Five-Brane Superpotentials and Heterotic/F-theory Duality
aa r X i v : . [ h e p - t h ] J un BONN-TH-2009-070912.2997 [hep-th]
Five-Brane Superpotentials and Heterotic/F-theory Duality
Thomas W. Grimm, Tae-Won Ha, Albrecht Klemm and Denis Klevers Bethe Center for Theoretical Physics, Universit¨at Bonn,Nussallee 12, 53115 Bonn, Germany
ABSTRACT
Under heterotic/F-theory duality it was argued that a wide class of heterotic five-branesis mapped into the geometry of an F-theory compactification manifold. In four-dimensionalcompactifications this identifies a five-brane wrapped on a curve in the base of an ellipticallyfibered Calabi-Yau threefold with a specific F-theory Calabi-Yau fourfold containing theblow-up of the five-brane curve. We argue that this duality can be reformulated by firstconstructing a non-Calabi-Yau heterotic threefold by blowing up the curve of the five-braneinto a divisor with five-brane flux. Employing heterotic/F-theory duality this leads us to theconstruction of a Calabi-Yau fourfold and four-form flux. Moreover, we obtain an explicitmap between the five-brane superpotential and an F-theory flux superpotential. The map ofthe open-closed deformation problem of a five-brane in a compact Calabi-Yau threefold intoa deformation problem of complex structures on a dual Calabi-Yau fourfold with four-formflux provides a powerful tool to explicitly compute the five-brane superpotential.December, 2009 grimm, tha, aklemm, [email protected] Introduction
The study of string compactifications leading to N = 1 supersymmetric four-dimensionallow-energy effective theories is of conceptual as well as of phenomenological interest. Twoprominent approaches to obtain such effective theories are either to consider heterotic E × E string theory on a Calabi-Yau manifold with non-trivial vector bundles, or to study F-theorycompactifications on singular Calabi-Yau fourfolds. At first, these two approaches appear tobe very different in nature, since the data determining the effective dynamics are encoded byseemingly different objects. However, at least if one focuses on certain compact geometries,the heterotic and F-theory picture are believed to be dual descriptions of the same physics[1, 2]. The dictionary of this duality not only contains the map for the vector bundles of theheterotic string, but also includes heterotic five-branes wrapped on curves in the Calabi-Yauthreefold [3, 4, 5]. These are often necessary in a consistent heterotic compactification toensure anomaly cancellation. Using this duality either of the two descriptions can be used toanswer specific questions about the four-dimensional physics. In this work we will focus onparts of the effective action which are efficiently calculable in F-theory, but admit a naturalphysical interpretation in the heterotic theory.An important question in the study of the four-dimensional N = 1 low-energy effectiveaction is the explicit computation of the superpotential and gauge-coupling functions whichdepend holomorphically on the chiral multiplets. In the following we will mainly focus on thestudy of the superpotential of a heterotic five-brane wrapped on a curve C in a Calabi-Yauthreefold Z . It was shown in ref. [6] that this superpotential depends on the deformationmodes of the curve C and the complex structure moduli of Z via the chain integral R Γ Ω,where Ω is the holomorphic three-form on Z , and Γ is a three-chain which admits C as aboundary component. We will argue by using heterotic/F-theory duality that this chain inte-gral is mapped to the flux superpotential of an F-theory compactification upon constructingan appropriate Calabi-Yau fourfold ˆ X encoding the five-brane dynamics, and the associ-ated four-form flux G . The F-theory flux superpotential can then be computed explicitlyby solving Picard-Fuchs differential equations determining the closed period integrals of theholomorphic four-form on ˆ X , and using mirror symmetry to identify the superpotential so-lution [7]. Earlier discussions and computations of the periods of the holomorphic four-formcan be found in refs. [8, 9, 10].The computation of brane superpotentials given by chain integrals has been of significantinterest in the D-brane literature. Starting with [11] the superpotential for D5-branes hasbeen studied intensively for non-compact Calabi-Yau threefolds [12, 13]. More recently, therehas been various attempts to extend this to compact threefolds [14, 15, 16, 17, 18, 19, 20, 21].1n particular, it was proposed in refs. [13] to use the variation of mixed Hodge structurefor an auxiliary divisor capturing the variations of the curve C . While initially studied innon-compact geometries, extensions to compact Calabi-Yau threefolds with D5-branes haveappeared in refs. [16, 19, 20]. In this proposal the deformations of an appropriately chosendivisor are effectively identified with the deformations of the curve. In contrast, it wasargued in [18] that the deformation problem of the curve C in Z admits a natural map toa geometric setup in which the curve is blown up into a divisor. In this case, the divisor isrigid and the deformations of the curve appear as new complex structure deformations ofthe blown-up threefold ˆ Z . In this work we will use the blow-up construction to study theduality of an heterotic five-brane to an F-theory compactification fourfold. Let us note thatin refs. [22, 19, 20, 23] it was proposed to use non-compact Calabi-Yau fourfolds to computethe D5-brane superpotential and a connection with F-theory was indicated. Let us stressthat the approach we are using here is different in nature, and rather completes the approachinitiated in our works [18, 7].The map of heterotic string theory on a Calabi-Yau threefold Z with five-branes to an F-theory compactification is best studied for elliptically fibered Z . It was shown in ref. [24] thatthere exist elegant constructions of heterotic vector bundles on these threefolds. Furthermore,a five-brane wrapped on a curve C in the base B of this elliptic fibration was argued to mapentirely into the geometry of an F-theory compactification. Using the adiabatic argument of[25] the heterotic string on Z is equivalent to F-theory on an elliptic K3- fibered Calabi-Yaufourfold X with base B . This implies, in particular, that the three-dimensional base B ofthe elliptically fibered F-theory fourfold X is a holomorphic P -fibration over B . It wasthen argued in refs. [4, 5, 26] that in the presence of a heterotic five-brane one has to blow upthe curve C into a rigid divisor in B . The deformations of the curve C then map to complexstructure deformations of the blown-up Calabi-Yau fourfold ˆ X , and hence can be constrainedby a calculable flux superpotential. Note that certain five-branes can also be interpreted asspecial gauge bundle configurations of the heterotic string, the so-called small instantons. Inthe small instanton/five-brane transition the deformation moduli of the curve C are identifiedwith heterotic bundle moduli. This yields yet another identification of superpotentials, sincethe five-brane superpotential arises as a localization of the Chern-Simons superpotentialfor the bundle moduli [11]. Both types of superpotentials are efficiently calculable on theF-theory side using the geometric tools for Calabi-Yau fourfolds.To study the duality map between the heterotic and F-theory setup, one can alternativelystart by blowing up the heterotic threefold Z along the five-brane curve C into ˆ Z [18].This can be made explicit by realizing ˆ Z as a complete intersection. The non-Calabi-Yau More accurately, such a divisor is described as an isolated divisor. Z contains the five-brane moduli as a subsector of its complex structure moduli.The heterotic superpotential crucially depends on the pull-back ˆΩ of the holomorphic three-form Ω to ˆ Z . Since ˆΩ vanishes along the blow-up divisor D , the heterotic flux, specifyingthe five-brane, localizes on elements in H ( ˆ Z − D, Z ). This is equivalent to consideringrelative three-forms in H ( ˆ Z , D, Z ). Identifying the elements of this relative group withelements in the dual fourfold cohomology, one finds an explicit map between the heteroticfive-brane and F-theory fluxes. We propose, and explicitly demonstrate for examples, thatthe F-theory geometry ˆ X can in turn be entirely constructed from ˆ Z . In particular, thisidentification becomes apparent when also realizing ˆ X as a complete intersection. In thisway the complex structure moduli of ˆ Z naturally form a subsector of the complex structuremoduli of ˆ X . In summary, the general idea of this discussion is to reformulate and slightlyextend the heterotic/F-theory duality map schematically as:Heterotic string on CY threefold Z ,vector bundle E , 5-brane on C ( ( RRRRRRRRRRRR o o / / F-theory on CY fourfold ˆ X blown up along C , G -fluxnon-Calabi-Yau ˆ Z blown up along C ,vector bundle ˆ E nnnnnnnnnnn where the horizontal arrow indicates the action of heterotic/F-theory duality.Following this general strategy the paper is organized as follows. In section 2 we firstrecall the connection between small instantons and heterotic five-branes. This allows us tointroduce the respective heterotic superpotentials. Moreover, we discuss the general blow-upprocedure of the heterotic Calabi-Yau threefold, and comment on the representation andproperties of the holomorphic three-form on the blown-up geometry ˆ Z . In section 3, wefirst review the heterotic/F-theory duality, highlighting the map of five-branes into a Calabi-Yau fourfold geometry. We then discuss the F-theory flux superpotential and describe howit is matched with its heterotic counterpart. In the last section we study two classes ofexamples. Firstly, we discuss the geometrical construction of the heterotic blow-up threefoldand its associated Calabi-Yau fourfold constructed as complete intersections. Secondly, weinvestigate an example for which we explicitly compute the superpotential and confirm themap between five-brane deformations and fourfold complex structure moduli. This should be compared with the use of relative cohomology for the auxiliary non-rigid divisor in theconstructions of refs. [13, 16, 19, 20]. Heterotic Five-Branes and Superpotentials
In this section we review the construction of N = 1 vacua by compactification of the heteroticstring on a Calabi-Yau threefold Z with vector bundle E and a number of space-time fillingfive-branes. We discuss the relation of the bundle moduli and five-brane deformations via asmall instanton transition in section 2.1. The heterotic superpotential for these moduli fieldswill be introduced in section 2.2. It will be argued that the motion of the five-brane inside Z is constraint by a superpotential given by the integral of the holomorphic three-form Ω overa chain ending on the five-brane. In section 2.3 we discuss the general blow-up procedureand some properties of the resulting geometry. Let us begin by reviewing the heterotic compactifications. Besides the choice of a Calabi-Yauthreefold Z , a consistent heterotic vacuum requires a choice of stable holomorphic vector-bundles E = E ⊕ E over Z which determine the gauge group preserved in the perturbative E × E of the heterotic theory. In general we can additionally have five-branes wrappingholomorphic curves C in the threefold Z . This setup is further constrained by the generalheterotic anomaly cancellation condition λ ( E ) + λ ( E ) + [ C ] = c ( Z ) , (2.1)where λ ( E ) is the fundamental characteristic class of the vector bundle E, which, for example,is c ( E ) for SU ( N ) bundles and c ( E ) /
60 for E bundles. This condition dictates consistentchoices of the cohomology class [ C ] of the curve C in the presence of non-trivial vector bundlesto match the curvature of the threefold Z as measured by the second Chern class c ( Z ).In particular, it implies that C corresponds to an effective class in H ( Z , Z ) [27].The analysis of the moduli space of the heterotic string on Z requires the study of threea priori very different pieces. Firstly, we have the geometric moduli spaces of the threefold Z consisting of the complex structure as well as K¨ahler moduli space. Secondly, there arethe moduli of the bundles E and E which parameterize different gauge-field backgroundson Z . Finally, if the five-brane is wrapped on a non-rigid curves C , the deformations of C within Z of the various five-branes have to be taken into account. The entire moduli spaceis in general very complicated and difficult to analyze. This problem, however, becomesmore tractable if one focuses on elliptically fibered Calabi-Yau threefolds Z . It was shownin ref. [24] that there exist elegant constructions of the vector bundle E on these threefolds.Moreover, the moduli space of five-branes on elliptically fibered Z has been discussed in4reat detail in ref. [28]. In general, it admits several different branches corresponding tothe number and type of five-branes present. However, there are distinguished points in themoduli space corresponding to enhanced gauge symmetry [29, 30] of the heterotic string thatallow for a clear physical interpretation and that we now discuss in more detail. It will turnout that at these points an interesting transition is possible where a five-brane completelydissolves into a finite size instanton of the bundle E and vice versa.Let us start with a threefold Z with c ( Z ) = 0 and no five-branes. Thus, the anomalycondition (2.1) forces us to turn on a background bundle E with non-trivial second Chernclass c ( E ) in order to cancel c ( Z ). Then the bundle is topologically non-trivial and carriesbundle instantons characterized by the topological second Chern number [31][ c ] = − Z Z J ∧ F ∧ F , (2.2)where J denotes the K¨ahler form on Z and F the field strength of the background bundle.The heterotic gauge group G in four dimensions is generically broken and given by thecommutant of the holonomy group of the bundle E in E . Varying the moduli of E one canask whether it is possible to restore parts or all of the broken gauge symmetry by flatteningout the bundle as much as possible [32]. To show how this can be achieved, one firstdecomposes c ( E ) into its components each of which being dual to an irreducible curve C i in Z . Since the invariant [ c ] has to be kept fixed, the best we can do is to consecutively splitoff the components of c ( E ) and to localize the curvature of E on the corresponding curves C i . This should be contrasted with the generic situation, where the curvature is smearedout all over Z . In the localization limit the holonomy of the bundle around each individualcurve C i becomes trivial and the gauge group G enhances accordingly. Having reached thisso-called small instanton configuration at the boundary of the moduli space of the bundle,the dynamics of (this part of) the gauge bundle can be completely described by a five-braneon C i [29].Small instanton configurations thus allow for transitions between branches of the modulispace with different numbers of five-branes, that consequently map bundle moduli to five-brane moduli and vice versa [33]. This is precisely what we need for our later F-theoryanalysis. Note that this transition is completely consistent with (2.1) since we have justshifted irreducible components between the two summands c ( E ) and [ C ]. Thus, we arein the following allowed to think about the small instanton configuration as the presenceof a five-brane. In particular, doing this transition for all components of c ( E ) the fullperturbative heterotic gauge group E × E can be restored. Turning this argument around,a heterotic string with full E × E gauge symmetry on a threefold Z with non-trivial c ( Z )has to contain five-branes to cancel the anomaly according to (2.1). In our concrete example5f section 4 we will precisely encounter this situation guiding us to the interpretation of theF-theory flux superpotential in terms of a superpotential for a particular class of five-branes.To precisely specify the five-branes we will consider later, we note that on an ellipticallyfibered Calabi-Yau threefold the five-brane class [ C ] can be decomposed as C = n f F + C B , (2.3)where C B denotes a curve in the base B of the elliptic fibration, F denotes the ellipticfiber, and n f is a positive integer. This is a split into five-branes vertical to the projection π : Z → B , where the integer n f counts the number of five-branes wrapping the ellipticfiber, and into horizontal five-branes on C B in the base B . Both cases are covered by (2.1),but lead to different effects in the F-theory dual theory. Vertical five-branes correspond tospacetime filling three-branes at a point in the base B of the F-theory fourfold X [3, 24].Conversely, horizontal five-branes on the curve C B map completely to the geometry of theF-theory side. They map to seven-branes supported on a divisor in the fourfold base B which projects onto the curve C in B [2, 34, 35] that has to be blown-up in B into a divisor D [4, 5, 26]. Of course, there can be mixed types of five-branes as well. It will be preciselythe horizontal five-branes corresponding to blow-ups into exceptional divisors D for whichour analysis and calculation of the superpotential will be performed. The small instanton transition implies a transition between bundle and five-brane mod-uli [33]. Since both types of moduli are generally obstructed by a superpotential also thesuperpotentials for bundle and five-brane have to be connected by the transition. As was ar-gued in [6] in the context of M-theory on a Calabi-Yau threefold, a spacetime-filling M5-branesupported on a curve C in general induces a superpotential W M5 = Z Γ Ω , (2.4)where Γ denotes a three-chain bounded by C and an unimportant reference curve C in thehomology class of C . It depends on both the moduli of the five-brane on C as well as thecomplex structure moduli of Z in the holomorphic three-form Ω. On the other hand, theperturbative superpotential for the heterotic bundle moduli is given by the holomorphicChern-Simons functional [31] W CS = Z Z Ω ∧ ( A ¯ ∂A + 23 A ∧ A ∧ A ) , (2.5)6here A denotes the gauge connection that depends on the bundle moduli. The dependenceon the complex structure moduli of Z is implicit through Ω.To see how the two superpotentials (2.4) and (2.5) are mapped onto each other in thetransition, let us assume a single instanton solution F with F ∧ F dual to an irreduciblecurve C . Displaying the explicit moduli dependence of the configuration F [36], in the smallinstanton limit F ∧ F reduces to the delta function δ C i of four real scalar parameters. Theydescribe the position moduli of the instanton normal to the curve in the class [ C i ] on whichit is localized. Inserting the gauge configuration F into W CS , the holomorphic Chern-Simonsfunctional is effectively dimensionally reduced to the curve C [11]. In the vicinity of C we maywrite the holomorphic three-form as Ω = dω which we insert into (2.5) in the background F ∧ F to obtain W CS = Z C ω (2.6)after a partial integration. Adding a constant given by the integral of ω over the referencecurve C this precisely matches the chain integral (2.4). Applying the above discussion, wecan think about the M5-brane moduli in W M5 as the bundle moduli describing the positionof the instanton configuration F , that in the small instanton limit precisely map to sections H ( C i , N Z C i ) of the normal bundle to C i .We will verify this matching of moduli explicitly from the perspective of the F-theory dualsetup later on. There we will on the one hand identify some of the fourfold complex structuremoduli with the heterotic bundle moduli, on the other hand, however, show that part ofthe F-theory flux superpotential depending on the same complex structure moduli reallycalculates the superpotential of a five-brane on a curve. This way, employing heterotic/F-theory duality, we show in the case of an example the equivalence of the small instanton/five-brane picture.To complete the discussion of perturbative heterotic superpotentials, let us also commenton the flux superpotential due to bulk fluxes. In general, the heterotic B -field can have anon-trivial background field strength H flux3 that has to be in H ( Z , Z ) due to the fluxquantization condition. The induced superpotential will be intimately linked to (2.4) and(2.5) due to the anomaly cancellation condition dH = Tr ( R ∧ R ) − Tr (
F ∧ F ) − X i δ C i , (2.7)which yields, with an appropriate definition of the traces, the condition (2.1) if one restrictsto cohomology classes. The superpotential in terms of this H reads [37, 38] W het = Z Z Ω ∧ H = W flux + W CS + W M5 , (2.8)7here the different terms can be associated to the various contributions in H in (2.7). Inorder to discuss the flux part, we expand H flux3 = N i α i − M i β i in the integral basis α i , β i of H ( Z , Z ) with integer flux numbers N i , M i . Then one can write the flux superpotential as W flux = Z Z Ω ∧ H flux3 = M i X i − N i F i , (2.9)where we introduced the period expansion Ω = X i α i − F i β i . In general, the periods ( X i , F i )admit a complicated dependence on the complex structure deformations of Z . It is thegreat success of algebraic geometry that this superpotential can be calculated explicitly fora wide range of examples, see [39] and [40, 41] for reviews. This is due to the fact that theperiods X i , F i obey differential equations, the so-called Picard-Fuchs equations , that canbe solved explicitly and thus allow to determine the complete moduli dependence of W flux .To end our discussion of the flux superpotential, let us stress that strictly speaking there isa back-reaction of H flux3 which renders Z to be non-K¨ahler [43]. Since our main focus willbe on the five-brane superpotential, we will not be concerned with this back-reaction in thefollowing. The form of the superpotential W M5 of (2.4) is rather universal. It occurs, for example, alsofor D5-branes on curves in Type IIB orientifold compactifications. In the following we willapply the blow-up procedure suggested in ref. [18] for the study of the chain integral forD5-branes to the heterotic setup. The idea is to find a purely geometric description thatputs the dynamics of the five-brane and the geometry of Z on an equal footing. To achievethis, we blow up the curve C into a rigid divisor D in a non-Calabi-Yau threefold ˆ Z . Thisembeds the deformation modes of C in Z as well as the complex structure deformations of Z into the deformation problem of only complex structures of ˆ Z . We will see explicitlylater that this alternative view on the heterotic string with five-branes allows for a directgeometric interpretation of the fate of the five-brane dynamics in heterotic/F-theory duality.Here we provide the geometrical tools to describe the blow-up of Z along a curve C whichwe will later use in the construction of explicit examples in section 4.For concreteness, let us consider a Calabi-Yau threefold Z described as the hypersurface { P = 0 } in a projective or toric ambient space V . Consider then a curve C specified by twoadditional constraints { h = h = 0 } in the ambient space intersecting transversally Z . For See for example [42] for a review. Similar expressions arise for higher dimensional branes with world-volume flux inducing D5-charge sup-ported on the same curves. C supporting horizontal five-branes in an elliptic Z , theconstraints take the form h ≡ ˜ z = 0 , h ≡ g = 0 , (2.10)where { ˜ z = 0 } restricts to the base B and g specifies C within B . In general, the constraints h , h describe divisors in the ambient space that descend to divisors in Z as well uponintersecting with { P = 0 } , called D and D . Locally, ( h , h ) can be considered as normalcoordinates to the curve C in Z . Thus, the normal bundle N Z C of the curve takes the form N Z C = O Z ( D ) ⊕ O Z ( D ), where O Z ( D i ) denotes the line bundle of D i as read of fromthe scalings of the section h i . As the divisors D i , also their line bundles O Z ( D i ) are inducedfrom the bundles O ( D i ) on the ambient space V .To describe the blown-up threefold ˆ Z , we introduce the total space of the projectivebundle P ( O ( D ) ⊕ O ( D )). This total space describes a P -fibration over the ambient space V on which we introduce the P -coordinates ( l , l ) ∼ λ ( l , l ). Then, the blow-up ˆ Z isgiven by the complete intersection [44] P = 0 , Q = l h − l h = 0 , (2.11)in the projective bundle. This is easily checked to describe ˆ Z . The first constraint dependingonly on the coordinates of the base V of the projective bundle restricts to the threefold Z .The second constraint then fibers the P non-trivially over Z to describe the blow-up along C . Away from h = 0 or h = 0 we can solve (2.11) for l or l respectively. Thus, (2.11)describes a point in the P -fiber for every point in Z away from the curve. However, if h = h = 0 the coordinates ( l , l ) are unconstrained and parameterize the full P , whichis fibered over C as its normal bundle N Z C . Thus, we have replaced the curve by theexceptional divisor D that is given by the projectivization of its normal bundle in Z , i.e. theruled surface D = P ( N Z C ) over C . We denote the blow-down map by π : ˆ Z / / Z . (2.12)Having described the construction of the blow-up, one can also determine details on thecohomology of D and ˆ Z [44]. For a single smooth curve C the non-vanishing Hodge numbersof D are determined to be h , = h , = 1 , h , = g , h , = 2 (2.13)as usual for a ruled surface D over a genus g curve C . One element, which we denote by η | D , of H , ( D ) is induced from the ambient space ˆ Z and given by η = c ( N ˆ Z D ). The The Lefshetz-Hyperplane theorem tells us that indeed any divisor and line bundle in Z is induced fromthe ambient space [44]. H , ( D ) is given by the Poincar´e dual [ C ] D of the curve C in D ,[ C ] D = c ( N D C ). It is related to the first Chern class c ( C ) and thus to the genus as c ( N D C ) = − c ( C ) − η , (2.14)by using the adjunction formula in ˆ Z . Note that as a blow-up divisor D is rigid in ˆ Z . Thefirst and second Chern class of ˆ Z are affected by the blow-up as c ( ˆ Z ) = π ∗ ( c ( Z )) − c ( N ˆ Z D ) , (2.15) c ( ˆ Z ) = π ∗ ( c ( Z ) + [ C ]) − π ∗ ( c ( Z )) D . (2.16)Clearly, if Z is a Calabi-Yau manifold one can use c ( Z ) = 0 to find c ( ˆ Z ) = − η , c ( ˆ Z ) = π ∗ ( c ( Z ) + [ C ]) , (2.17)in particular that ˆ Z is no more Calabi-Yau.It was argued in [18] that the complex structure moduli space of ˆ Z contains the complexstructure moduli of Z as well as the deformation of C within Z . The basic reason forthis is roughly that the complex structure deformations of the rigid divisor D contain thedeformation moduli of the curve C and thus embed them into the complex structure ofˆ Z . This way the deformations of the pair ( Z , C ) form a subsector of the geometricaldeformations of ˆ Z . This allows for the study of the combined superpotential of five-brane(2.4) and flux (2.9) as well. First we use the formal unification of the two superpotentials interms of the relative homology group H ( Z , C , Z ) consisting of three-cycles H ( Z , Z ) andthree-chains Γ C ending on the curve C . Then the superpotential can be written as [13] W flux + W M5 = X i ˜ N i Z Γ i C Ω (2.18)with respect to an integral basis Γ i C of the relative group H ( Z , C , Z ). Here the integers ˜ N i correspond to the three-form flux quanta ( M i , N i ) in (2.9) and the five-brane windings. Inparticular Ω has to be interpreted as a relative form.It has been argued in ref. [18] that in the blow-up π : ˆ Z → Z the superpotential (2.18)is lifted to ˆ Z as follows. First we have to replace Ω by its equivalent on ˆ Z , the pullbackform ˆΩ = π ∗ (Ω) , ˆΩ | D = 0 (2.19)that can be shown to vanish on D , see [18] for details and references. Consequently we canwrite the heterotic superpotentials as W flux + W M5 = Z ˆ Z H ∧ ˆΩ = Z ˆ Z − D H ∧ ˆΩ = Z Γ H ˆΩ (2.20)10uch that it only depends on the topology of the open manifold Z − C = ˆ Z − D . Here, wenaturally obtain Γ H as the Poincar´e dual of the flux H in the group H ( ˆ Z − D, Z ).These replacements can also be understood in the language of relative (co)homology. Onthe one hand we can treat ˆΩ as a relative form exploiting the fact that any element in therelative group H ( ˆ Z , D, Z ) can be represented by a form vanishing on D . On the otherhand the element Γ H maps to the relative homology since Lefshetz and Poincar´e dualityrelate the de Rham homology of the open manifold to the relative homology as H ( ˆ Z − D, Z ) = H ( ˆ Z , D, Z ) (2.21)This identification of (co-)homology groups gets completed by the equivalence H ( Z , C , Z ) = H ( ˆ Z , D, Z ) telling us that we have consistently replaced all relevant topological quantitieson Z by those on the blow-up ˆ Z . Finally, we expand the element Γ H in a basis Γ iD of H ( ˆ Z − D, Z ) = H ( ˆ Z , D, Z ) to obtain an expansion of the superpotential by relativeperiods of ˆΩ as W flux + W M5 = X i ˜ N i Z Γ iD ˆΩ = X i ˜ N i Z ˆ Z ˆΩ ∧ γ Di . (2.22)Here γ Di are the Poincar´e duals in H ( ˆ Z , D, Z ).Similar to the Calabi-Yau threefold case where every element in H ( Z , Z ) can be ob-tained upon differentiating Ω with respect to the complex structure, it is possible to obtaina basis of H ( ˆ Z , D, Z ) the same way. More precisely we can write the basis elements γ Di asdifferentials of ˆΩ evaluated at the large complex structure point, γ Di = R i ˆΩ | z =0 . (2.23)The operators R i are polynomials in the differentials θ a = z a ddz a . Such a representation canbe made explicit by noting that ˆΩ can be written as a residue integral [45]ˆΩ = Z ǫ Z ǫ ∆ P Q , (2.24)where
P, Q are the two constraints (2.11) which define ˆ Z . The form ∆ denotes a top-formon the five-dimensional ambient space P ( O ( D ) ⊕ O ( D )) that is invariant under its torusactions and the ǫ i are loops around { P = 0 } , { Q = 0 } . For the type of ambient space weconsider, the measure ∆ takes the schematic form [46]∆ = ∆ V ∧ ( l dl − l dl ) , (2.25)where ∆ V denotes the invariant form on the toric base V and ( l , l ) the coordinates ofthe P -fiber. This makes it possible to study some of the afore-mentioned properties of ˆΩexplicitly. 11he crucial achievement of the blow-up to ˆ Z is the fact that all moduli dependence ofthe superpotential is now contained in the complex structure dependence of ˆΩ. Thus it ispossible, analogous the Calabi-Yau case, to derive Picard-Fuchs type differential equationsfor ˆΩ by studying its complex structure dependence explicitly. Upon the algebraic represen-tation of ˆ Z by the complete intersection (2.11) it is now possible to find an explicit residuerepresentation of ˆΩ such that Griffiths-Dwork reduction can be used to derive the desireddifferential equations for ˆΩ, among whose solutions we find the superpotential W .So far the discussion of the blow-up procedure and the determination of the brane andflux superpotential was entirely in the heterotic theory Z . However, we will shed more lighton the connection between the brane geometry of ( Z , C ) and the classical complex geometryof the blow-up ˆ Z in the context of heterotic/F-theory duality. More precisely, we argue thatthe five-brane superpotential is mapped to a flux superpotential for F-theory compactifiedon a dual Calabi-Yau fourfold ˆ X . Starting with ˆ Z , the fourfold ˆ X can be representedas a complete intersection generalizing (2.11). However, in contrast to ˆ Z the fourfold ˆ X can also be represented as a hypersurface. This fact allows us to directly compute the fluxsuperpotential. Such a computation has been performed in ref. [7] for a set of examples,and confirmed that the five-brane superpotential is naturally contained in the F-theory fluxsuperpotential. In the next section we will discuss this duality in detail and outline theconstruction of ˆ X and the F-theory flux G . Here we turn to the discussion of F-theory compactifications on elliptic Calabi-Yau fourfolds X yielding N = 1 effective theories in four dimensions. We will discuss the basic geometricingredients encoding the seven-brane content as well as the three-brane tadpole in its mostgeneral form including G -flux in section 3.1. There, we will readily restrict to F-theoryfourfolds X with a heterotic dual on an elliptic threefold Z . The F-theory dual to an E × E heterotic string with small instantons/five-branes is discussed in section 3.2 requiringa blow-up in the F-theory base B along curves C in B . We will argue that the five-branemoduli and superpotential are mapped to complex structure moduli of X and the fluxsuperpotential. Finally in section 3.4, we will construct the appropriate G -flux inducing theflux superpotential dual to the heterotic brane superpotential.12 .1 F-Theory and Heterotic/F-Theory Duality We prepare for our further discussion by briefly reviewing the necessary aspects of F-theoryand heterotic/F-theory duality.An F-theory compactification to four dimensions is in general defined by an ellipticallyfibered Calabi-Yau fourfold X with a section. This section can be used to express thefourfold X as an analytic hypersurface in the projective bundle P ( O B ⊕ L ⊕ L ) withcoordinates ( z, x, y ) for which the constraint equation can be brought to the Weierstrassform y = x + f xz + gz . (3.1)The Calabi-Yau condition on X implies L = K − B and f , g have to be sections of L and L for the constraint (3.1) to transform as a section of L . F-theory defined on X automaticallytakes care of a consistent inclusion of spacetime-filling seven-branes. These are supportedon the in general reducible divisors ∆ in the base B determined by the degeneration loci of(3.1) given by the discriminant ∆ = { ∆ = 27 g + 4 f = 0 } . (3.2)The degeneration type of the fibration specified by the order of vanishing of f , g and ∆along the irreducible components ∆ i of the discriminant have an ADE–type classificationthat physically specifies the four-dimensional gauge group G [34].There are further building blocks necessary to specify a consistent F-theory setup. Thisis due to the fact that a four-dimensional compactification generically has a three-branetadpole of the form [47, 48, 49] χ ( X )24 = n + 12 Z X G ∧ G . (3.3)In the case that the Euler characteristic χ ( X ) of X is non-zero a given number n ofspacetime-filling three-branes on points in B and a specific amount of quantized four-formflux G have to be added in order to fulfill (3.3).For a generic setup with three-branes and flux, the four-dimensional gauge symmetryas determined by the seven-branes is not affected. However, if the three-brane happens tocollide with a seven-brane, it can dissolve, by the same transition as discussed in section 2.1,into a finite-size instanton on the seven-brane worldvolume that breaks the four-dimensionalgauge group G . During this transition the number n of three-branes jumps and a flux G is generated describing the gauge instanton on the seven-brane worldvolume [41]. In par-ticular, in case of a heterotic dual theory the three-branes on the F-theory side precisely13orrespond to vertical five-branes on the heterotic threefold [24]. Thus, under duality thethree-brane/instanton transition is precisely the F-theory dual of the transition of a verticalfive-brane into a finite size instanton breaking the gauge group on the heterotic side accord-ingly. However, we will not encounter this any further since we restrict our discussion to thecase that the gauge bundle on those seven branes dual to the perturbative heterotic gaugegroup is trivial and no three-branes sit on top of their worldvolumes.Let us now come to a more systematic discussion of heterotic/F-theory duality. Thefundamental duality that underlies it in any dimensions is the eight-dimensional equiva-lence of the heterotic string compactified on T and F-theory on elliptic K3 [1]. The eight-dimensional gauge symmetry G is determined in the heterotic string as the commutant of an E × E -bundle on T with structure group H . This precisely matches the singularity type G of the elliptic fibration of K3 in the F-theory formulation. Using the adiabatic argument[25] it is possible to consider a family of dual eight-dimensional theories parameterized bya base manifold B n to obtain dualities between the heterotic string and F-theory in lowerdimensions.This way a four-dimensional heterotic string on the elliptic threefold Z is equivalentto F-theory on the elliptic K3-fibered Calabi-Yau fourfold X . Consequently, the three-dimensional base B of the elliptic fibration of X has to be ruled over the base B of theheterotic threefold Z , i.e. B is a holomorphic P -fibration over B . It turns out thatprecisely this fibration data of B is crucial for the construction of the dual heterotic theory,in particular the stable vector bundle E on Z that determines the four-dimensional gaugegroup G . To analyze this issue in a more refined way it is necessary to use the methodsdeveloped in [24], in particular the spectral cover. However, instead of delving into thetechnical details, we will focus on the results essential for our further discussion.The basic strategy of the spectral cover is to obtain the stable holomorphic bundle E onthe elliptic threefold Z roughly speaking by fibering the stable bundles on the fiber torus sothat they globally fit into a stable bundle on the threefold Z [24]. This way, the topologicaldata of the bundle E can be determined in terms of the cohomology of the two-dimensionalbase B . For example, for our case of interest, H = SU ( n ) and E , the second Chern class c ( E ) of the bundle E schematically reads λ ( E ) = ησ + π ∗ ( ω ) , (3.4)where η and ω are up to now arbitrary classes in H ( B , Z ) and σ = c ( O ( σ )) is Poincar´edual to the section σ of π : Z → B . The class η is essential in the general construction of Strictly speaking, there is no spectral cover description of E bundles. However, upon application of themethod of parabolics very similar results to the SU ( n ) case can be obtained [24]. B of the dual F-theory.Consider the heterotic string with an E × E -bundle on Z . Besides the required sin-gularities of the elliptic fibration of X to match the heterotic gauge group G only the part B of the F-theory geometry is fixed by duality. The threefold B can be freely specified bychoosing the P -fibration over B as follows. Fixing a line bundle L over B the threefold B is described as the total space of the projective bundle P ( O ⊕ L ). There are two dis-tinguished classes in H ( B , Z ), namely the B -independent class of the hyperplane of the P -fiber denoted by r = c ( O (1)) and the line bundle L with c ( L ) = t . Then, the heteroticbundle E = E × E is specified by [24] η ( E ) = 6 c + t , η ( E ) = 6 c − t , (3.5)meaning that the choice of P -fibration uniquely determines the η -classes of the two bundles.In particular, we note that the heterotic anomaly (2.1) is trivially fulfilled without theinclusion of any horizontal five-branes.So far, the above discussion is not the most general setup possible since it does not allowfor the presence of horizontal five-branes. It turns out that the F-theory dual to the E × E heterotic string has to a be analyzed more thoroughly in order to naturally include horizontalfive-branes to the setup. In this section we will discuss the F-theory dual of horizontal five-branes [5, 26] as will beessential for our understanding of the five-brane superpotential.Thus, we now restrict our considerations completely to F-theory compactifications witha heterotic dual. Then B is the total space of the projective bundle P ( O B ⊕ L ) where wenow assume L = O B ( − Γ) for an effective divisor Γ in B . This fibration p : B → B hastwo holomorphic sections denoted C , C ∞ with C ∞ = C + p ∗ Γ . (3.6)Then, the perturbative gauge group G = G × G , where we denote the group factors fromthe first E as G and from the second E as G , is realized by seven-branes over C and C ∞ with singularity type G and G , respectively [2, 35]. On the other hand, components of thediscriminant on which ∆ vanishes of order greater than one that project onto curves C i in B correspond to heterotic five-branes on the same curves in Z [2, 34, 35]. Consequently,15he corresponding seven-branes induce a gauge symmetry that is a non-perturbative effectdue to five-branes on the heterotic side.Since the understanding of horizontal five-branes is the central point of our discussionlet us analyze the consequences of these vertical components of the discriminant for the F-theory geometry more thoroughly. Guided by our example of section 4.3, we will consider theenhanced symmetry point with G = E × E due to small instantons/five-branes such thatthe heterotic bundle is trivial. In general, an analysis of the local F-theory geometry nearthe five-brane curve C is possible [26] applying the method of stable degeneration [50, 24].However, since the essential point in the analysis is the trivial heterotic gauge bundle, theresults of [26] carry over to our situation immediately.As follows in general using (3.6) the canonical bundle of the ruled base B reads K B = − C + p ∗ ( K B − Γ) = − C − C ∞ + p ∗ ( K B ) . (3.7)From this we obtain the classes F , G and ∆ of the divisors defined by f , g and ∆ as sectionsof K − B , K − B and K − B , respectively. To match the heterotic gauge symmetry G = E × E ,there have to be II ∗ fibers over the divisors C , C ∞ in B . Since II ∗ fibers require that f , g and ∆ vanish to order 4, 5 and 10 over C and C ∞ , their divisor classes split accordinglywith remaining parts F ′ = F − C + C ∞ ) = − p ∗ ( K B ) ,G ′ = G − C + C ∞ ) = C + C ∞ − p ∗ ( K B ) , (3.8) ∆ ′ = ∆ − C + C ∞ ) = 2 C + 2 C ∞ − p ∗ ( K B ) . This generic splitting implies that the component ∆ ′ can locally be described as a quadraticconstraint in a local normal coordinate k to C or C ∞ , respectively. Thus, ∆ ′ can beunderstood locally as a double cover over C respectively C ∞ branching over each irreduciblecurve C i of ∆ ′ · C and ∆ ′ · C ∞ . In fact, near one irreducible curve C i intersecting say C thesplitting (3.9) implies that the sections f, g take the form f = k f ′ , g = k ( g + kg ) ≡ k g ′ (3.9)with f ′ denoting a section of KB − and g , g sections of KB − ⊗ L , KB − , respectively.The discriminant then takes the form ∆ = k ∆ ′ where ∆ ′ is calculated from f ′ and g ′ . Thus,the intersection curve is given by g = 0 and the degree of the discriminant ∆ rises by twoover C i with f ′ and g ′ vanishing of order zero and one. Precisely the singular curves C i in X that appear in g as above are the locations of the small instantons/horizontal five-branes in Z [5, 26] on the heterotic side. In the fourfold X the collision of a II ∗ and a I singularityover C i induces a singularity of X exceeding Kodaira’s classification of singularities. Thus, it16equires a blow-up π : ˜ B → B in the three-dimensional base of the curves C i into divisors D i .This blow-up can be performed without violating the Calabi-Yau condition since the shift inthe canonical class of the base, K ˜ B = π ∗ K B + D i , can be absorbed into a redefinition of theline bundle L ′ = π ∗ L − D i entering (3.1) such that K X = p ∗ ( K B + L ) = p ∗ ( K ˜ B + L ′ ) = 0.To describe this blow-up explicitly let us restrict to the local neighborhood of one irre-ducible curve C i of the intersection of ∆ and C . We note that the curve C i in B is givenby the two constraints h ′ ≡ k = 0 , h ′ ≡ g = 0 , (3.10)for k and g being sections of the normal bundle N B C and of KB − ⊗ L , respectively. Thenif X is given as a hypersurface P ′ = 0 we obtain the blow-up as the complete intersection P ′ = 0 , Q ′ = l h ′ − l h ′ = 0 , (3.11)where, as in (2.11), we have introduced coordinates ( l , l ) parameterizing the P -fiber.However, at least in a local description, we can introduce a local normal coordinate t to C i in B such that g = tg ′ for a section g ′ which is non-vanishing at t = 0. Then bychoosing a local coordinate k of the P -fiber we can solve the blow-up relation Q ′ of (3.11)to obtain k = k t . This coordinate transformation can be inserted into the constraint P ′ = 0of X to obtain the blown-up fourfold ˆ X as a hypersurface. The f ′ , g ′ of this hypersurfaceare given by f ′ = k f , g ′ = k ( g + k t g + . . . ) (3.12)In particular, calculating the discriminant ∆ ′ of ˆ X it can be demonstrated that the I singularity no longer hits the II ∗ singularity over C [26]. This way we have one descriptionof ˆ X as the complete intersection (3.11) and another as a hypersurface. Both will be ofimportance for the explicit examples discussed in sections 4.1, 4.3 and in particular section4.2.To draw our conclusions of this blow-up, we summarize what we just discussed. TheF-theory counterpart of a heterotic string with full perturbative gauge group is given by afourfold with II ∗ fibers over the sections C , C ∞ in B . The component ∆ ′ of the discriminantenhances the degree of ∆ on each intersection curve C i such that a blow-up in B becomesnecessary. On the other hand, each blow-up corresponds to a small instanton in the heteroticbundle [2, 32], that we previously described in section 2.1 as a horizontal five-brane on thecurve C i in the heterotic threefold Z . Indeed, this can be viewed as a consequence ofthe observation mentioned above that a vertical component of the discriminant with degreegreater than one corresponds to a horizontal five-brane [35] as the degree of ∆ ′ on C and C ∞ is two. 17e finish this discussion by a brief look at the moduli map between F-theory and itsheterotic dual, where we focus on the fate of the five-brane moduli in the just mentionedblow-up process. The first step in the moduli analysis is to relate the dimensions of thevarious moduli spaces in both theories and to point to possible mismatches where moduliof some ingredients are missing. In particular, this happens in the presence of heteroticfive-branes. Indeed it was argued in [5] that the relation of the fourfold Hodge numbers h , ( X ) and h , ( X ) counting complex structure and K¨ahler deformations, respectively, to h , ( Z ), h , ( Z ) and the bundle moduli and characteristic data has to be modified in thepresence of five-branes. The extra contribution is due to deformation moduli of the curve C i supporting the five-brane counted by h ( C i , N Z C i ) as well as the blow-ups in B increasing h , ( B ) such that we obtain [5] h , ( X ) = h , ( Z ) + I ( E ) + I ( E ) + h , ( X ) + 1 + X i h ( C i , N Z C i ) ,h , ( X ) = 1 + h , ( B ) + rk(G) . (3.13)Here the sum index i runs over all irreducible curves C i and we denote the rank of the four-dimensional gauge group by rk( G ). The index I ( E , ) counts a topological invariant of thebundle moduli and is given by [24, 51] I ( E i ) = rk( E i ) + Z B (4( η i σ − λ i ) + η i c ( B )) . (3.14)The map for h , ( X ) reflects the fact that the four-dimensional gauge symmetry G ison the heterotic side determined by the gauge bundle E whereas on the F-theory side G isdue to the seven-brane content defined by the discriminant ∆ that is sensitive to a changeof complex structure. For an explicit demonstration of this map exploiting the techniques of[4] we refer to our work [7].Let us now discuss how (3.13) changes during the blow-up procedure. To actually performthe blow-up along the curve C i it is necessary to first degenerate the constraint of X such that X develops the singularity over C i described above. This requires a tuning of the coefficientsentering the fourfold constraint thus restricting the complex structure of X accordinglywhich means h , ( X ) is lowered. Then, we perform the actual blow-up by introducing thenew K¨ahler class associated to the complexified volume of the exceptional divisor D i . Thus,we end up with a new fourfold with decreased h , and h , ( ˜ B ) increased by one. Thisis also clear from the general argument [35] that, enforcing a given gauge group G in fourdimensions, the complex structure moduli have to respect the form of ∆ dictated by thesingularity type G . Since the blow-up which is dual to the heterotic small instanton/five-brane transition enhances the gauge symmetry G , the form of the discriminant becomes18ore restrictive, thus fixing more complex structures. In this picture the blow-down can beunderstood as switching on moduli decreasing the singularity type of the elliptic fibration.Similarly, we can understand (3.13) from the heterotic side. For each transition betweensmall instanton and five-brane, the bundle loses parts of its moduli since the small instanton ison the boundary of the bundle moduli space. Consequently, the index I reduces accordingly.In the same process, the five-brane in general gains position moduli counted by h ( C i , N Z C i ),that have to be added to (3.13).We close the discussion of moduli by making a more refined and illustrative statementabout the heterotic meaning of the K¨ahler modulus of the exceptional divisors D i . To do sowe have to consider heterotic M-theory on Z × S / Z . In this picture the instanton/five-brane transition can be understood [52] as a spacetime-filling five-brane wrapping C i andmoving on S / Z onto the end-of-the-world brane where one perturbative E gauge groupis located. There, it dissolves into a finite size instanton of the heterotic bundle E . Withthis in mind the distance of the five-brane on the interval S / Z away from the end-of-worldbrane precisely maps [26] to the K¨ahler modulus of the divisor D i resolving C i in B . In this section we discuss the F-theory flux superpotential and recall how mirror symmetryfor Calabi-Yau fourfolds allows to compute its explicit form [8, 9, 10, 7]. Recall, that theF-theory superpotential is induced by four-form flux G and given by [53] W G ( t ) = Z X G ∧ Ω ( t ) = N a Π b ( t ) η ab , a, b = 1 , . . . b ( X ) , (3.15)where t collectively denote the h , ( X ) complex structure deformations of X . Note thatin order to compute W G it is necessary to expand in a basis γ a of the integral homologygroup H ( X , Z ). The N a = R γ a G ∈ Z / a ( t ) = R γ a Ω ( t ) are the periods of the holomorphic (4 , . The constantintersection matrix η ab = Z X ˆ γ a ∧ ˆ γ b , Z γ a ˆ γ b = δ ab , (3.16)is defined for the integral basis ˆ γ a of the cohomology group H ( X , Z ) which is dual to γ a . Note that in contrast to H ( Z , Z ) of Calabi-Yau threefolds the fourth cohomologygroup of X does not carry a symplectic structure which necessitates the introduction of η ab .The last expression in formula (3.15) is therefore obtained by expanding G = N a ˆ γ a andΩ ( t ) = Π a ( t )ˆ γ a in the cohomology basis. 19n the F-theory side one has the following consistency condition on the flux. The firstconstraint comes from the quantization condition for G , which depends on the second Chernclass of X in the following way [54] G + c ( X )2 ∈ H ( X , Z ) . (3.17)More restrictive is the condition that G has to be primitive, i.e. orthogonal to the K¨ahlerform of X . In the F-theory limit of vanishing elliptic fiber this yields the constraints Z X G ∧ J i ∧ J j = 0 . (3.18)for every generator J i , i = 1 , . . . , h , ( X ) of the K¨ahler cone. To discuss the two conditionsfurther it is useful to remind us of the fact that the (co)homology of a Calabi-Yau splits intoa horizontal and a vertical subspace H H ( X , Z ) = M k =0 H − k,kH ( X , Z ) , H V ( X , Z ) = M k =0 H k,kV ( X , Z ) . (3.19)Since we have an even number of complex dimensions the group H , ( X , C ) contains bothparts and splits accordingly into the vertical and the horizontal subspace as [8] H , ( X ) = H , V ( X ) ⊕ H , H ( X ) . (3.20)Analogous to the two-dimensional case of K3 and in contrast to the Calabi-Yau threefoldcase, the derivatives of Ω with respect to the complex structure modulo the differential idealgiven by the Picard-Fuchs operators generate only the horizontal subspace. The remainingpart is the vertical subspace which is the natural ring of polynomials in the K¨ahler conegenerators J i modulo the ideal defining the intersection ring. Mirror symmetry exchangesthe vertical and the horizontal subspace. A corollary of these statements is that the allowedfluxes in the superpotential (3.15) are in the horizontal subspace. On the other hand Chernclasses are in the vertical subspace, so that half integral flux quantum numbers are notallowed if condition (3.18) is met. Now, the most important task on the fourfold side is tofind the periods which correspond to the integrals over an integral basis of H ( X , Z ).The first step to determine the periods is to determine the Picard-Fuchs equations L κ Π a ( t ) = 0 satisfied by the periods. The Picard-Fuchs operators L κ are differential op-erators in the complex structure moduli t . In general, the L κ can be determined by applyingGriffiths-Dwork reduction [45]. One identifies the L κ which yield exact forms when appliedto Ω , i.e. L κ Ω = dw κ , (3.21)20here w κ are three-forms on X . To derive the Picard-Fuchs operators L κ one uses anexplicit expression for the holomorphic four-form Ω via the Griffiths residuum expression[45]. For Calabi-Yau fourfold hypersurfaces and complete intersections { P = . . . = P s = 0 } with dP ∧ . . . ∧ dP s = 0 in toric varieties P ∆ of dimensions s + 4 the four-form Ω can beexpressed as Ω = Z ǫ . . . Z ǫ s s Y k =1 a ( k )0 P k ∆ . (3.22)Here ǫ i are paths in P ∆ , which encircle P i = 0 and ∆ is an measure invariant under the torusaction. The parameter a ( k )0 denotes a distinguished coefficient in the defining constraint P k as introduced below. This method is general but tedious. However, the operators L κ can alsobe determined by the toric data. They are related to the scaling relations of the dual toricvariety P ˜∆ that happens to be the ambient space of the mirror fourfold ˜ X of X . This nicelyconnects to the framework of toric mirror symmetry [55, 42, 56] where the charge vectors ℓ ( a ) ,defining the K¨ahler cone of the mirror ˜ X , determine a canonical set of differential operators,the GKZ-system, from which the Picard-Fuchs system for the complex structure of X isobtained. From these operators L κ one can evaluate a finite set of solutions Π a ( t ).In a second step, one has to identify the solutions corresponding to the integral basisof H ( X , Z ). A strategy to do this was outlined in [7] (see also refs. [9, 10]) and madeconcrete in simple examples. The key idea is to use the structure of the solution nearconifold divisors in the moduli space, where a four-cycle ν and therefore the correspondingperiod R ν Ω vanishes. The vanishing cycle ν can often be identified directly with generatorsof H ( X , Z ). Associated to each vanishing cycle, there will be a monodromy action on theperiod vector that is generated by encircling the divisor in the moduli space and is patchingthe, in general redundant, generators of these monodromies globally together.Most information comes form the large complex structure, i.e. the point of maximalunipotent monodromy whose location is the origin in the Mori cone coordinate system z a =( − ℓ ( a )0 Q mj =0 a ℓ ( a ) j j for a toric hypersurface X . For every entry ℓ ( a ) j of the Mori vectors ℓ ( a ) there are parameters a j that are just the coefficients of the constraint P = 0 defining X .At the point z = 0 several cycles γ a vanish and we have one analytic solution X ( z ) = R γ Ω and h , ( X ) logarithmic periods X a ( z ) = R γ a Ω = X ( z ) log( z a ) + Σ a ( z ). Then the mirrormap is given by t a = X a X . (3.23)Noting that t a ∼ log( z a ) at this point we can use these flat coordinates to write the leadinglogarithmic structure of the period vector asΠ T = (cid:16) Z γ Ω , . . . , Z γ b H Ω (cid:17) = X (cid:0) , t a , C δab t a t b , C abcd t b t c t d , C abcd t a t b t c t d (cid:1) . (3.24)21n particular, the grading ( { k } ) = (0 , , , ,
4) in powers of t a corresponds to a grading of γ a ∈ H ( X ). In the complex structure given by the point z the dual cohomology group hasthe natural grading H H ( X , Z ). Mirror symmetry maps this group to the vertical cohomology H V ( ˜ X , Z ). Thus, the Greek indices in (3.24) run from 1 to h , H ( X ) = h , V ( ˜ X ), the Latinindices from 1 to h , ( X ) = h , ( ˜ X ). Note that we have introduced the constant coefficients C δab = η (2) δγ C abγ , C eabc = η (1) ed C abcd that are related to the classical intersection numbers C abγ and C abcd . These are calculated in the classical geometry of ˜ X as follows. Let us denotea basis of H V ( ˜ X , Z ) by A ( k ) p k = a i ,...,i k p k ˜ J i ∧ . . . ∧ ˜ J i k , (3.25)where the ˜ J i n are the generators of the K¨ahler cone of the mirror ˜ X . Then one has C abcd = Z ˜ X A (1) a ∧ A (1) b ∧ A (1) c ∧ A (1) d , C abγ = Z ˜ X A (1) a ∧ A (1) b ∧ A (2) γ (3.26)and η (1) ab = R ˜ X A (1) a ∧ A (3) b as well as η (2) γδ = R ˜ X A (2) γ ∧ A (2) δ denote subblocks of η ab at grade k = 1 and k = 2 respectively whose inverses are indicated by upper indices. By formallyreplacing the ˜ J i with θ i = z i ddz i , we get a map µ : H V ( ˜ X , Z ) / / H H ( X , Z ) (3.27)given by µ : A ( k ) p k (cid:31) / / R ( k ) p k Ω (cid:12)(cid:12)(cid:12) z =0 := a i ,...,i k p k θ i · · · θ i k Ω (cid:12)(cid:12)(cid:12)(cid:12) z =0 , (3.28)which preserves the grading. This implies that one can think of the integral basis ˆ γ a in termsof their corresponding differential operators R ( k ) p k acting on Ω .The representation of the integral basis as differential operators will be particularly use-ful in the identification of the heterotic and F-theory superpotential. In particular, thisformalism allows us to express the flux G in an integral basis in the form G = X k =0 X p k N p k ( k ) R ( k ) p k Ω (cid:12)(cid:12) z =0 . (3.29)In the next section we will argue that the heterotic/F-theory duality map is obtained by amatching of the operators R ( k ) p k with their analogs in the heterotic blow-up. Let us finally turn to the matching of the heterotic and F-theory superpotentials. Recall,that the heterotic superpotential (2.8), is formally given by W het ( t c , t g , t o ) = W flux ( t c ) + W CS ( t c , t g ) + W M5 ( t c , t o ) , (3.30)22here t c , t g and t o denote the complex structure, bundle and five-brane moduli respectively.The last two terms are not inequivalent, since tuning the t g or t o moduli one can condense orevaporate five-branes and explore different branches of the heterotic moduli space. Clearlythe moduli spaces parametrized by t c and t g do not factorize globally in complex structureand bundle moduli since the notion of a holomorphic gauge bundle on Z depends on thecomplex structure of Z . Similarly, t c and t o do not factorize as the notion of a holomorphiccurve in Z does depend on the complex structure of Z . This is also reflected in the factthat flux and brane superpotential can be unified into one superpotential (2.18) for whichthe splitting into W M5 and W flux is just a matter of basis choice of H ( Z , C , Z ).The key point of the construction is of course that we can map the heterotic moduli( t c , t g , t o ) to the complex structure moduli t of X which are encoded in the fourfold periodintegrals. To make the equivalence W het ( t c , t g , t o ) = W G ( t ) , (3.31)precise, we need to establish a dictionary between the topological data on the heterotic side,which consist of the heterotic flux quanta, the topological classes of gauge bundles and theclass of the curves C , and the F-theory flux quanta.In order to study the duality map, we will restrict our considerations to the map betweenfive-brane moduli and complex structure deformations of Z to complex structure deforma-tions of ˆ X . This can be achieved by restricting the heterotic gauge bundle E to be of trivial SU (1) × SU (1) type. In this case one needs to include heterotic five-branes to satisfy theanomaly cancellation condition (2.1). In accord with the discussion of section 3.2 the dualfourfold ˆ X can be realized as a complete intersection blown up along the five-brane curves.As above, we will restrict the discussion to a single five-brane. We want to match thisdescription with the heterotic theory on ˆ Z . One can now identify the blow-up constraints Q = l g ( u ) − l ˜ z (cid:31) / / Q ′ = l g ( u ) − l k , ˜ z (cid:31) / / k , (3.32)where u denote coordinates on the base B , { ˜ z = 0 } defines the base B in Z , and { z =0 } ∩ { k = 0 } defines the base B in X . The map (3.32) is possible since both Z and X share the twofold base B with the curve C . The identification of ˜ z with k corresponds tothe fact that in heterotic/F-theory duality the elliptic fibration of Z is mapped to the P -fibration of B . Clearly, the map (3.32) identifies the deformations of C realized as coefficientsin the constraint { Q = 0 } of ˆ Z with the complex structure deformations of ˆ X realized ascoefficient in { Q ′ = 0 } . We also have to match the remaining constraints { P = 0 } and Note that the P -fibration B → B has actually two sections. As in section 3.2, k = 0 is one of the twosections, say, the zero section. P ′ = 0 } of ˆ Z and ˆ X , respectively. Clearly, there will not be a general match. However, aswas argued in ref. [4] for Calabi-Yau fourfold hypersurfaces, one can split P ′ = 0 as P + V E yielding a map P + V E (cid:31) / / P ′ , (3.33)where V E is describing the spectral cover of the dual heterotic bundles E = E ⊕ E . Again,this requires an identification of ˜ z and k . For SU (1) bundles this map was given in (3.32),but can be generalized for non-trivial bundles. Note that the maps (3.32) and (3.33) canalso be formulated in terms of the GKZ systems of the complete intersections ˆ Z and ˆ X .It implies that the ℓ ( a ) i of ˆ X contain the GKZ system of Z and the five-brane ℓ -vectors,similar to the situation encountered in refs. [19, 20, 7, 23, 21].To match the superpotentials as in (3.31) one finally has to identify the integral basis of H ( ˆ Z , D, Z ) with elements of H ( ˆ X , Z ) and show that the relative periods of ˆΩ can beidentified with a subset of the periods of Ω . In order to do that, one compares the residueintegrals (2.24) and (3.22) for ˆΩ and Ω represented as complete intersections. Using themaps (3.32) and (3.33) one then shows that each Picard-Fuchs operator annihilating ˆΩ isalso annihilating Ω . Hence, also a subset of the solutions to the Picard-Fuchs equationscan be matched accordingly. As a minimal check, one finds that the periods of Ω on Z before the blow-up arise as a subset of the periods of Ω is specific directions [9, 7]. Themap between the cohomologies H ( ˆ Z , D, Z ) ֒ → H ( ˆ X , Z ) is also best formulated in termsof operators R ( i ) p applied to the forms ˆΩ and Ω , R ( i ) p ˜Ω ( z c , z o ) (cid:12)(cid:12)(cid:12) z c = z o =0 (cid:31) / / R ( i ) p Ω ( z ) (cid:12)(cid:12)(cid:12) z =0 . (3.34)Note that the preimage of this map will in general contain derivatives with respect to thevariables z o and hence is an element in relative cohomology. It was shown in refs. [16] thatone can find differential operators R ( i ) p which span the full space H ( ˆ Z , D, Z ). One nowfinds that by identifying the heterotic and F-theory moduli at the large complex structurepoint z = 0, one obtains an embedding map of the integral basis.One immediate application of this formalism is that if we know the classical quadraticterms in W het we can fix the dual G -flux and use the periods of the fourfold to determinethe instanton parts. In particular, for the five-brane superpotential W ˆ M ( t o , t c ) one findsthat the dual flux G M54 can be expressed as G M54 = X p N p (2) R (2) p Ω (cid:12)(cid:12) z =0 (3.35)Note that for G fluxes generated by operators R (2) the superpotential yields an integralstructure of the fourfold symplectic invariants at large volume of the mirror Mir( ˆ X ) of ˆ X
24s [8, 9, 57] W inst G = X β ∈ H ( Mir ( ˆ X ) , Z ) n β ( γ G )Li ( q β ) , n β ∈ Z , (3.36)where γ G is co-dimension two cycle specified by the flux [7], and q β = exp( R β ˜ J ) is theexponential of the mirror K¨ahler form ˜ J integrated over classes β . This integrality structure isinherited to the heterotic superpotentials in geometric phases of their parameter spaces. Forsuperpotential from five-branes wrapped on a curve C this matches naturally the disk multi-covering formula of [58], since this part is mapped by mirror symmetry to disk instantonsending on special Lagrangians L mirror dual to C . It would be interesting to explore ageneralization of this integral structure to the gauge sector of the heterotic theory.Finally, there is geometric way to identify the flux which corresponds to a chain integral R Γ Ω . The three-chain Γ can be mapped to a three-chain Γ in B whose boundary two-cycles lie in the worldvolume of a seven-brane over which the cycles of the F-theory ellipticfiber degenerates. By fibering the one-cycle of the elliptic fiber which vanishes at the seven-brane locus over Γ, one gets a transcendental cycle in H ( X , Z ). Its dual form lies thenin the horizontal part H H ( X , Z ) and therefore yields the flux (see ref. [41] for a reviewon such constructions). For a recent very explicit construction of these cycles in F-theorycompactifications on elliptic K3 surfaces and Calabi-Yau threefolds see refs. [59]. In this section we study concrete examples to demonstrate the concepts discussed in theearlier sections. We will examine two geometries in detail. The first F-theory Calabi-Yaufourfold geometry, discussed in section 4.1 and 4.2, will have few K¨ahler moduli and manycomplex structure moduli. In this case we can use toric geometry to compute explicitly theintersection numbers, evaluate both sides of the expression (3.13) yielding the number of de-formation moduli of the five-brane curve, and check the anomaly formula (2.1). We also showthat the Calabi-Yau fourfold can be explicitly constructed from the heterotic non-Calabi-Yauthreefold obtained by blowing up the five-brane curve in section 4.2. The second Calabi-Yaufourfold example, introduced in section 4.3, will admit few complex structure moduli andmany K¨ahler moduli. This allows us to identify the bundle moduli and five-brane moduliunder duality by studying the Weierstrass constraint. The F-theory flux superpotential forthis configuration was already evaluated in ref. [7], and we will discuss its heterotic dual insection 4.3. 25 .1 Example 1: Five-Branes in the Elliptic Fibration over P We begin the discussion of our first example of heterotic/F-theory dual theories by definingthe setup on the heterotic side. Following section 2.1 the heterotic theory is specified byan elliptic Calabi-Yau threefold Z with a stable holomorphic vector bundle E = E ⊕ E obeying the heterotic anomaly constraint (2.1).We choose the threefold Z as the elliptic fibration over the base B = P with generictorus fiber P , , [6]. It is given as a hypersurface P = 0 in the toric ambient space∆( Z ) = -1 0 0 0 3 B + 9 H B + 6 H B H H H (4.1)with the class of the hypersurface Z given by[ Z ] = X D i = 6 B + 18 H . (4.2)Here we denoted the two independent toric divisors D i by H and B , the pullback of thehyperplane class of the P base respectively the class of the base itself. From the toric datathe basic topological numbers of Z are obtained as χ ( Z ) = 540 , h , ( Z ) = 2 , h , ( Z ) = 272 . (4.3)The second Chern-class of Z is in general given in terms of the Chern classes c ( B ), c ( B )and the section σ : B → Z of the elliptic fibration by c ( Z ) = 12 c ( B ) σ + 11 c ( B ) + c ( B ). Here we have σ = B and thus obtain c ( Z ) = 36 H · B + 102 H . (4.4)To satisfy the heterotic anomaly formula (2.1), we have to construct the heterotic vec-tor bundle E ⊕ E and compute the characteristic classes λ ( E i ). Since Z is ellipticallyfibered the classes λ ( E i ) can be constructed using the basic methods of [24] that werebriefly mentioned in section 3.1. According to (3.4), we first need to specify the classes η , η ∈ H ( B , Z ) essential in the spectral cover construction. We furthermore restrict E ⊕ E to be an E × E bundle over Z and choose both classes as η = η = 6 c ( B ).Then, we use the formula for the second Chern class of E -bundles λ ( E i ) = c ( E i )60 = η i σ − σ + 135 η i c ( B ) − c ( B ) (4.5)26o obtain λ ( E ) = λ ( E ) = 18 H · B − H . The anomaly condition (2.1) then leadsto conditions on the coefficients of the independent classes in H ( Z ). For the class H · B contributed by the base via σ · H ( P , Z ) this is trivially satisfied by the choice of λ ( E i ). Thisimplies that no horizontal five-branes are present. For the class of the fiber F the anomalyforces the inclusion of vertical five-branes in the class C = c ( B )+91 c ( B ) = 822 H ≡ n f F .Since F is dual to the base B the number n f of vertical branes is determined by integrating C over P , n f = Z P C = 822 . (4.6)To conclude the heterotic side we compute the index I ( E i ) since it appears in the identifica-tion of moduli (3.13) and thus is crucial for the analysis of heterotic/F-theory duality. For Z we use the formula (3.14) to obtain that I ( E ) = I ( E ) = 8 + 4 ·
360 + 18 · η i appropriately.We achieve this by putting η = 6 c ( B ) − H . The class of the five-brane C can then bedetermined analogous to the above discussion by evaluating (4.5) and imposing the anomaly(2.1). It takes the form C = 91 c ( B ) + c ( B ) − c ( B ) · H + 15 H + H · B = 702 H + H · B , (4.7)which means that we have to include five-branes in the base on a curve C in the class H ofthe hyperplane of P . Additionally the number of five-branes on the fiber F is altered to n f = 702. Accordingly, the shifting of η changes the second index to I = 1019, whereas I = 1502 remains unchanged.Let us now turn to the dual F-theory description. We first construct the fourfold X dualto the heterotic setup with no five-branes. In this case the base B of the elliptically fiberedfourfold is B = P × P . This can be seen from the relation (3.5) of the classes η i and thefibration structure of B for E -bundles. Since both classes equal 6 c we have t = 0 and thusthe bundle L = O P is trivial as well as the projective bundle B = P ( O P ⊕ O P ). Thenthe fourfold X is constructed as the elliptic fibration over B with generic fiber given by P , , [6]. Again X is described as a hypersurface in a five-dimensional toric ambient space V as described by the toric data in (4.11) if one drops the point (3 , , − , ,
1) and sets thedivisor D to zero. The class of X is then given by[ X ] = X i D i = 6 B + 18 H + 12 K , (4.8)where the independent divisors are the base B denoted by B , the pullback of the hyperplane H in P and of the hyperplane K in P . Then, the basic topological data reads χ ( X ) = 19728 , h , ( X ) = 3 , h , ( X ) = 3277 , h , ( X ) = 0 . (4.9)27ow we have everything at hand to discuss heterotic/F-theory duality along the lines ofsection 3.2, in particular the map of moduli (3.13). As discussed there, the complex structuremoduli of the F-theory fourfold are expected to contain the complex structure moduli of Z on the heterotic side as well as the bundle and brane moduli of possible horizontal five-branes.Indeed we obtain a complete matching by adding up all contributions in (3.13), h , ( X ) = 3277 = 272 + 1502 + 1502 + 1 , (4.10)where it is crucial that no horizontal five-branes with possible brane moduli are are present.To obtain the F-theory dual of the heterotic theory with horizontal five-branes, we haveto apply the recipe discussed in section 3.2. We have to perform the described geometrictransition of first tuning the complex structure moduli of the fourfold X such that it becomessingular over the curve C which we then blow up into a divisor D . This way we obtain a newsmooth Calabi-Yau fourfold denoted by ˆ X . The toric data of this fourfold are given by∆( ˆ X ) = − D + 3 B + 9 H + 6 K D − D + 2 B + 6 H + 4 K D B D H D − H − D D − H D K D − K + D D − D D . (4.11)where we included the last point (3 , , − , ,
1) and a corresponding divisor D = D toperform the blow-up along the curve C as follows.Since the curve C on the heterotic theory is in the class H we have to blow-up over thehyperplane class of P in B . First we project the polyhedron ∆( ˆ X ) to the base B which isdone just by omitting the first and second column in (4.11). Then the last point maps to thepoint ( − , ,
1) that subdivides the two-dimensional cone spanned by ( − , ,
0) and (0 , , B . Thus, upon adding this point the curve C = H in B correspondingto this cone is removed from B and replaced by the divisor D corresponding to the newpoint. Thus we see that the toric data (4.11) contain this blown-up base B in the last threecolumns.The fourfold is then realized as a generic constraint P = 0 in the class[ ˆ X ] = 6 B + 18 H + 12 K + 6 D . (4.12)28ote that this fourfold has now three different triangulations which correspond to the variousfive-brane phases on the dual heterotic side. The topological data for the new fourfold ˆ X are given by χ ( ˆ X ) = 16848 , h , ( ˆ X ) = 4 , h , ( ˆ X ) = 2796 , h , ( ˆ X ) = 0 , (4.13)where the number of complex structure moduli has reduced in the transition as expected.If we now analyze the map of moduli (3.13) in heterotic/F-theory duality we observethat we have to put h ( C , N Z C ) = 2 in order to obtain a matching. This implies, fromthe point of view of heterotic/F-theory duality, that the horizontal five-brane wrapped on C has to have two deformation moduli. Indeed, this precisely matches the fact that thehyperplane class of P has two deformations since a general hyperplane is given by the linearconstraint a x + a x + a x = 0 in the three homogeneous coordinates x i of P . Upon theoverall scaling it thus has two moduli parameterized by the P with homogeneous coordinates a i . This way we have found an explicit construction of an F-theory fourfold with complexstructure moduli encoding the dynamics of heterotic five-branes.In section 4.2 we provide further evidence for this identification by showing that onecan also construct ˆ X as a complete intersection starting with a heterotic non-Calabi-Yauthreefold. Unfortunately, it will be very hard to compute the complete superpotential forthe fourfold ˆ X since it admits such a large number of complex structure deformations. Itwould be interesting, however, to extract the superpotential for a subsector of the moduliincluding the two brane deformations. Later on we will take a different route and considerexamples with only a few complex structure moduli which are constructed by using mirrorsymmetry.
In this section we discuss the example of section 4.1 employing the blow-up proposal ofref. [18] as discussed in section 2.3. More precisely, we will explicitly construct a non-Calabi-Yau threefold ˆ Z which is obtained by blowing up the horizontal five-brane curve into adivisor. This translates the deformations of C into new complex structure deformations ofˆ Z . The F-theory Calabi-Yau fourfold ˆ X is then naturally obtained from the base of ˆ Z byan additional P fibration. ˆ X is identical to the fourfold considered in section 4.1, despitethe fact that it is now realized as a complete intersection. If one considers exactly the mirror of ˆ X , as we will in fact do in section 4.3, it might be possible toembed this reduced deformation problem into the complicated deformation problem of ˆ X constructed inthis section.
29s in section 4.1 the starting point is the elliptic fibration Z over B = P with a five-brane wrapping the hyperplane class of the base. Let us describe the explicit constructionof ˆ Z . The blow-up geometry ˆ Z is given by P ( N Z C ). Z is a hypersurface { P = 0 } ina toric variety V and the curve C is given as a complete intersection of two hypersurfacesin Z , i.e. C = { h = 0 } ∩ { h = 0 } ⊂ Z . The charge vectors of V are given by { ℓ ( i ) } with i = 1 , . . . , k . We are aiming to construct a five-dimensional toric variety which is givenby ˆ V = P ( N V C ) and use the blow-up equation described in section 2.3. Let us denotethe divisor classes defined by h i by H i and the charges of h i by µ i = ( µ (1) i , . . . µ ( k ) i ). Then,the coordinates l i of N V H i transform with charge µ ( m ) i under the k scaling relations. Thenormal bundle N V C is given by N V H ⊕ N V H . Since we have to projectivize N V C , wehave to include another C ∗ -action with charge vector ℓ ( k +1)ˆ V acting non-trivially only on thenew coordinates l i . The new charge vectors of ˆ V are thus given by the following tablecoordinates of V l l ℓ (1)ˆ V ℓ (1) µ (1)1 µ (1)2 ... ... ... ... ℓ ( k )ˆ V ℓ ( k ) µ ( k )1 µ ( k )2 ℓ ( k +1)ˆ V Z is now given as a complete intersection P = 0 , l h − l h = 0 . (4.14)To apply this to the elliptic fibration over P with the polyhedron (4.1), one picks the curve C given by { ˜ z = 0 } and { x = 0 } . C is a genus zero curve and we will find that the exceptionaldivisor D will be the first del Pezzo surface dP in accord with the discussion of section 2.3.We construct the five-dimensional ambient manifold as explained above,∆( ˆ Z ) = -1 0 0 0 0 3 B + 3 D + 9 H B + 2 D + 6 H B H H H D H − D . (4.15)Note that one has to include the inner point (3 , , , ,
0) which corresponds to the base ofthe elliptic fibration ˆ Z . Furthermore, one shows that the point (0 , , , , Z . Explicitly the complete intersection ˆ Z is given by a generic constraint inthe class ˆ Z : (6 B + 6 D + 18 H ) ∩ H , (4.16)where
H, B, D are the divisor classes of the ambient space (4.15). The first divisor in (4.16)is the sum of the first seven divisors in (4.15) and corresponds to the original Calabi-Yauconstraint P = 0 in (4.14). The second divisor in (4.16) is the sum of the last two divisorsand is the class of the second equation of (4.14). This complete intersection threefold has χ ( ˆ Z ) = −
538 = χ ( Z ) − χ ( P ) + χ ( dP ), and one checks that the exceptional divisor D has the characteristic data of a del Pezzo 1 surface. This means that we have replaced thehyperplane isomorphic to P in the base with the exceptional divisor which is dP . It canbe readily checked that the first Chern class of ˆ Z is non-vanishing and equals − D .Having described the blow-up geometry, we now turn to the construction of the fourfoldˆ X for F-theory. This fourfold will also be constructed as complete intersection, but it willbe the same manifold as the fourfold described in section 4.1, equation (4.11). We fiber anadditional P over P (∆( ˆ Z )) which is only non-trivially fibered along the exceptional divisor.This is analogous to the construction of the dual fourfold in the heterotic/F-theory dualitywhere one also fibers P over the base twofold of the Calabi-Yau threefold to obtain theF-theory fourfold. Here we proceed in a similar fashion but construct a P -fibration overthe base of the non-Calabi-Yau manifold ˆ Z . This base is a complete intersection and thusleads to a realization of ˆ X as a complete intersection. Concretely, we have the followingpolyhedron∆( ˆ X ) = − D + 3 B + 9 H + 6 K D − D + 2 B + 6 H + 4 K D B D H D − H D − H D K D − K + D D − D D − H − D D . (4.17)The fourfold ˆ X is given as the following complete intersectionˆ X : (6 B + 6 D + 18 H + 12 K ) ∩ H . (4.18)Note that this fourfold is indeed Calabi-Yau as can be checked explicitly by analyzing thetoric data (4.17). For complete intersections the Calabi-Yau constraint is realized via the two31artitions, so-called nef partitions, in (4.17) as in refs. [62]. The first nef partition yields thesum of the first eight divisors P i =1 D i in (4.17) and gives the first constraint in (4.18). Thesecond nef partition yields the sum of the last two divisors D + D in (4.17) and yield thesecond constraint in (4.18). The divisors D and D correspond to the P fiber in the baseof ˆ X obtained by dropping the first two columns in (4.17). This fibration is only non-trivialover the exceptional divisors D = D in the second nef partition of (4.17). Note that if onesimply drops K from the expression (4.18) one formally recovers the constraint (4.16) of ˆ Z .To check that the complete intersection ˆ X is precisely the fourfold constructed in section4.1, one has to compute the intersection ring and Chern classes. In particular, it is not hardto show that also (4.17) has three triangulations matching the result of section 4.1.In summary, we have found that there is a natural construction of ˆ X as complete in-tersection with the base obtained from the heterotic non-Calabi-Yau threefold ˆ Z . Let usstress that this construction will straightforwardly generalize to dual heterotic/F-theory se-tups with other toric base spaces B and different types of bundles. For example, to studythe bundle configurations on Z of section 4.1 with η , = 6 c ( B ) ± kH, k = 0 , , D → (3 , , , , k ) , D → (3 , , , , , k ) , (4.19)in the polyhedra (4.11) and (4.17), respectively. Moreover, also bundles which are not ofthe type E × E can be included by generalizing the form of the P fibration just as in thestandard construction of dual F-theory fourfolds. Let us now discuss a second example for which the F-theory flux superpotential can becomputed explicitly since the F-theory fourfold admits only few complex structure moduli.Clearly, using mirror symmetry such fourfolds can be obtained as mirror manifolds of ex-amples with few K¨ahler moduli. To start with, let us consider heterotic string theory onthe mirror of the Calabi-Yau threefold which is an elliptic fibration over P . This mirror isthe heterotic manifold Z . One shows by using the methods of ref. [60], that this Z is alsoelliptically fibered, such that, at least in principle, one can construct the bundles explicitly.The polyhedron of Z is the dual polyhedron to (4.1) and the Weierstrass form of Z is asfollows µ = x + y + xy ˜ za u u u + ˜ z ( a u + a u + a u + a u u u ) . (4.20)The coordinates { u i } are the homogeneous coordinates of the twofold base B . Note thatone finds that the elliptic fibration of this Z is highly degenerate over B . The threefoldis nevertheless non-singular since the singularities are blown up by many divisors in the32oric ambient space of Z . In writing (4.20) many of the coordinates parameterizing theseadditional divisors have been set to one. Turning to the perturbative gauge bundle E ⊕ E we will restrict in the following to the simplest bundle SU (1) × SU (1) which thus preservesthe full perturbative E × E gauge symmetry in four dimensions. To nevertheless satisfythe anomaly condition (2.1) one also has to include five-branes. In particular, we consider afive-brane in Z given by the equations h = b u + b u u u = 0 , h = ˜ z = 0 . (4.21)The curve C wrapped by the five-brane is thus in the base B of Z . Unfortunately, it is hardto check (2.1) explicitly as in the example of section 4.1 since there are too many K¨ahlerclasses in Z . However, one can proceed to construct the associated Calabi-Yau fourfold ˆ X which encodes a consistent completion of the setup.The associated fourfold ˆ X cannot be constructed as it was done in section 4.1. However,one can employ mirror symmetry to first obtain the mirror fourfold Mir( ˆ X ) of ˆ X as Calabi-Yau fibration Mir( Z ) / / Mir( ˆ X ) (cid:15) (cid:15) P (4.22)where Mir( Z ) is the mirror of the heterotic threefold Z [4]. This naturally leads us toidentify ˆ X as the mirror to the fourfold (4.11) from section 4.1. This fourfold is also themain example discussed in detail in ref. [7]. In the following we will check that this is indeedthe correct identification by using the formalism of refs. [4, 26]. The Weierstrass form of ˆ X can be computed using the dual polyhedron of (4.11) yielding µ = y + x + m ( u i , w j , k m ) xyz + m ( u i , w j , k m ) z = 0 , (4.23)where m ( w j , u i ) = a u u u w w w w w w k k ,m ( w j , u i ) = a ( k k ) u w w w w + a ( k k ) u w w + a ( k k ) u w w + a ( k k ) ( u u u w w w w w w ) (4.24)+ b k u w w w w + b k ( u u u ) ( w w w ) + c k ( u u u ) ( w w w ) . The coordinates u i are the coordinates of the base twofold B as before and w i , k , k arethe additional coordinates of the base threefold B . Again, note that we have set many Note that the blow-down of these divisors induces a large non-perturbative gauge group in the heteroticcompactification. µ . The chosen coordinates correspond to divisors whichinclude the vertices of ∆( X ) and hence determine the polyhedron fully. In particular, onefinds that k , k are the coordinates of the fiber P over B . The coefficients a i , b , b , c denotecoefficients encoding the complex structure deformations of ˆ X . However, since h , ( ˆ X ) = 4,there are only four complex structure parameters rendering six of the a i redundant. As thefirst check that ˆ X is indeed the correct geometry, we use the stable degeneration limit [24]and write µ in a local patch with appropriate coordinate redefinition as follows [4] µ = p + p + + p − , (4.25)where p = x + y + xy ˜ za u u u + ˜ z (cid:0) a u + a u + a u + a u u u (cid:1) ,p + = v ˜ z (cid:0) b u + b u u u (cid:1) , (4.26) p − = v − ˜ z c u u u . The coordinate v is the affine coordinate of the fiber P . In the stable degeneration limit { p = 0 } describes the Calabi-Yau threefold of the heterotic string. In this case p coincideswith µ which means that the heterotic Calabi-Yau threefold of ˆ X is precisely Z . Thisshows that the geometric moduli of Z are correctly embedded in ˆ X . The polynomials p ± encode the perturbative bundles, and the explicit form (4.26) shows that one has a trivial SU (1) × SU (1) bundle. This fact can also be directly checked by analyzing the polyhedronof ˆ X using the methods of [34, 61]. One shows explicitly that over each divisor k i = 0 in B a full E gauge group is realized. Since the full E × E gauge symmetry is preserved weare precisely in the situation of section 3.2, where we recalled from ref. [26] that a smoothˆ X contains a blow-up corresponding to a heterotic five-brane. We will now check that thisallows us to identify the brane moduli in the duality.Let us now make contact to section 3.2. To make the perturbative E × E gaugegroup visible in the Weierstrass equation (4.23), we have to include new coordinates (˜ k , ˜ k )replacing ( k , k ). This can be again understood by analyzing the toric data using themethods of [61, 34]. We denote by (3 , , ~µ ) the toric coordinates of the divisor correspondingto ˜ k in the Weierstrass model. Then the resolved E singularity corresponds to the points (3 , , n~µ ) , n = 1 , ..., , (2 , , n~µ ) , n = 1 , ..., , (4.27)(1 , , n~µ ) , n = 1 , , , (1 , , n~µ ) , n = 1 , , (0 , , ~µ ) Note that we have chosen the vertices in the P , , [6] to be ( − , , (0 , − , (3 ,
2) to match the discussionin refs. [34, 61]. However, if one explicitly analyses the polyhedron of ˆ X one finds that one has to apply a Gl (2 , Z ) transformation to find a perfect match. This is due to the fact that ˆ X , in comparison to its mirrorMir( ˆ X ), actually contains the dual torus as elliptic fiber. , , ~µ ) corresponding to k is a vertex of the polyhedron, (3 , , ~µ ) corresponding to˜ k is an inner point. Using the inner point for ˜ k , the Weierstrass form µ changes slightly,while the polynomials p , p + and p − can still be identified in the stable degeneration limit.To determine g in (3.9), we compute g of the Weierstrass form in a local patch where ˜ k = 1 g = ˜ k (cid:16) b u + b u u u + ˜ k (cid:0) a u + a u + . . . (cid:1)(cid:17) . (4.28)The dots contain only terms of order zero or higher in ˜ k . Comparing this with (3.12), g isgiven by g = b u + b u u u . (4.29)This identifies { g = 0 } with the curve of the five-brane in the base B of Z and is inaccord with (4.21). One concludes that ˆ X is indeed a correct fourfold associated to Z with the given five-brane. As we can see from (4.29), the five-brane has one modulus. If wecompare g with p + , we see that p + = v ˜ z g . This nicely fits with the bundle description. Inour configuration, p + and p − should describe SU (1)-bundle since we have the full unbrokenperturbative E × E -bundle as described above. The SU (1)-bundles do not have any moduli,such that the moduli space corresponds to just one point [24]. In the explicit discussion ofthe Weierstrass form in our setting, p + has one modulus which corresponds to the modulusof the five-brane. Note that the Calabi-Yau fourfold ˆ X is already blown up along the curve˜ k = g = 0 in the base of ˆ X . This blow-up can be equivalently described as a completeintersection as we discussed in the previous sections. A simple example of such a constructionwas presented in section 4.2.Finally, we consider the computation of the flux superpotential. Here, we do not need torecall all the details, since the superpotential for this configuration was already studied inref. [7]. The different triangulations of Mir( ˆ X ) correspond to different five-brane configu-rations. The four-form flux, for one five-brane configurations, was shown to be given in thebasis elementsˆ γ (2)1 = θ ( θ + θ )Ω | z =0 , ˆ γ (2)1 = θ ( θ − θ + 6 θ − θ )Ω | z =0 , (4.30)where the θ i = z i ddz i are the logarithmic derivatives as introduced in (3.28). The moduli z , z can be identified as the deformations of the complex structure of the heterotic threefold Z ,while z corresponds to the deformation of the heterotic five-brane. A non-trivial checkof this identification was already provided in [7], where it was shown that the F-theoryflux superpotential in the directions (4.30) matches with the superpotential for a five-braneconfiguration in a local Calabi-Yau threefold obtained by decompactifying Z . This non-compact five-brane can be described by a point on a Riemann surface in the base B of The deformation z describes the change in p − . . Using heterotic F-theory duality as in section 3, one can now argue that the flux (4.30)actually describes a compact heterotic five-brane setup. In this work we have studied the dynamics of heterotic five-branes using the duality betweenthe heterotic string and F-theory compactifications. In particular, we have exploited thefact that five-branes wrapped on the base of an elliptically fibered Calabi-Yau threefold Z map under duality into the geometry of the F-theory Calabi-Yau fourfold X . Thisimplies that the heterotic five-brane superpotential has to be identified with a F-theory fluxsuperpotential. On the heterotic side the five-brane superpotential is given by a chain integral R Γ Ω over the holomorphic three-form of Z . Upon identifying the F-theory four-form fluxwhich corresponds to this three-chain Γ, the determination of the superpotential becomesa tractable task [7]. This is due to the fact that the deformation moduli of the five-braneare mapped to complex structure deformations of the dual Calabi-Yau fourfold X . Theirdynamics is then captured by the periods of the holomorphic four-form on X .The construction of the F-theory fourfold dual to a five-brane has been argued to involvea blow-up of the five-brane curve [5, 4]. We have provided further evidence for this proposalby noting that this blow-up can also be performed in the heterotic Calabi-Yau threefold.Following our discussion in ref. [18], the deformation moduli of the five-brane curve becomenew complex structure deformation of the blown-up K¨ahler threefold. This space is nolonger Calabi-Yau and the vanishing of ˆΩ implies, that the heterotic flux naturally mapsto the relative cohomology of ˆ Z . This allows for an equal treatment of the different partsof the superpotential by expressing the complete heterotic flux supporting both the five-brane and flux superpotential as derivatives of ˆΩ with respect to the complex structure ofˆ Z . Finally, we were able to explicitly show that there exists a natural map of this non-Calabi-Yau threefold to the F-theory Calabi-Yau fourfold. In an upcoming publication [63]such maps from a more general class of non-Calabi-Yau threefolds to Calabi-Yau fourfoldsis constructed and verified by explicit computations on both geometries.By the identification of the fourfold variables (3.23) with the heterotic variables in thesuperpotentials W G ( t ) = W het ( t c , t g , t o ), the integral structure (3.36) of the fourfold sym-plectic invariants at large volume [8, 57] is now inherited to the heterotic superpotentials ingeometric phases of their parameter spaces. For the superpotential from five-branes wrappedon a curve C this matches naturally the disk multi-covering formula of [58], since this partis mapped by mirror symmetry to disk instantons ending on special Lagrangians L mirror36ual to C . Similar for the heterotic flux superpotential it matches the expectations fromthe rational curve counting on threefolds as encoded in the period of the threefold. For theChern-Simons part of the potential we obtain by our construction integer geometric invariantsfor the gauge bundles on Calabi-Yau threefolds whose precise relation to Donaldson-Thomasinvariants is an interesting subject of research. Finally the Picard-Fuchs system of the four-fold allows to analytically continue the superpotential away from the geometric phases of theopen/gauge and closed moduli space into the interior of this moduli space and to find thecorrect open and closed flat coordinates in these regions, see e.g. for the orbifold points [64].Let us point out some applications of our results. Firstly, the computation of the su-perpotential is crucial in the study of moduli stabilization. The F-theory fourfold setupprovides powerful tools to determine heterotic vacua in which five-brane and bundle moduliare stabilized. As the F-theory flux superpotential can be determined at an arbitrary pointin the moduli space, one is able to study a landscape of heterotic vacua with five-branes andgauge-bundle configurations far inside the moduli space. Since the F-theory K¨ahler potentialfor the complex structure moduli of the Calabi-Yau fourfold is computable as a function ofthe periods of the holomorphic four-form, one also expects to determine the kinetic termsof the five-brane and bundle moduli at different points in the moduli space. It will be aninteresting task to explicitly determine the heterotic K¨ahler potential close to singular con-figurations and to search for interesting supersymmetric and non-supersymmetric vacua inanalogy to the Type IIB analysis [39, 40, 41].A second application will be the study of the heterotic compactifications which are dualto phenomenologically appealing F-theory vacua. Recently, in refs. [65], a promising classof Calabi-Yau fourfolds for GUT model building was constructed by blowing up singularcurves in the base of an elliptic fourfold. The geometries were explicitly realized as completeintersections in a toric ambient space. Remarkably, these manifolds share various proper-ties with the geometries constructed in this work. To explore this relation and the use ofheterotic/F-theory duality in more detail will be an interesting and important task [66]. Acknowledgments
We gratefully acknowledge discussions with Ralph Blumenhagen, Johannes Walcher andespecially Timo Weigand. T.G. would like to thank the MPI Munich for hospitality. Thiswork was supported in parts by the European Union 6th framework program MRTN-CT-2004-503069 “Quest for unification”, MRTN-CT-2004-005104 “ForcesUniverse”, MRTN-CT-2006-035863 “UniverseNet”, SFB-Transregio 33 “The Dark Universe” by the DFG. The workof T.-W.H. and D.K. is supported by the German Excellence Initiative via the graduate37chool “Bonn Cologne Graduate School”. The work of D.K. is supported by a scholarshipof the “Deutsche Telekom Stiftung”.