogeneous coordinates ( y, x, s, µ ). The powers of the line bundles appearing in thesesections are p X y x s µ f α ( x k ) g α ( x k ) L M α α N . p X ∈ Γ( L ⊗ M ⊗ N ).The hypersurface X has a positive first Chern class c ( X ) = c ( N ) and the CY3-fold Z B is embedded in X as the divisor µ = 0, p Z B = p X ∩ { µ = 0 } = p , verifying a claim that was needed in the argument of sect. 2.4. According to the picture ofF-theory/heterotic duality developped in [15,12], the polynomial p + containing the positivepowers in s describes part of the bundle data in a single E factor of the heterotic stringcompactified on Z B . Using a different argument, based on the type IIA string compactifiedon fibrations of ADE singularities, more general n -fold geometries ˆ X of the general form(4.9) have been obtained in [22,23] as local mirror geometries of bundles with arbitrarystructure group on elliptic fibrations. Mirror symmetry gives an entirely explicit mapbetween the moduli of a given toric n -fold and the geometric data of a G bundle on a toric n − Z B , which applies to any geometry ˆ X of the form (4.9) [23]. The application ofthese methods will be illustrated at the hand of selected examples in sects. 6 and 7.A special case of the above discussion is the one, where the heterotic gauge sectoris not a smooth bundle, but includes also non-perturbative small instantons [49]. TheF-theory interpretation of these heterotic 5-branes as a blow up of the base of ellipticfibration X B → B has been studied in detail in [15,39,13]; see also refs. [23,73] for detailsin the case of toric hypersurfaces and ref. [74] for an elegant discussion of the moduli spacein M-theory.From the point of Hodge variations and brane superpotentials this is in fact the mostsimple case, starting from the approach of [8,9,11], as the brane moduli of the type II sidemap to moduli of the heterotic 5-brane. An explicit example from [10] will be discussed insect. 7 33 .3. Type II / heterotic map The above argument also provides a means to describe an explicit map between atype II brane compactification on Z B and a heterotic bundle compactification on Z B . Thekey point is again the afore mentioned relation ( C
2) between the large volume limit of thefibration π : X A → P and the s.d. limit of the F-theory 4-fold X B . The relation betweenthe F-theory 4-fold geometry, the heterotic bundle on Z B and the type II branes on Z B isconcisely summarized by the following diagram: Z A → X A → P mirror symmetry (cid:15) (cid:15) large base + local limit / / Z A → X ncA ( L ) → C local mirror symmetry (cid:15) (cid:15) X B stable deg + local limit / / ˆ X ( E ) (4 . X ncA ( L ) of an A -type bundle L onthe 3-fold Z A sits in the compact 4-fold X A mirror to X B . The details of the bundle L areencoded in the toric resolution of the central fiber Z A at the origin 0 ∈ C , as described interms of toric polyhedra in refs. [16,18,10]. The limit consists of concentrating on a localneighbourhood of the point 0 ∈ P and taking the large volume limit of P base.The lower row describes how the heterotic bundle E on the elliptic manifold Z B dualto F-theory on X B is captured by a local mirror geometry of the form (4.9). Assumingthat the large base/local limit commutes with mirror symmetry, the diagram is completedto the right by another vertical arrow, which represents local mirror symmetry of the non-compact manifolds. The mirror of the open closed dual X ncA ( L ) has been previously called X ncB ( E ), and we see that commutativity of the diagram requires that the open-closed dual X ncB ( E ) is the same as the heterotic dual ˆ X ( E ). Indeed, the hypersurface equations for G = SU ( N ) given in ref. [23] for the heterotic 4-fold ˆ X and in ref. [10] for the open-closed4-fold X ncB can be both written in the form p ( ˆ X ) = p ( Z B ) + v p + (Σ) (heterotic/F-theory duality) p ( X ncB ) = P ( Z B ) + v Q ( D ) (open-closed duality) (4 . v is a local coordinate defined on the cylinder related to s in (4.9). In both cases, the v term specifies the 3-fold Z B on which the type II/heterotic string is compactified. Inthe type II context, Q ( D ) = 0 is the hypersurface D ⊂ Z B , which is part of the definition34f the B -type brane [16,18,10]. In the heterotic dual of [23], p + (Σ) = 0 specifies the SU ( n )spectral cover [12].The agreement of the local geometries dual to the type II/heterotic compactificationon Z B predicted by the commutativity of (4.11) is now obvious with the identification type II/heterotic map: P ( Z B ) = p ( Z B ) , Q ( D ) = p + (Σ) . (4 . H ( Z B , D ) to the periods of the 4-fold X ncB in the context of open-closedduality, carry also over to the heterotic string setting for G = SU ( N ). More ambitiously,one would like to have an explicit relation between the 4-fold periods and the holomorphicChern Simons integral also for a heterotic bundle with general structure group G . Theapproach of refs. [22,23] gives an explicit map from the the moduli of a G bundle on Z B to alocal mirror geometry ˆ X for any G and evaluation of the periods of ˆ X gives the 4-fold side.A computation on the heterotic side could proceed by a generalization of the arguments ofsect. 2.3, e.g. by constructing the sections of the bundle from the more general approachesto G bundles described in [12,31]. In sect. 8 we outline a possible alternative route, usinga conjectural relation between two two-dimensional thories associated with the 3-fold andthe 4-fold compactification.
5. Type II/heterotic duality in two space-time dimensions
In the previous sections we demonstrated the chain of dualities in eq. (3.3) by match-ing the holomorphic superpotentials of the various dual theories. In this section we furthersupplement this analysis by relating the two-dimensional low energy effective theories ofthe type IIA compactificatons on the 4-folds X A and X B with the dual heterotic compact-ification on T × Z B . Many aspects of the type II/heterotic duality on the level of the lowenergy effective action are already examined in ref. [44]. We further extend this discussionhere. 35or the afore mentioned string compactifications the low energy effective theory isdescribed by two-dimensional N = (2 ,
2) supergravity. Chiral multiplets ϕ and twistedchiral multiplets ˜ ϕ comprise the dynamical degrees of freedom of these supergravity the-ories [75,76]. In a dimensional reduction of four-dimensional N = 1 theories the two-dimensional chiral multiplets/twisted chiral multiplets arise from four-dimensional chiralmultiplets/vector multiplets, respectively.The scalar potential of the two-dimensional N = (2 ,
2) Lagrangian arises from theholomorphic chiral and twisted chiral superpotentials W ( ϕ ) and f W ( ˜ ϕ ), and the kineticterms are specified by the two-dimensional K¨ahler potential K (2) ( ϕ, ¯ ϕ, ˜ ϕ, ¯˜ ϕ ) = K (2) ( ϕ, ¯ ϕ ) + e K (2) ( ˜ ϕ, ¯˜ ϕ ) . (5 . K (2) and e K (2) can be thought of individual K¨ahler potentials for the chiral andtwisted chiral sectors. In this section we mainly focus on the K¨ahler potential (5.1) tofurther establish the type II/heterotic string duality of eq. (3.3). The low energy degrees of freedom of type IIA compactifications on the Calabi-Yau4-fold X are the twisted chiral multiplets T A , A = 1 , . . . , h , ( X ) and the chiral multiplets z I , I = 1 , . . . , h , ( X ). They arise from the K¨ahler and the complex structure moduli ofthe 4-fold X . Then the tree-level K¨ahler potential is given by [44] K (2)IIA = K (2)CS ( z, ¯ z ) + e K (2)K ( T, ¯ T ) = − ln Y IIACS ( z, ¯ z ) − ln e Y IIAK ( T, ¯ T ) , (5 . K (2)CS for the complex structure moduli is determinedby Y IIA CS ( z, ¯ z ) = Z X Ω( z ) ∧ ¯Ω(¯ z ) , (5 . Note that these two-dimensional theories describe the effective space-time theory and notthe two-dimensional field theory of the underlying microscopic string worldsheet. This splitting of the K¨ahler potential does not represent the most general form. In fact ingeneral the target space metric need not even be K¨ahler [75]. The given ansatz, however, sufficesfor our purposes. In two dimensions the graviton and the dilaton are not dynamical [77]. For h , ( X ) = 0 there are additional h , chiral multiplets, which we do not take into accounthere. With these multiplets the simple splitting ansatz (5.1) ceases to be sufficient [44].
36n terms of the holomorphic (4 ,
0) form Ω of the Calabi-Yau X . In the large radius regimethe twisted potential e K (2)K for the K¨ahler moduli reads e Y IIAK = 14! Z X J = 14! X A,B,C,D K ABCD ( T A − ¯ T A )( T B − ¯ T B )( T C − ¯ T C )( T D − ¯ T D ) , (5 . K ABCD the topological intersection numbers of the 4-fold X . The K¨ahler moduli T A appear in the expansion of the complexified K¨ahler form B + iJ = T A ω A , ω A ∈ H ( X, Z ),where B and J are the NS 2-form and the real K¨ahler form, respectively. Finally, in thepresence of background fluxes, we obtain the holomorphic superpotentials [24,50] W ( z ) = Z X Ω ∧ F hor , f W ( t ) = Z X e B + iJ ∧ F ver . (5 . F hor ∈ H hor ( X ) is a non-trivial horizontal RR 4-form flux, whereas F ver ∈ H ev ver ( X )is a non-trivial even-dimensional vertical RR flux. The twisted chiral superpotential f W receives non-perturbative worldsheet corrections away from the large radius point [78,79]. X A and X B We now turn to the type IIA compactification on the special Calabi-Yau 4-fold X A .As discussed in sect. 4.1. the 4-fold geometry X A is a fibration over the P base, wherethe generic fiber is the Calabi-Yau 3-fold Z A . Geometries of this type have been studiedpreviously in [79,44] and we extend the discussion here to fibrations with singular fibers,which support the brane/bundle degrees of freedom in the context of open-closed/heteroticduality.For the divisor D S dual to the base this implies Z D S c ( X A ) = χ ( Z A ) . (5 . c ( X A ) is the third Chern class of the 4-fold X A and χ ( Z B ) is the Euler characteristicof 3-fold Z A . Hence the divisor D S is homologous to the generic (non-singular) fiber Z A .For type IIA compactified on the 4-fold X A we are interested in the twisted chiralsector, and hence in the twisted K¨ahler potential (5.4). This means we need to obtain theintersection numbers of the fibered 4-fold X A . We use similar arguments as in ref. [56],where the intersection numbers of K The 6- and 8-forms are the magnetic dual fluxes to the RR 4- and 2-form fluxes in type IIA.
37e denote by S the (complexified) K¨ahler modulus that measures the volume of the P base, which is dual to the divisor D S representing the generic fiber Z A . Considernow a divisor H a of the generic fiber Z A . As we move this divisor about the base bymapping it to equivalent divisors in the neighboring generic fibers, we define a divisor D a in the Calabi-Yau 4-fold X A . The remaining (inequivalent) divisors of the 4-fold X A areassociated to singular fibers, and we denote them by ˆ D ˆ a .The 2-forms ω S , ω a and ˆ ω ˆ a , which are dual to the divisors D S , D a and ˆ D ˆ a , furnishnow a basis of the cohomology group H ( X A , Z ), and we denote the corresponding (com-plexified) K¨ahler moduli by S , t a and ˆ t ˆ a . They measure the volume of the P -base, thevolume of the 2-cycles in the generic 3-fold fiber Z A , and the volume of the remaining2-cycles arising from the degenerate fibers.From this analysis we can extract the structure of intersection numbers. Since D S is ahomology representative of the generic fiber it intersects only with the Calabi-Yau divisors D a according to the triple intersection numbers κ abc of the 3-fold Z A . The intersectionnumbers for divisors, which do not involve D S , cannot be further specified by these generalconsiderations. Therefore we find14! K ABCD T A T B T C T D = 13! κ abc S t a t b t c + 14! K ′ αβγδ t ′ α t ′ β t ′ γ t ′ δ , (5 . t ′ α are the K¨ahler moduli ( t a , ˆ t ˆ a ) with their quartic intersection numbers K ′ αβγδ .The twisted K¨ahler potential for the 4-fold X A then reads e Y K ( X A ) = 13! ( S − ¯ S ) X κ abc ( t a − ¯ t a )( t b − ¯ t b )( t c − ¯ t c )+ 14! X K ′ αβγδ ( t ′ α − ¯ t ′ α )( t ′ β − ¯ t ′ β )( t ′ γ − ¯ t ′ γ )( t ′ δ − ¯ t ′ δ ) . (5 . S involves only the moduli t a associated with the bulk fields in the dual compactifications, whereas the brane/bundledegrees of freedom appear in the subleading term. In the decoupling limit Im S → ∞ , Due to monodromies with respect to the degenerate fibers, it may happen that two inequiv-alent divisors H a and H b are identified globally, and hence yield the same divisor D a = D b . Thenwe work on the 3-fold Z A with monodromy-invariant (linear combinations of) divisors such thatonly inequivalent divisors D a are generated on the 4-fold X A . G A ¯ B ( T C ) ∂ µ T A ∂ µ ¯ T ¯ B → G bulka ¯ b ( t c ) ∂ µ t a ∂ µ ¯ t ¯ b + 1Im S G bundleα ¯ β ( t c , t γ ) ∂ µ t α ∂ µ ¯ t ¯ β , illustrating the separation of the physical scales at which the fields in the two sectorsfluctuate. In this limit, the backreaction of the (dual) bulk fields to the (dual) bundlefields vanishes and the latter fluctuate in the fixed background determined by the bulkfields. A more detailed treatment of the heterotic dual will be given below.Analogously to the three contributions to H ( X A , Z ) distinguished above, we candecompose the even-dimensional fluxes F V into three distinct classes F V = f (1) + f (2) ∧ ω S + f (3) , (5 . f (1) and f (2) pull back to even-forms in H ev ( Z B ), while the fluxes f (3) vanish upon pullback to the regular 3-fold fiber Z A . With the vertical fluxes (5.9) the(semi-classical) twisted chiral superpotential f W ( X A ) simplifies to f W ( X A ) = Z Z B e P a t a ω a ∧ ( Sf (1) + f (2) ) + Z X A e P α t ′ α ω ′ α ∧ ( f (1) + f (3) ) , (5 . ω a , ˆ ω ˆ a ) collectively denoted by ω ′ α .Next we turn to the chiral sector of type IIA strings compactified on the mirror 4-fold X B . The K¨ahler potential (5.3) is then expressed in terms of the periods Π Σ = R γ Σ Ω (4 , of the Calabi-Yau 4-fold X B Y CS ( X B ) = X γ Σ ,γ Λ ∈ H ( X B ) Π Σ ( z ) η ΣΛ ¯Π Λ (¯ z ) , (5 . η ΣΛ is the topological intersection paring on H ( X B ). The horizontal backgroundfluxes F H induce the chiral superpotential W ( X B ) given in eq. (3.1), where the quanta N Σ correspond to the integral flux quanta of 4-form flux F H .By 4-fold mirror symmetry the superpotential f W ( X A ) and W ( X B ) are equal on thequantum level. In comparing the semi-classical expression (5.10) for the twisted superpo-tential to the structure of the chiral superpotential (3.1) in the stable degeneration limit(3.2), we observe that the vertical fluxes f (1) , f (2) and f (3) give rise to the flux quanta M Σ , N Σ and ˆ N Σ , respectively. 39 .3. Heterotic string on T × Z B The low energy effective action of the heterotic string compactified on the 4-fold T × Z B together with a (non-trivial) gauge bundle V has in the large radius regime thestructure [44] K (2)het = K (4)het (Φ , ¯Φ) + e K (2)het ( ˜Φ , ¯˜Φ) . (5 . K (4)het coincides with the four-dimensional K¨ahler potential ofthe heterotic string compactified on the Calabi-Yau 3-fold Z B with the gauge bundle V .Apart from the heterotic dilaton, which is not a dynamic field in two dimensions [77],it comprises all the kinetic terms for both the chiral multiplets of the K¨ahler/complexstructure moduli of the 3-fold Z B and the chiral multiplets from the gauge bundle V . TheK¨ahler potential e K (2)het of the twisted chiral multiplet consists of the modes arising fromthe torus T and the gauge fields, which correspond to the vector multiplets in higherdimensions.For heterotic Calabi-Yau compactifications with the standard embedding of the spinconnection the K¨ahler potential K (4)het splits further according to K (4)het = K (4)CS ( z, ¯ z ) + K (4)K ( t, ¯ t ) + . . . , where K (4)CS and K (4)K are the K¨ahler potentials for the chiral complex structure and K¨ahlermoduli z and t of the Calabi-Yau Z B . For a general heterotic string compactification, we donot know of any generic model independent properties of the K¨ahler potential. However,in the context of type IIA/heterotic duality (3.3), we expect a special subsector associatedwith the kinetic terms of the complex structure moduli z a of the 3-fold together with thespecific moduli fields ˆ z ˆ a of the bundle captured by the dual 4-fold.In order to infer some qualitative information about the relevant kinetic terms of themoduli z a and ˆ z ˆ a we briefly discuss the general structure of the bosonic part of the four-dimensional low-energy effective heterotic action in the four-dimensional Einstein frame S (4)het = 12 κ Z d x √ g (cid:18) R (4) − (cid:16) C a ¯ b ∂ µ z a ∂ µ ¯ z ¯ b (cid:17) − (cid:18) B ˆ a ¯ˆ b ∂ µ ˆ z ˆ a ∂ µ ¯ˆ z ¯ˆ b (cid:19) + . . . (cid:19) . (5 . R (4) is the Einstein-Hilbert term, κ is the four-dimensional gravitational couplingconstant. C a ¯ b and B ˆ a ¯ˆ b denote the K¨ahler metrics of the chiral fields z a and ˆ z ˆ a . Forsimplicity cross terms among bulk and bundle moduli and the kinetic terms of other40oduli fields are omitted. Note that the α ′ dependence of the bundle moduli is absorbedinto the K¨ahler metric B ˆ a ¯ˆ b .From a dimensional reduction point of view the bundle moduli ˆ z ˆ a arise from a Kaluza-Klein reduction of the ten-dimensional vector field A (10) , which in terms of four-dimensionalcoordinates x and internal coordinates y enjoys the expansion A (10) ( x, y ) = A (4) µ ( x ) dx µ + X ˆ a (ˆ z ˆ a ( x ) v ˆ a ( y ) + c . c . ) + . . . . The four-dimensional vector A (4) gives rise to the Yang-Mills kinetic term, while the in-ternal vectors fields v ˆ a are integrated out in the dimensional reduction process and yieldthe metric B ˆ a ¯ˆ b B ˆ a ¯ˆ b = 1 V ( Z B ) Z Z B d y √ g α ′ g i ¯ Tr (cid:16) v ˆ a,i ¯ v ¯ˆ b, ¯ (cid:17) . (5 . V ( Z B ) arises due to the Weyl rescaling to the four-dimensional Ein-stein frame, and it compensates the scaling of the (internal) measure d y √ g . Thus thedimensionless quantity α ′ ℓ , where ℓ is the length scale of the internal Calabi-Yau manifold Z B , governs the magnitude of the kinetic terms B ˆ a ¯ˆ b .As discussed in sect. 4.1., the decoupling limit Im S → ∞ defined in ref. [11] is mappedon the heterotic side to the large fiber limit of the elliptically fibered Calabi-Yau 3-fold Z B → B . In order to work in at semi-classical regime, the volume V ( B ) of the base B ,common to the K3 fibration X B → B and the elliptic fibration Z B → B , has to be takenof large volume as well, due to the relations [44] λ − II, d = λ − het, d , V het ( B ) · V II ( B ) = λ − II, d , which follow from the relations λ II, d = λ − het, d , g het = λ − II, d g II in six dimensions [58].As we move away from the stable degeneration point in the dual type IIA description, thevolume of the elliptic fiber in the 3-fold Z B becomes finite while we keep the volume ofthe base large 0 ≪ ℓ F ≪ ℓ B . (5 . ℓ F is the length scale for the generic elliptic fiber and ℓ B is the length scale for thebase. 41s a consequence, as we move away from the stable degeneration point, the bundlecomponents, which scale with the dimensionless quantity g F ≡ α ′ ℓ F , are the dominant contributions to the metric (5.14). The moduli of the spectral cover cor-respond on the (dual) elliptic fiber to vector fields v ˆ a , which are contracted with the metriccomponent scaling as g F . Therefore the bundle moduli ˆ z ˆ a associated to the subbundle E of the spectral cover become relevant.Thus for the heterotic string compactification on the 3-fold Z B with gauge bundle thecomplex structure/bundle moduli space of the pair ( Z B , E ) is governed by the deformationproblem of a family of Calabi-Yau 3-folds Z B together with a family of spectral covers Σ + .As proposed in (3.1), this moduli dependence is encoded in the relative periods Π Σ ( z, ˆ z ) ofthe relative three forms H ( Z B , Σ + ), and therefore in the semi-classical regime the K¨ahlerpotential of the complex structure/bundle moduli space ( Z B , E ) is expressed explicitly by[11,80] K (4)CS ,E = − ln Y CS ,E ( Z B , Σ + ) , Y CS ,E ( Z B , Σ + ) = X γ Σ ,γ Λ ∈ H ( Z B , Σ + ) Π Σ ( z, ˆ z ) η ΣΛ ¯Π Λ (¯ z, ¯ˆ z ) . (5 . η ΣΛ arises from the intersection matrix of the relative cycles γ Σ .This intersection matrix has the form [11] (cid:0) η (cid:1) = (cid:18) η Z B i g F ˆ η Σ + (cid:19) , where η Z B is the topological metric of the absolute cohomology H ( Z B ) and ˆ η Σ + is thetopological metric of the variable cohomology sector H (Σ + ) of the relative cohomologygroup H ( Z B , Σ + ).Note that the structure of the K¨ahler potential (5.16) is also in agreement with themirror K¨ahler potential of type IIA compactified on the 4-fold X A . By the arguments ofsect. 4, the K¨ahler modulus S of the P base of the 4-fold X A is related to the heteroticvolume modulus of the elliptic fiber of the fibration Z B → B . In the large base limit of X A /bundle decoupling limit of ( Z B , V ) the leading order terms are the K¨ahler moduli ofthe 3-fold fiber Z A /complex structure moduli of the 3-fold Z B . These moduli spaces areidentified by mirror symmetry of the 3-fold mirror pair ( Z A , Z B ). The subleading terms42or type IIA on X A in eq. (5.8) should be compared to the subleading bundle moduli termsin eq. (5.16) on the heterotic side.Finally we remark that since the chiral sector of the heterotic string compactificationon T × Z B and on Z B are equivalent ( cf. eq. (5.12)), the identification of the chiral K¨ahlerpotentials in the type IIA/heterotic duality in two space-time dimensions carries over tothe analog identification of K¨ahler potentials in the F-theory/heterotic dual theories infour space-time dimensions discussed in sect. 4.
6. A heterotic bundle on the mirror of the quintic
Our first example will be an N = 1 supersymmetric compactification on the quintic in P and its mirror. This was the first compact manifold for which disc instanton correctedbrane superpotentials have been computed from open string mirror symmetry in [29,30].This computation was confirmed by an A model computation in [81]. An off-shell versionof the superpotential was later obtained in [9,10,11,17], both in the relative cohomologyapproach, eq.(2.1), as well as from open-closed duality, eq.(2.3). Here we follow the treatment in [10,11], In the framework of [82], the mirror pair( X A , X B ) of toric hypersurfaces can be defined by a pair (∆ , ∆ ∗ ) of toric polyhedra, givenin app. B.1 for the concrete example. The h , = 3 K¨ahler moduli t a , a = 1 , ,
3, of thefibration Z A → X A → P describe the volume t = t + t of the generic quintic fiber ofthe type Z A , the volume S = t of the base P and one additional K¨ahler volume ˆ t = t measuring the volume of an exceptional divisor intersecting the singular fiber Z A . Thisdivisor is associated with the vertex ν ⊂ ∆ in eq.(B.1) and its K¨ahler modulus representsan open string deformation of a toric A brane geometry ( Z A , L ) of the class considered in[7]. The hypersurface equation for the mirror 4-fold X B is given by the general expression P ( X B ) = N X i =0 a i M Y j =0 x h ν i ,ν ⋆j i +1 j . (6 . i and j run over the relevant integral points of the polyhedra ∆ and ∆ ∗ ,respectively, and a i are complex coefficients that determine the complex structure of X B .43 similar expression holds for the hypersurface equation of the mirror manifold X A , withthe roles of ∆ and ∆ ∗ exchanged.Instead of writing the full expression, which would be too complicated due to the largenumber of relevant points of ∆ ∗ , we first write a simplified expression in local coordinatesthat displays the quintic fibration of the mirror: P ( X B ) = p + v p + + v − p − , (6 . p = x + x + x + x + x − ( z z ) − / x x x x x ,p + = x + z ( z z ) − / x x x x x , p − = z x . (6 . v is a local coordinate on C ∗ and z a the three complex structure moduli of X B relatedto the afore mentioned K¨ahler moduli of X A by the mirror map, t i = t i ( z . ). In the largevolume limit the leading behavior is t i ( z . ) = πi ln( z i ) + O ( z . ). The special combination z z appearing above is mirror to the volume of the quintic fiber of π : X A → P , Werefer to app. B.1 for further details of the parametrization used here and in the following.Although the above expression for P ( X B ) is oversimplified (most of the coordinates x j in (6.1) have been set to one), it suffices to illustrate the general structure and to sketchthe effect of the decoupling limit, which, again simplifying, corresponds to setting z = 0,removing the term ∼ p − in (6.3). This produces a hypersurface equation of the promisedform (4.12). In particular, p ( Z B ) = 0 defines the mirror of the quintic, which has a singlecomplex structure deformation parametrized by z = z z . The hypersurface D for therelative cohomology space H ( Z B , D ), which specifies the Hodge variation problem, isdefined by p + = 0, that is Z B ⊃ D : x + z ( z z ) − / x x x x x = 0 . (6 . x i = 0 ∀ i and passing to appropriate local coordinates for this patch,the Hodge variation on D is equivalent to that on a quartic K3 surface in P [10]. A more precise description of this process as a local mirror limit is given in ref. [23]. X B and its heterotic dual are exposedin different local coordinates on the ambient space, which put the hypersurface equationinto the form studied in the context of F-theory/heterotic duality in [23]: p = Y + X + Y X Z ( stu + s + t ) − z z Z ( s t u ) ,p + = X − z Y X Z ( stu ) , p − = z X . (6 . Y, X, Z ) are the coordinates on the elliptic fiber, a cubic in P . Again the zero set p = 0 defines the 3-fold geometry Z B , while the polynomials p ± specify the componentsΣ ± of the spectral cover of the heterotic bundle in the two E factors. While p − correspondsto the trivial spectral cover, p + describes a non-trivial componentΣ + : X − z Y Z ( stu ) = 0 . (6 . SU (2) as fol-lows. The intersection of the equation Σ + with the cubic elliptic equation gives six zeros.However these zeros are identified by the Greene-Plesser orbifold group Z , acting on thecoordinates { Z, Y, X } according to { Z, Y, X } → { ρ Z, ρ Y, X } , ρ = 1 , (6 . ρ is a third root of unity. Note that the spectral cover Σ + represents the mostgeneral polynomial of degree two invariant with respect to the orbifold group (6.7). As aconsequence the six zeros become just two distinct zeros in the elliptic fiber E , adding upto zero. Therefore the spectral cover describes a SU (2) bundle on the heterotic manifold Z B . Alternatively one may study the perturbative gauge symmetry of the heterotic com-pactification from studying the singularities of the elliptic fibration X B . The result ofthis procedure, described in detail in the appendix, is that the bundle leads to the gaugesymmetry breaking pattern E × E / / SU (6) × E (6 . SU (2).45 lux superpotential in the decoupling limit To be more precise, the above discussion describes only the data of the bundle geometrizedby F-theory and ignores the ’non-geometric’ part of the bundle arising from fluxes on the7-branes, which may lead to a larger structure group of the bundle, and thus smaller gaugegroup of the compactification then the one described above [13].In particular, to compute the heterotic superpotential (2.7), we have to specify theclass γ of sect. 2.2, which determines the flux number ˆ N Σ in (3.1), and thus the super-potential as a linear combination of the 4-fold periods. This is the heterotic analogue ofchoosing the 5-brane flux on the type II brane (6.4). Since eq.(4.13) identifies the type IIopen string brane modulus z literally with the heterotic bundle modulus in the decou-pling limit Im S → ∞ , the relative cohomology space and the associated Hodge variationproblem is identical to the one studied in the context of type II branes in [11]. Using theidentification γ = ˜ γ between the classes defined in (2.5) and (2.6), the heterotic superpo-tential in the decoupling limit is identical to that for the type II brane computed in sect. 5of [11], see eq. (5.3). We now discuss the corrections to this result for finite Im S . X B According to the arguments of sect. 3, Hodge theory on the F-theory 4-fold X B com-putes further corrections to the superpotential of the type II/heterotic compactification forfinite S . We will now perform a detailed study of the periods of X B using mirror symmetryof the 4-folds ( X A , X B ).Mirror symmetry is vital in two ways. Firstly, it allows to determine the geometricperiods on H ( X B , Z ), appearing as the coefficients of the flux numbers N Σ in (3.1), froman intersection computation on the mirror X A . Secondly, the mirror map t ( z ) can be usedto define preferred local coordinates on the complex structure moduli space M CS ( X B )near a large complex structure point. In the context of open-closed string duality thesetwo steps are central to extracting the large volume world-sheet instanton expansion of theperiods for the mirror A -model geometry X A , as they yield the disc instanton expansionof the superpotential for A -type brane geometry ( Z A , L ) by open-closed duality [10,11]. Inthe present context we use this A model expansion to describe the superpotential W F ( X B )near a large complex structure limit of X B , which by the previous arguments describes46he decoupling limit Im S → ∞ of the dual heterotic compactification ( Z B , E ) near largecomplex structure of Z B . The methods of mirror symmetry for toric 4-fold hypersurfaces used in the followinghave been described in detail in [83,79,84] and we refer to these papers to avoid excessiverepetitions. We work at the large complex structure point of X B defined by the values z a = 0 , a = 1 , , t a ∼ πi ln( z a ) → i ∞ in the K¨ahler moduli of the mirror manifold X A generated by the charge vectors l = ( − − ,l = ( − − ,l = ( 0 − . (6 . F = 14! Z X c J = 14! X a,b,c,d K αβγδ t α t β t γ t δ = 56 ( t + t ) t + 512 ( t + t ) − t = 56 ˇ t ˇ t + (cid:0)
512 ˇ t −
16 ˇ t (cid:1) . (6 . J = P a t a J a = P a ˇ t a ˇ J a denotes the K¨ahler form on X A , with J a , a = 1 , , H , ( X A ) dual to the Mori cone defined by (6.9). In the above, we have introduced thelinear combinationsˇ t = t = t + t , ˇ t = ˆ t − t = − t , ˇ t = S = t , (6 . { ˇ J a } of H , ( X A ) to expose the simple dependence on theK¨ahler modulus ˇ t = Vol( Z A ) of the generic quintic fiber of π : Z A → X A → P .The leading terms of the period vector Π Σ = R γ Σ Ω for X B in the limit z a → X A Π Σ ( X B ) = Z γ Σ Ω( z ) ∼ q ! Z ˜ γ Σ J q , The fact that the large complex structure limit of the 4-fold X B implies a large structurelimit of the dual heterotic 3-fold Z B follows already from the hypersurface equation, eq.(6.5), andis explicit in the monodromy weight filtration of the 4-fold periods discussed below. γ Σ ∈ H ( X B , Z ) refers to a basis of primitive 4-cycles in X B and ˜ γ Σ a basis for the2 q dimensional algebraic cycles in H q ( X A ) , q = 0 , ...,
4, related to the former by mirrorsymmetry. Except for q = 2, there are canonical basis elements for H q ( X A , Z ), given bythe class of a point, the class of X A , the divisors dual to the generators ˇ J a and the curvesdual to these divisors, respectively. On the subspace q = 2 we choose as in [11] the basis γ = D ∩ D , γ = D ∩ D , γ = D ∩ D . Here the D i = { x i = 0 } , i = 0 , .., x i on the ambient space for X A (cpw. eq. (6.1)),which correspond to the vertices of the polyhedron ∆ in (B.1). The classical volumes ofthese basis elements computed from the intersections (6.10) areΠ = 1 , Π ,i = ˇ t i , Π , = 5ˇ t ˇ t , Π , = 52 ˇ t , Π , = 2ˇ t , Π , = 52 ˇ t ˇ t + 53 ˇ t , Π , = −
23 ˇ t , Π , = 56 ˇ t , Π = F , (6 . q on Π q,. denotes the complex dimension of the cycle.The entries of the period vector Π( X B ) are solutions of the Picard-Fuchs system forthe mirror manifold X B with the appropriate leading behavior (6.12) for z a →
0. ThePicard-Fuchs operators can be derived from the toric GKZ system [79,84] and are given ineq. (A.6) in the appendix.The Gauss-Manin system for the period matrix imposes certain integrability condi-tions on the moduli dependence of the periods of a CY n -fold. For n = 2 these conditionsimply that there are no instanton corrections on K3 and for n = 3 they imply the existenceof a prepotential F for the periods. For n = 4 the periods can no longer be integrated toa prepotential, but still satisfy a set of integrability conditions discussed in ref. [11].Applying the integrability condition to the example the leading behavior of Π nearˇ t = i ∞ , is captured by only seven functions denoted by (1 , ˇ t , ˇ t , ˜ F t , ˜ W , ˜ F , ˜ T ). The elevensolutions can be arranged into a period vector of the formΠ = 1Π , = ˇ t , Π , = ˇ t , Π , = ˇ t , Π , = 5ˇ t ˇ t + π , , Π , = − ˜ F t , Π , = − ˜ W , Π , = ˇ t ˜ F t + π , , Π , = ˜ T , Π , = − ˜ F , Π = ˇ t ˜ F + π , (6 . q on Π q,. now labels the monodromy weight filtration w.r.t. to the largevolume monodromy ˇ t a → ˇ t a + 1. 48ince the decoupling limit sends the compact 4-fold X B to its non-compact open-closeddual X ncB , these functions should reproduce the relative 3-fold periods on H ( Z B , D ) invirtue of eq. (2.3). Indeed the four functions (1 , ˇ t , ˜ F t , ˜ F ) converge to the four periods on H ( Z B ) lim ˇ t → i ∞ (1 , ˇ t , ˜ F t , − ˜ F ) = (1 , t, ∂ t F ( t ) , − F ( t ) + t∂ t F ( t )) , (6 . F ( t ) = t + O ( e πit ) is the closed string prepotential on the mirror quintic. The remaining three functions reproduce the three chain integrals on H ( Z B , D ) withnon-trivial ∂γ ∈ H ( D ): lim ˇ t → i ∞ (ˇ t , ˜ W , ˜ T ) = (ˆ t − t, W ( t, ˆ t ) , T ( t, ˆ t )) , (6 . W ( t, ˆ t ) = − t + O ( e πi ˇ t k ), T ( t, ˆ t ) = ˇ t + O ( e πi ˇ t k ), k = 1 ,
2. Inthe context of open-closed duality, the double logarithmic solution W ( t, ˆ t ) of the 4-foldis conjectured [16] to be the generating function of disc instantons in the type II mirrorconfiguration ( Z A , L ), W ( t, ˆ t ) = − t + X β ∞ X k =1 N β q kβ k , similarly as F ( t ) is the generating function of closed string sphere instantons [85]. Inthe above formula, β denotes the homology class of the disc and the N β are the integralOoguri-Vafa disc invariants [86].Since the closed string period vector (6.11) appears twice in (6.13), with coefficients1 and ˇ t = S , respectively, the leading terms of the eleven periods on X B are proportionalto the seven relative periods on H ( Z B , D )lim Im S →∞ Π q,. ∼ (1 , S ) × (1 , t, ∂ t F , − F + t∂ t F )(ˆ t − t, W ( t, ˆ t ) , T ( t, ˆ t )) . A linear combination of these leading terms gives a large S expansion for the superpotentialof the form (3.1). Here and in the following we neglect terms in the geometric periods from polynomials oflower degree in ˇ t i . .3. Finite S corrections: perturbative contributions There are two types of finite S contributions in the 4-fold periods, which correct the3-fold result: linear corrections ∼ S − and exponential corrections ∼ e πiS . In the type IIorientifold where Im S ∼ /g s , the first should correspond to perturbative corrections.These linear corrections are described by the three additional functions π , , π , , π in (6.13) with leading behaviorlim ˇ t → i ∞ π , = f , (ˇ q , ˇ q ) , lim ˇ t → i ∞ π , = −
53 ˇ t + f , (ˇ t , ˇ t , ˇ q , ˇ q )lim ˇ t → i ∞ π = 512 ˇ t −
16 ˇ t + f (ˇ t , ˇ t , ˇ q , ˇ q ) , (6 . S -duality symmetryof the type II string (and the T -duality of the heterotic string) even in the large S limitwhere one ignores the D-instanton corrections ∼ e πiS . The above functions f q,. vanishexponentially in the ˇ q i = e πi ˇ t i for i = 1 , Z B ,but contribute in the interior of the complex structure moduli space of Z B .E.g., the ratio of two periods corresponding to the central charges of an ’ S -dual’ pairof BPS domain walls with classical tension ∼ ˜ F t is Z /Z = S ˜ F t + π , ˜ F t = S + 23 t + ˜ f (ˇ t k , ˇ q k ) + O ( e − π/g s ) . In principle there are various possibilities regarding the fate of S duality. Firstly, therecould be a complicated field redefinition which corrects the relation Im S = g s awayfrom the decoupling limit such that there is an S duality for a redefined field ˜ S includingthese corrections. Such a redefinition is known to be relevant in four-dimensional N = 2compactifications of the heterotic string, where one may define a perturbatively modularinvariant dilaton [87]. On the other hand, duality transformations often originate frommonodromies of the periods in the Calabi-Yau moduli space, which generate simple trans-formations at a boundary of the moduli space, such as Im S = ∞ , but correspond tocomplicated field transformations away from this boundary. Again, such a ’deformation’of a duality transformation is known to happen in the heterotic string [88]. At this pointwe can not decide between these options, or a simple breaking of S -duality, without adetailed study of the monodromy transformations in the three-dimensional moduli spaceof the 4-fold, which beyond the scope of this work.50 .4. D-instanton corrections and Gromov–Witten invariants on the 4-fold There are further exponential corrections ∼ e πiS to the moduli dependent functions ineqs. (6.13). Recall that we are considering here the classical periods of X B , which describethe complex structure moduli space of the 4-fold X B and complex deformations of the dualheterotic bundle compactification on Z B . From the point the type IIA compactificationon X B , obtained by compactifying F-theory on X B × T , these are B model data and donot have an immediate instanton interpretation.However, according to the identification of the decoupling limit in sect. 2, we expectthese B model data to describe D-instanton corrections ∼ e − π/g s to the type II orientifoldon the 3-fold, see (3.3). Lacking a sufficient understanding of the afore mentioned issueof field redefinitions, we will express the expansion in exponentials ∼ e πiS in terms ofGromov–Witten invariants, or rather in terms of integral invariants of Gopakumar–Vafatype, using the multi-cover formula for 4-folds given in [83,79]. These invariants capturethe world-sheet instanton expansion of the A -model on the mirror X A of X B . Note that ifmirror pair ( X A , X B ) supports a duality of the type (3.16), then this expansion capturesworld-sheet and D-instanton corrections computed by the twisted superpotential f W ( X A ),according to the arguments in sect 3.5. However, according to eq. (3.6) such a duality canonly exist if the mirror 4-fold X A is given in terms of a suitable fibration structure, whichis not true for the quintic example of this section (since X A is neither elliptically nor K3fibered), but for other examples considered in sect. 7.The integral A model expansion of the 4-fold is defined by [83,79] Π ,γ = p γ ( t a ) + X β X k> N γβ q β · k k , (6 . ,γ is one of the periods in the q = 2 sector, double logarithmic near the largecomplex structure limit z a = 0, and p a degree two polynomial in the coordinates t a defined by (6.9). Moreover β is a label, which in the A model on the mirror X A specifiesa homology class β ∈ H ( X A , Z ) with exponentiated K¨ahler volume q β = Q a q n a a , q a = e πit a . As discussed above, these K¨ahler moduli of X A map under mirror symmetry to The fact that this multi-cover formula for spheres in a 4-fold is formally the same as themulti-cover formula for discs in a 3-fold [86] is at the heart of the open-closed duality of [16,10,17]. X B , and we use these coordinates to write an expansion for the B model on X B .We restrict here to discuss only the few leading coefficients N γβ for the three linearlyindependent q = 2 periods of X B . We label the ’class’ β by tree integers ( m, n, k ), suchthat N γβ is the coefficient of the exponential exp(2 πi ( mt + nt + kt ) in the basis (6.9).Thus k is the exponent of e πiS in the expansion. Deformation of the closed string prepotential F t The leading term of the period Π , is the closed string prepotential (6.13). This periodis mirror to a 4-cycle in the quintic fiber of X A and depends only on the closed stringvariable t = t + t in the limit Im S → ∞ . The leading terms in the expansion (6.17) ofthe 4-fold period are k = 0 0 1 2 30 0 0 0 01 0 2875 0 02 0 0 1218500 03 0 0 0 951619125 k = 1 0 1 2 30 5 20 0 01 0 8895 33700 6002 0 19440 16721375 630718003 0 − k = 2 0 1 2 30 0 0 0 01 0 0 3060 37502 0 0 5038070 986495003 0 0 19074160 47957485000 k = 3 0 1 2 3 40 0 0 0 0 01 0 0 0 − − . m ( n ). The k = 0 expansion isa power series in the closed string exponential, which displays the independence of theclosed string prepotential on the open string sector. This independence is lost taking intoaccount e πiS corrections, as is expected from the backreaction of the closed string to theopen string degrees of freedom at finite g s .The mixture between the closed and open string sector at finite S is already visible inthe definition of mirror map. In [89,8] it had been observed, that the definition of the flatclosed string coordinate does not depend on the open string moduli in the non-compact The t a are the distinguished flat coordinates of the Gauss-Manin connection. t = t ( z ) for the closed string modulus t = t + t isthe same as in the theory without branes, with z = z z . This is no longer the case forfinite S , as there are corrections to the mirror map of the form t ( z a ) = t ( z ) + e πiS f ( z, ˆ z ). Deformation of disc superpotential W ( t, ˆ t )The leading term of the period Π , is the brane superpotential of [11], which conjecturallycomputes the disc instanton expansion of an A type brane on the quintic. The leadingterms in the expansion (6.17) of the 4-fold period with respect to the corrections e πikS are k = 0 0 1 2 3 4 50 0 20 0 0 0 01 −
320 1600 2040 − −
802 13280 − − − − − k = 1 0 1 2 3 4 50 0 20 0 0 0 01 0 1600 30640 3180 − − − − k = 2 0 1 2 3 4 5 60 0 0 0 0 0 0 01 0 0 2040 3180 480 −
40 02 0 0 679600 55277220 151559040 10282300 − − . Deformation of Π , As discussed in the previous subsections, the corrections to the third period Π , contain S − corrections and are in this sense the most relevant. The leading terms of the expansion536.17) are k = 0 0 1 2 30 0 20 0 01 0 6020 3060 − − − k = 1 0 1 2 30 − −
20 0 01 0 − − − . k = 0 corrections capture the linear corrections discussed in sect. 6.3. These shouldarise from a one-loop effect on the brane; it would be interesting to verify this by anindependent computation.
7. Heterotic five-branes and non-trivial Jacobians
In this section we discuss a number of further examples to illustrate the duality re-lations and the application of the method. The geometries are mostly taken from [10],where the brane superpotential for B -type branes has been already computed. Since thesuperpotential (2.7) for the heterotic compactification on Z B with the appropriate bundle E agrees with the brane superpotential in the decoupling limit, the explicit heterotic su-perpotential in this limit can be read off from the results of [10]. We have performed alsoa computation of the finite S corrections to the heterotic superpotential for the examplesbelow, by the methods described in detail the previous section. The results are of a similargeneral structure as in the quintic case. Detailed expressions for the examples are availableupon request.The main focus of this section will be to describe some additional aspects arisingfrom the point of F-theory and the heterotic compactification on Z B . Let us recall thefollowing basic result on F-theory/heterotic duality which will help to understand thedifferent outcomes in the following examples. The elements of the Hodge group H , ( X B )54f the 4-fold can be roughly divided into the following sets w.r.t. their meaning in the dualheterotic compactification on the CY 3-fold Z B with bundle E (see [15,90,13,]): Generic classes:
The first set arises from the two generic classes from the K3 fiber Y of the K3 fibration X B → B :1. The class E of the fiber of the elliptic fibration Y → P , which is also the ellipticfiber of X B . This curve shrinks in the 4d F-theory limit and does not lead to a fieldin four dimensions;2. The class F of the section of the elliptic fibration Y → P , which provides the universaltensor multiplet associated with the heterotic dilaton. Geometry of Z B : h , ( B ) classes of the base of the K3 fibration X B → B with K3 fiber Y .4. h , ( Z B ) − h , ( B ) − Z B → B . Gauge fields & 5-branes: h , ( Y ) − G pert classes from singular fibers of the elliptic fibration Y → P ,corresponding to the Cartan subgroup of the perturbative gauge group G pert .6. h , ( B ) − h , ( B ) − P bundle B → B withfiber of class F . These blow ups correspond to heterotic 5-branes wrapping a curve C ∈ B .7. The remaining rank G non − pert classes of X B arise from extra singularities of the ellipticfibration, which correspond to the Cartan subgroup of a non-perturbative gauge group G non − pert .Fixing the heterotic 3-fold Z B , one can still vary the 4-fold data in the last group, to choosea bundle E . In the framework of toric geometry, this step can be made very explicit byusing local mirror symmetry of bundles [22]. Starting from the toric 3-fold polyhedronfor Z B one may to ’geometrically engineer’ the bundle in terms of a 4-fold polyhedron,by appropriately adding or removing exceptional divisors, as described in great detail in[23,73]. By the type II/heterotic map (4.13), this is the complement of adding singularfibers to the mirror fibration X A → P in (3.5), to define a toric A type brane on the3-fold mirror Z A [10]. 55he items 5.-7. in the above describe, how an element of H , ( X B ) added in theengineering of the bundle falls into one of the three classes in the last set, depending on therelative location of the exceptional divisor w.r.t. the fibration structure. It follows that the B -type branes in the type II compactification may map to quite different heterotic degreesof freedom under the type II/heterotic map (4.13): perturbative gauge fields, heteroticfive-branes and non-perturbative gauge fields. This variety can be seen already in theexamples of [10], as discussed below. SU (1) : Heterotic five-branes As seen in the previous section, the quintic example of [29,9,10] corresponds to aperturbative heterotic bundle with structure group SU (2). Another example of a branecompactification taken from ref. [10] turns out to have a quite different interpretation.In this case, the brane deformation of the type II string does not translate to a bundlemodulus on the heterotic side under the type II/heterotic map (4.13), but rather to a branemodulus. On the heterotic side, this is a 5-brane representing a small instanton [49].Let us first recall the brane geometry on the type II side, which is defined in [10]as a compactification of a non-compact brane in the non-compact CY O ( − P , i.e. theanti-canonical bundle of P . This example has been very well studied in the context ofopen string mirror symmetry in [89,18,91]. The non-compact CY can be thought of as thelarge fiber limit of an elliptic fibration Z A → P which gives the interesting possibilityto check the result obtained from the compact 4-fold against the disc instanton corrected3-fold superpotential computed by different methods in [89,18,91]. Indeed it was shown in[10], that 4-fold mirror symmetry reproduces the known results for the non-compact branein the large fiber limit, including the normalization computed from the intersections of the4-fold X A . The result for the local result is corrected by instanton corrections for finitefiber volume. Two different 3-fold compactifications of O ( − P were considered in [10], with adifferent model for the elliptic fiber. As the two examples produce very similar results,we discuss here the degree 18 case of [10] in some detail and only briefly comment on thedifference for the degree 9 hypersurface, below. Note that this is a large fiber limit in the type IIA theory compactified on Z A , not thepreviously discussed large fiber limit of the heterotic string compactified on Z B . A cubic in P for the degree 9 and a sixtic in P (1 , ,
3) for the degree 18 hypersurface. B -type brane is defined in [10] by adding a new vertex ν = ( − , , , , −
1) (7 . X B obtained in this way are X B : h , = 4 , h , = 0 , h , = 2796 , χ = 16848(= 0 mod 24) . We refer the interested reader again to app. B for the details on the toric geometry andthe parametrizations used in the following and continue with a non-technical discussion ofthe geometry. The addition of the vertex ν corresponds to the blow up of a divisor in thesingular central fiber of the 4-fold fibration X A → P . The new element in H , ( X A ) isidentified as the deformation parameter of the A -brane on the 3-fold Z A , via open-closedduality.On the mirror side, the blow up modulus corresponds to a new complex structuredeformation parametrizing a holomorphic divisor in Z B . As will be explained now, thisdeformation maps in the heterotic compactification to a modulus moving a heterotic 5-brane that wraps a curve C in the base B of the 3-fold Z B .In appropriate local coordinates, the form (6.2) of the hypersurface equation, exposingthe elliptic fibration of both, Z B and X B , is p = Y + X + ( z z z ) − / Y X Z stu + Z (( z z ) − / ( stu ) + s + t + u ) ,p + = Z (( stu ) + ˆ zs ) , p − = Z ( stu ) . (7 . Z B , reducing to the mirror of the non-compact brane in O ( − P of [89], is defined by the hypersurface D : p + = 0 within Z B defined by p = 0 [10].The hypersurface constraint (7.2) is already in the form to which the methods of [23]can be applied. The relevant component of p + deforming with the modulus ˆ z lies in apatch with s, t, u = 0 and is given byΣ + : Z ( t u + ˆ zs ) = 0 . (7 . z does not involve the coordinates of the elliptic fiber, and thereforeit does not correspond to a bundle modulus. Instead this F-theory geometry describesheterotic 5-branes wrapping a curve C in the base B of the heterotic compactification.As described in detail in [15,39,13] (see also ref. [74]), F-theory describes these heterotic5-branes by a blow ups of the the P bundle B → B .57he toric 4-fold singularities associated with heterotic five-branes of type (7.2) werealso studied in great detail in [23,73]. In the present case, the 5-branes wrap a set ofcurves C in the elliptic fibration Z B → B , defined by the zero of the function f ( s, t, u ) = s ( t u + ˆ zs ). The deformation ˆ z moves the branes on the second component, similarlyas it moves the type II brane in the dual type II compactification on Z B .By the F-theory/heterotic dictionary developed in refs. [15,39,13], the above singular-ity describes a small E instanton, which can be viewed as an M-theory/type IIA 5-brane[49]. Note that there are also exceptional blow up divisors in X B associated with the 5-brane wrapping, which support the elements in H , ( X B ) dual to the world-volume tensorfields on the 5-branes [15,39,13]. However, these K¨ahler blow ups are not relevant for thepurpose of computing the superpotential W ( X B ).The above conclusions may again be cross-checked by analyzing the perturbative gaugesymmetry of the heterotic compactification, which does not changes in this case for ˆ z = 0 E × E / / E × E , (7 . O ( − P in the degree 9 hypersur-face leads to similar results. The 4-fold considered in [10] has the Hodge numbers X B : h , = 6(2) , h , = 0 , h , = 586 , χ = 3600(= 0 mod 24) . and describes a heterotic compactification with 5-branes wrapping a curve given by theequation Σ + : Z s ( t u + ˆ zs ) = 0 . (7 . E × E → E × E .In the decoupling limit Im S → ∞ limit, the heterotic superpotential for the 5-branesin these two cases agrees with the type II brane superpotential computed in sect. 3.2 andapp. B of [10], respectively. See also sect. 5 of [19] for a reconsideration of the first case,with an identical result (Tab.3a/5.2). 58 .2. Non-trivial Jacobians: SU (2) bundle on a degree 9 hypersurface A new aspect of another example of [10] is the appearance of a non-trivial Jacobian J (Σ) of the spectral surface, corresponding to non-zero h , [12]. In this case there areadditional massless fields associated with the Jacobian J ( X B ) = H ( X B , R ) /H ( X B , Z )in the F-theory compactification, and the non-trivial Jacobian of Σ in the heterotic dual[12,31,32].The present example has been considered in sect. 3.3 of [10] and describes a branecompactification on the same degree nine hypersurface Z A as in the previous section,but with a different gauge background. Z A is defined as a hypersurface in the weightedprojective space P (1 , , , ,
3) with hodge numbers and Euler number Z A : h , = 4(2) , h , = 112 , χ = − , (7 . Z A , which are unavailablein the given hypersurface representation.As familiar by now, the technical details on toric geometry are relegated to app. B.The Hodge numbers of the dual F-theory 4-fold X B are X B : h , = 4 , h , = 3 , h , = 246(11) , χ = 1530 = 18 mod 24 . The local form (6.2) of the hypersurface equation for X B , exposing the elliptic fibrationand the hypersurface Z B is p = a Y + a X + Z ( a ( stu ) + a s + a t + a u ) + a Y X Z stu,p + = Y ( a Y + a X Zstu ) , p − = a Y . (7 . p = 0 defines the 3-fold geometry Z B for the compactification of thetype II/heterotic string, while the brane geometry considered in [10] is defined by thehypersurface D : p + = 0. By the type II/heterotic map (4.13), we reinterprete theseequations in terms of a heterotic bundle on Z B . While p − corresponds to the trivialspectral cover, p + describes a component with non-trivial dependence on a single modulusˆ z : Σ + : Y + ˆ zX Zstu = 0 , (7 . z is the brane/bundle deformation. As in the quintic case, Σ + may be identified witha component with structure group SU (2). This is confirmed by a study of the perturbativegauge symmetry of the heterotic compactification, which changes for ˆ z = 0 as E × E / / SU (6) × E . (7 . S → ∞ limit of the heterotic superpotential for this bundle coincides with thetype II result computed in [10]. 59 . ADE Singularities, Kazama-Suzuki models and matrix factorizations In the above we have described how 4-fold mirror symmetry computes quantum correc-tions to the superpotential and the K¨ahler potential of supersymmetric compactificationsto four and lower dimensions with four supercharges. Specifically, these corrections areexpected to correspond to D( − Z B and to world-sheet and space-time instanton correctionsto a (0 ,
2) heterotic string compactification on the same manifold. At present, it is hardto concretely verify these predictions by an independent computation. A particularly neatway to find further evidence for our proposal (in the N = 2 supersymmetric situation)would be to establish a connection with refs. [92]. In these works, considerable progress hasbeen made in understanding corrections to the hyper-multiplet moduli, especially the in-teraction with mirror symmetry. It would be very interesting to study the overlap with thenon-perturbative corrections discussed in the present paper. In this section, we discuss adifferent application of heterotic/F-theory duality which might be viewed as an interestingcorroboration of our main statements, and is also of independent interest. N = 2 supersymmetry It is best again to begin with 8 supercharges. Consider a heterotic string compactificationon a K3 manifold near an ADE singularity with a trivial gauge bundle on the blown up2-spheres. The hypermultiplet moduli space of this heterotic compactification is correctedby α ′ corrections from perturbative and world-sheet instanton effects. It has been shown in[93] that for an A singularity, the heterotic moduli space space in the hyperk¨ahler limit isgiven by the Atiyah-Hitchin manifold, which is also the moduli space of three-dimensional N = 4 SU (2) Yang-Mills theory. This relation between the moduli space of the heteroticstring on a singular K3 and the moduli space of a three-dimensional gauge theory canbe derived and generalized by studying the stable degeneration limit of the dual typeIIA/F-theory 3-fold. Specifically it is shown in refs. [94,95], that the 3-fold X B dual to theheterotic string on an ADE singularity of type G and with a certain local behavior of thegauge bundle V develops a singularity, which ’geometrically engineers’ a three-dimensionalgauge theory of gauge group and matter content depending on G and V , see ref. [96]. Inconnection with the N = 2 version of the decoupling limit Im S → ∞ , eq.(3.11), thisleads to a very concrete relation between the 3-fold period and the world-sheet instantoncorrections to the heterotic hypermultiplet space in the hyperk¨ahler limit. This could be60xplicitly checked against the known result, at least in the case dual to 3d SU (2) SYMtheory. N = 1 supersymmetry The above situation has an interesting N = 1 counter part. Namely, it has been con-jectured in [95] that one may use the heterotic string on a certain 3-fold singularity togeometrically engineer (the moduli space of) interesting 2-dimensional field theories. The3-fold singularities are of the type y + H ( x k ) = 0 , (8 . H ( x k ) describes an ADE surface singularity. The idea is the obvious generalizationof the above, by first applying heterotic/F-theory duality and then exploiting the relationof ref. [24] between similar 4-fold singularities and Kazama-Suzuki models. We here makethis correspondence more precise.Recall that the identification of [24] proceeded through the comparison of the vac-uum and soliton structure of a type IIA compactification on Calabi-Yau four-fold withits superpotential from four-form flux, and the Landau-Ginzburg description [97] of thedeformed Kazama-Suzuki coset models [98]. The four-folds relevant for this connectionare local manifolds that are fibered by singular 2-dimensional ALE spaces and their de-formations. The ADE type of the singularity in the fiber determines the numerator G ofthe N = 2 coset G/H , while the flux determines the denominator H and the level. Moreprecisely, the fluxes studied in [24] are the minimal fluxes corresponding to a minusculeweight of G . These give rise to the so-called SLOHSS models (simply-laced, level one,Hermitian symmetric space), which is the subset of Kazama-Suzuki models admitting aLandau-Ginzburg description. This identification was checked for the A -series in ref. [24]and worked out in detail for D and E in ref. [25]. It has remained an interesting questionto identify the theories for non-minimal flux, see e.g., the conclusions of [25].An important clue to address this question has come from the study of matrix factor-izations and their deformation theory. In particular, it was observed in ref. [26], see alsoref. [99], that the superpotential resulting from the deformation theory of certain matrixfactorization in N = 2 minimal models coincides with the Landau-Ginzburg potential ofa corresponding SLOHSS model. More precisely, the matrix factorizations are associated61ith the fundamental weights of ADE simple Lie algebras via the standard McKay cor-respondence, and the relevant subset are those matrix factorizations corresponding to theminuscule weights. We argue that this coincidence of superpotentials can be explained viaheterotic/F-theory/type II duality.The missing link is provided by ref. [100]. Among the results of this work is thatthe matrix factorizations of ADE minimal models can be used to describe bundles onpartial resolutions (Grassmann blowups) of the threefold singularities of ADE type (8.1)that appear in the above-mentioned conjecture of ref. [95]. The bundles have support onlyon the smooth part of the partial blowup, which is important to apply the arguments ofref. [93].The combination of the last three paragraphs suggests that we should couple theheterotic worldsheet to the matrix factorizations of ref. [100]! This can be implemented byusing the (0 ,
2) linear sigma model [76] resp. (0 ,
2) Landau-Ginzburg models [101], along thelines of [102], in combination with an appropriate non-compact Landau-Ginzburg modelto describe the fibration structure. The resulting strongly coupled heterotic worldsheettheories are conjectured to be dual to those 2-d field theories that are engineered on thefour-fold side. The ADE type of the minimal model is that of the fiber of the four-fold,while the fundamental weight specifies the choice of four-form flux.As formulated, the above conjecture makes sense for all, fundamental weights. Themain testable prediction is thus the coincidence of the deformation superpotentials of thehigher rank matrix factorizations corresponding to non-minuscule fundamental weightswith the appropriate periods of the four-folds of refs. [24,25]. Note that the Kazama-Suzuki models only appear for the minuscule weights, and that we have not covered thecase of fluxes corresponding to non-fundamental weights. We plan to return to thesequestions in the near future.
9. Conclusions
In this note we study the variation of Hodge structure of the complex structure mod-uli space of certain Calabi-Yau 4-folds. These moduli spaces capture certain effectivecouplings of the N = 1 supergravity theory arising from the associated F-theory 4-foldcompactification. Furthermore, through a chain of dualities we relate such F-theory sce-narios to heterotic compactifications with non-trivial gauge bundle and small instanton5-branes and to type II compactifications with branes.62he connection to the heterotic string is made through the stable degeneration limitof the F-theory 4-fold [15,12,39]. Taking this limit specifies the corresponding heteroticgeometry. Due to the employed F-theory/heterotic duality the resulting heterotic geometryis given in terms of elliptically fibered Calabi-Yau 3-folds. Furthermore, in the simplestcases, the geometric bundle moduli are described in terms of the spectral cover, which isalso encoded in the 4-fold geometry in the stable degeneration limit [12]. Alternatively,depending on the details of the F-theory 4-fold, we describe the moduli space of heterotic5-branes instead of bundle moduli. On the other hand the link to the open-closed type IIstring theories is achieved through the weak coupling limit [11], and it realizes the open-closed duality introduced in ref. [16,18,17].We argue that the two distinct limits to the heterotic string and to the open-closedstring map the variation of Hodge structure of the F-theory Calabi-Yau 4-fold to thevariation of mixed Hodge structure of the corresponding Calabi-Yau 3-fold relative to acertain divisor. For the heterotic string this divisor is either identified with the spectralcover of the heterotic bundle or with the embedding of small instantons. In the contextof open-closed type II geometries the divisor encodes a certain class of brane deformationsas studied in refs. [8,9,10,17,11,19,20,103,104].Starting from the F-theory 4-fold geometry we discuss in detail non-trivial backgroundfluxes and compute the N = 1 superpotential, which couples to the moduli fields describedby the variation of Hodge structure. We trace these F-terms along the chain of dualitiesto the open-closed and heterotic string compactifications. For the heterotic string we findthat, depending on the characteristics of the 4-fold flux quanta, these fluxes either deformthe bulk geometry of the heterotic string to generalized Calabi-Yau manifolds [69,70,71],or they give rise to superpotential terms for the bundle/five-brane moduli fields. Thesuperpotentials associated to the flux quanta encode obstructions to deformations of thespectral cover. Furthermore, we show that in the stable degeneration limit the holomorphicChern-Simons functional of the heterotic gauge bundle gives rise to these F-terms for thegeometric bundle moduli.The underlying 4-fold description of the heterotic and the type II strings allows us toextract (non-perturbative) corrections to the stable degeneration limit and the weak cou-pling limit respectively. We discuss the nature of these corrections, and we find that theyencode world sheet instanton, D-instanton and space-time instanton corrections dependingon the specific theory in the analyzed web of dualities. In order to exhibit the origin of63hese corrections we compare our analysis with the analog N = 2 scenarios, which havebeen studied in detail in refs. [23,36].Apart from these F-term couplings we demonstrate that our techniques are also suit-able to extract the K¨ahler potentials for the metrics of the studied moduli spaces in ap-propriate semi-classical regimes. In ref. [11] the connection to the open-closed K¨ahlerpotential for 3-fold compactifications with 7-branes has been developed. Here, startingfrom the K¨ahler potential of the complex structure moduli space of the Calabi-Yau 4-fold,we also extract the corresponding K¨ahler potential associated to the combined modulispace of the complex structure and certain moduli of the heterotic gauge bundle. In lead-ing order these K¨ahler potentials are in agreement with the results obtained by dimensionalreduction of higher dimensional supergravity theories [44,80]. In addition our calculationpredicts subleading corrections.Thus, the used duality relations together with the presented computational techniquesoffer novel tools to extract (non-perturbative) corrections to N = 1 string compactifica-tions arising from F-theory, from heterotic strings or from type II strings in the presenceof branes. It would be interesting to confirm the anticipated quantum corrections byindependent computations and to understand in greater detail the physics of various (non-perturbative) corrections discussed here. In particular, our analysis suggests a connectionto the quantum corrections in the hypermultiplet sector of N = 2 compactifications ana-lyzed in refs. [92].Our techniques should also be useful to address phenomenological interesting ques-tions in the context of F-theory, type II or heterotic string compactifications. As discussedin sects. 5,6, the finite S corrections to the superpotential capture the backreaction of thegeometric moduli to the bundle moduli. Such corrections are a new and important ingre-dient in fixing the bundle moduli in phenomenological applications, as emphasized, e.g., inref. [35]. Thus the calculated (quantum corrected) superpotentials provide a starting pointto investigate moduli stabilization and/or supersymmetry breaking for the class of modelsdiscussed here. In the context of the heterotic string it seems plausible that our approachcan be extended to more general heterotic bundle configurations, which can be describedin terms of monad constructions [101,105]. Such an extension is not only interesting froma conceptual point of view, but in addition it also gives a handle on analyzing the effectivetheory of phenomenologically appealing heterotic bundle configurations as discussed, forinstance, in ref. [106]. 64n section 8, we propose an explanation, and conjecture an extension of, an observa-tion originally made by Warner, which relates the deformation superpotential of matrixfactorizations of minimal models to the flux superpotential of local four-folds near an ADEsingularity. One of the results of this connection is the suggestion that (higher rank) ma-trix factorizations should also play a role in constructing the (0 ,
2) worldsheet theories ofheterotic strings.The presented approach to calculate deformation superpotentials by studying ad-equate Hodge problems is ultimately linked to the derivation of effective obstruc-tion superpotentials with matrix factorization or, more generally, worldsheet techniques[107,108,109,110,111,112]. While the latter approach leads to effective superpotentials upto field redefinitions, our computations give rise to effective superpotentials in terms offlat coordinates due to the underlying integrability of the associated Hodge problem. Itwould be interesting to explore the physical origin and the necessary conditions for theemergence of such a flat structure in the context of the deformation spaces studied in thisnote.
Acknowledgements:
We would like to thank Mina Aganagic, Ilka Brunner, Shamit Kachru, Wolfgang Lerche,Jan Louis, Dieter L¨ust, Masoud Soroush and Nick Warner for discussions and correspon-dence. We would also like to thank Albrecht Klemm for coordinating the submission ofrelated work. The work of H.J. is supported by the Stanford Institute of TheoreticalPhysics and by the NSF Grant 0244728. The work of P.M. is supported by the program“Origin and Structure of the Universe” of the German Excellence Initiative.
Appendix A. Some toric data for the examples
A.1. The quintic in P (1 , , , , Parametrization of the hypersurface constraints
The toric polyhedra for the example considered in sect. 6 are defined as the convex hull ofthe vertices 65 ν = ( 0 , , , ,
0) ∆ ∗ ν ⋆ = ( 0 , , , , ν = ( − , , , , ν ⋆ = ( 1 , − , , , ν = ( 0 , − , , , ν ⋆ = ( 1 , , − , , ν = ( 0 , , − , , ν ⋆ = ( 1 , , , − , ν = ( 0 , , , − , ν ⋆ = ( 1 , , , , ν = ( 1 , , , , ν ⋆ = ( − , , , , ν = ( 0 , , , , − ν ⋆ = ( − , , , , − ν = ( − , , , , − ν ⋆ = ( 0 , − , , , ν = ( − , , , , ν ⋆ = ( 0 , , − , , ν ⋆ = ( 0 , , , − , ν ⋆ = ( 0 , , , ,
1) (A.1)The local coordinates in the expressions (6.3) and (6.5) are defined by the following selec-tions Ξ and Ξ of points of ∆ ∗ , respectively:Ξ x ′ − x − x − x − x a − − b − Z Y − X ′ − s − t − u a − b − − { x i } and { Z, Y, X ′ , s, t, u } may be associatedwith the ’heterotic’ manifold Z B encoded in the F-theory 4-fold X B . In the example, Z B is the mirror quintic, which is embedded in a toric ambient space with a large number h , = 101 of K¨ahler classes, resulting in 101 coordinates x k in the hypersurface constraint(6 . { x i } and { Z, Y, X ′ , s, t, u } are special selections of these 101 coordinates, where thelatter display (one of) the elliptic fibration(s) of Z B .On the other hand ( a, b ) are coordinates inherent to the 4-fold X B , parametrizing aspecial P , F , which plays the central role in the stable degeneration limit of [12,39] andthe local mirror limit of [22,23]. F is the base of the elliptic fibration of a K3 Y , which inturn is the fiber of the K3 fibration of X B : Y → F, Y → X B → B . In the above example, B can be thought of as a blow up of P . The stable degenerationlimit of the toric hypersurface can be defined as a local mirror limit in the complex structuremoduli of X B , where one passes to new coordinates [23](6 .
3) : x = x ′ ab, v = a/b , (6 .
5) : X = X ′ ab, v = a/b . v = a/b on C ∗ parametrizes a patch near the localsingularity associated with the bundle/brane data for a Lie group G [22]. For G = SU ( n ), v appears linearly, which leads to a substantial simplification of the Hodge variation problem,as described in the appendices of refs. [16,17]. Perturbative gauge symmetry of the heterotic string
The perturbative gauge symmetry of the dual heterotic string is determined by the singu-larities in the elliptic fibration of the K3 fiber Y [15]. There is a simple technique to read offfibration structures for the CY 4-fold X B from the toric polyhedra described in refs. [113].Namely a fibration of X B → B − n with fibers a Calabi-Yau n -fold Y n corresponds to theexistence of a hypersurface H of codimension 4 − n , such that the integral points in theset H ∩ ∆ ∗ define the toric polyhedron of Y n .In the present case, the toric polyhedron ∆ ∗ K for the K3 fiber Y is given as the convexhull of the points in ∆ ∗ lying on the hypersurface H : { x = x = 0 } :∆ K µ = ( 0 , ,
0) ∆ ∗ K µ ⋆ = ( 0 , , µ = ( 0 , − , µ ⋆ = ( − , , − µ = ( 1 , , µ ⋆ = ( − , , µ = ( 0 , , − µ ⋆ = ( 0 , − , µ = ( − , , − µ ⋆ = ( 0 , , µ = ( − , , µ ⋆ = ( 1 , − , µ ⋆ = ( 1 , ,
0) (A.3)where the zero entries at the 3rd and 4th position have been deleted and ∆ K is thedual polyhedron of ∆ ∗ K . The elliptic fibration of Y is visible as the polyhedron ∆ ∗ E =∆ ∗ K ∩ { x = 0 } of the elliptic curve∆ E = Conv { ( − , , (0 , − , (1 , } , ∆ ∗ E = Conv { ( − , , (1 , − , (1 , } . Since the model for the elliptic fiber is not of the standard form, but the cubic in P orbifolded by the action (6.7), the application of the standard methods to determine thesingularity of the elliptic fibration and thus the perturbative heterotic gauge group shouldbe reconsidered carefully. The singularity of the elliptic fibration can be determined directlyfrom the hypersurface equation of X of the elliptically fibered K3 polynomial p ( K
3) = Z + Y + X ′ ( a b + a b + a b ) + ZY X ′ ( ab + b ) , (A.4)67hich is associated to the toric data (A.3). The Z orbifold singularity is captured by r = p q in terms of the invariant monomials p = Y X ′ , q = Z X ′ and r = ZYX ′ . Then,to leading order, the singularities of the elliptic fiber E in the vicinity a = 0 and in thevicinity b = 0 are respectively given by p a → ( K
3) = a q + q + qr + r , p b → ( K
3) = b q + q + bqr + r . Using a computer algebra system, such as ref. [114], it is straight forward to check thatthe polynomials p a → ( K
3) and p b → ( K
3) correspond to the ADE singularities SU (6) and E . In fact it turns out, that the same answer is obtained by naively applying the methoddeveloped in refs. [115,116] for the standard model of the elliptic fiber, which implementsthe Kodaira classification of singular elliptic fibers in the language of toric polyhedra, suchthat the orbifold group is taken into account automatically. The polyhedron ∆ ∗ K splitsinto a top and bottom piece Ξ + and Ξ − with the pointsΞ + − − − − − − − − − − − − − − − − −
10 1 − SU (6) and E , respectively. As asserted in[90,115,116], these toric vertices corresponds to two ADE singularities of the same type,in agreement with the direct computation. Moreover, deleting the vertex ν ∈ ∆ which isassociated with the exceptional toric divisor that described the brane/bundle modulus ˆ z ,the same analysis produces a K3 fiber with two ADE singularities of type E , leading tothe pattern (6.8). Moduli and Picard-Fuchs system
The moduli z a are related to the parameters a i in (6.1) by z a = ( − ) l a Y i a l ai i , (A.5)68here l ai are the charge vectors that define the phase of the linear sigma model for the mirror X A . For the phase considered in [10,11], these are given in (6.9). The complex structuremodulus z ∼ e πit mirror to the volume of the generic quintic fiber, the brane/bundlemodulus ˆ z ∼ e πi ˆ t and the distinguished modulus z S ∼ e πiS capturing the decouplinglimit are given by z = z z = − a a a a a a , ˆ z = z = − a a a a , z S = z = a a a . The GKZ system for CY 4-folds has been discussed in the context of mirror symmetrye.g. in [79,84,10]. A straightforward manipulation of it leads to the following system ofPicard-Fuchs operators for the above example: L = θ ( θ + θ − θ ) − z ( − θ + θ )(4 θ + 1 + θ )(4 θ + 2 + θ )(4 θ + 3 + θ )(4 θ + 4 + θ ) , L = ( θ + θ − θ ) θ − z (2 θ − θ )(2 θ + 1 − θ ) , L = − (2 θ − θ )( − θ + θ ) − z ( θ + θ − θ )(4 θ + 1 + θ ) , L = ( − θ + θ ) θ + z z (2 θ − θ )(4 θ + 1 + θ ) , L = − (2 θ − θ ) θ − z z (4 θ + 1 + θ )(4 θ + 2 + θ )(4 θ + 3 + θ )(4 θ + 4 + θ )(4 θ + 5 + θ ) , L = − (2 θ − θ ) θ − z z (4 θ + 1 + θ )(4 θ + 2 + θ )(4 θ + 3 + θ )(4 θ + 4 + θ ) − z θ ( θ + θ − θ ) . (A.6)Here θ a = z a ∂z a are the logarithmic derivatives in the coordinates z a , a = 1 , , A.2. Heterotic 5-branesDegree 18 hypersurface in P (1 , , , , X A , X B ) of 4-folds dual to the 3-fold compactificationson ( Z A , Z B ) are defined as the convex hull of the points:∆ ν ν − ν − ν ν − ν − ν ν − ν − − ν ∗ − −
12 1 1 06 −
12 1 1 −
60 6 1 1 60 0 1 − − − −
12 6 1 1 6 −
12 6 1 1 − x i Ξ Y − X − Z ′ s −
12 6 1 1 0 t −
12 1 1 0 u a − b X ncA in Table 2 of [10], with the point ν added in thecompactification X A of X ncA . The polyhedron ∆ for the 3-fold Z A defined as a degree 18hypersurface in P (1 , , , ,
9) is given by the points on the hypersurface ν i, = 0, withthe last entry deleted. The vertices of the dual polyhedron ∆ ∗ of ∆ are given by thepoints of ∆ ∗ with ν ⋆i, = 0 and on extra vertex ( − , , , ∗ used to define local coordinates in (7.2). The relation tothe coordinates used there is Z = Z ′ ab, v = a/b . The relevant phase of the K¨ahler cone considered in [10,19] is l = ( − ,l = ( 0 0 0 − − ,l = ( 0 0 0 − − .l = ( 0 0 0 − − . (A.8)In the coordinates (A.5), the brane modulus in (6.6) is given by ˆ z = z / z − / . Degree 9 hypersurface in P (1 , , , , X A , X B ) of 4-folds dual to the 3-fold compactificationson ( Z A , Z B ) are defined as the convex hull of the points:∆ ν ν − ν − ν ν − ν − ν ν − ν − − ν ∗ − − − − − − −
33 3 1 1 03 − − − x i Ξ Y − X − Z ′ s − t − u a − b for the 3-fold Z A defined as a degree 9 hypersurface in P (1 , , , , ν i, = 0. On the r.h.s we have given theselection Ξ of points in ∆ ∗ used in (7.5), with the redefinitions Z = Z ′ ab , v = a/b . Thephase of the K¨ahler cone considered in [10] is l = ( − ,l = ( 0 0 0 − − ,l = ( 0 0 0 − − .l = ( 0 0 0 − − . (A.10)In the coordinates (A.5), the brane modulus in (7.5) is given by ˆ z = z / z − / .70 .3. SU (2) bundle of the degree 9 hypersurface in P (1 , , , , X A , X B ) of 4-folds dual to the 3-fold compactifi-cations on ( Z A , Z B ) are defined as the convex hull of the points:∆ ν ν − ν − ν ν − ν − ν ν − ν − − ν − ∗ − − − − −
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