5.4 Basics of Enumerative Geometry
5.4.4 Flux Superpotentials on Calabi-Yau Fourfolds
In this section we discuss the F-theory flux superpotential and recall how mirror symmetry for Calabi-Yau fourfolds [79, 93–95] allows to relate it to the enumerative geometry of the A-model. In this route we demonstrate some general features of the flux superpotential.
Recall, that the F-theory superpotential is induced by four-form fluxG4 and given by [14]
WG4(z) =X
k
N(k)aΠ(k)b(z)ηab(k)=NΣΠ , (5.98) where z collectively denote the h(3,1)(X4) complex structure deformations of X4, Π(k)a the periods and the integersN(k)athe flux quanta. Both periods and flux quanta are summarized in vectorsN, Π, where Σ denotes ah4H×h4H-matrix containing the topological metric28 η(k)ab in (5.44). Here we further used the expansions into an integral basis ˆγa(k) of HH4(X4,Z) as
Ω4 =X
k
Π(k)aγˆ(k)a, Π(k)a= Z
γa(k)
Ω4, G4 =X
k
N(k)a ˆγa(k) , N(k)a= Z
γa(k)
G4. (5.99) We refer to section 5.3 and in particular (5.45) for more details on the notation.
In F-theory setups the fluxG4 is restricted by the two conditions (4.35) and (4.36). The latter condition implies thatG4 is an element in the primary horizontal subgroup
H(2,2)(X4) =HV(2,2)(X4)⊕HH(2,2)(X4). (5.100) A corollary of this statement is that the Chern classes are in the vertical subspace, so that half integral flux quantum numbers are not allowed if condition (4.36) is met. In general, it is an important open problem to have a description of four-flux G4 on a generic Calabi-Yau fourfold. Formally, this can be solved by mirror symmetry established via the map (5.65). This implies that one can think of the integral basis ˆγain terms of their corresponding differential operatorsR(k)a acting on Ω4. In particular, this formalism allows us to express the fluxG4 in an integral basis in the form
G4= X4
k=0
X
pk
Npk(k)R(k)pkΩ4
z=0 . (5.101)
28Recall that in contrast toH3(Z3,Z) of Calabi-Yau threefolds the fourth cohomology group ofX4does not carry a symplectic structure which necessitates the introduction ofη(k)ab.
5.4. BASICS OF ENUMERATIVE GEOMETRY 131 The representation of the integral basis as differential operators will be particularly useful in the identification of the heterotic and F-theory superpotential, cf. chapter 6 and section 8.4.2.
Once the fluxG4 is fixed, also in the fourfold case the flux superpotential (5.98) is com-pletely determined in terms of the periods Π. As we have discussed in section 5.3.3 the periods Π can in principle be determined from the Picard-Fuchs differential system which allows for an analytic continuation ofWG4 deep into the complex structure moduli space of X4. However, it is the most important task on the B-model side to find the fourfold periods which are evaluated with respect to an integral basis ofH4H(X4,Z). Moreover, an intrinsic definition of the integral basisγa(k)seems to be technically impossible, due to the absence of a symplectic basis as in the threefold case, and mirror symmetry and analytic continuation, like the monodromy analysis at the conifold in section 5.3.4, have to be used in order to construct an integral basis.
In the context of mirror symmetry it is meaningful to comment on the structure of (5.98) at distinguished points in the complex structure moduli space. Again the large complex structure/large volume point is of particular importance since an interpretation as classical volumes and quantum instantons on the A-model side is possible.
For a toric hypersurface X4 the point of maximal unipotent monodromy is the origin in the Mori cone coordinate system z introduced in section 5.2.4 as in the threefold case.
Geometrically at the pointz= 0 several cyclesγa(k)hierarchically vanish which is encoded in the grading of the solutions to the Picard-Fuchs system by powers in (log(zi))k,k= 0,4, see (5.63). According to the map (5.65) and the condition (5.66) there is one analytic solution X0(z) = R
γ0Ω4 corresponding to the fundamental period, h(3,1)(X4) logarithmic periods Xa(z) = R
γaΩ4 ∼ X0(z) log(za), h(2,2)H (X4) double logarithmic solutions, h(3,1)(X4) triple logarithms and one quartic logarithms. Noting that ta ∼ log(za) at this point we can use these flat coordinates to write the leading logarithmic structure of the period vector as
ΠT = Z
γ(0)
Ω4, . . . , Z
γ(4)
b4 H
Ω4
∼X0 1, ta, 12Cab0δtatb, 3!1Cbcd0atbtctd, 4!1Cabcd0 tatbtctd
. (5.102)
Here we have introduced the constant coefficients Cab0δ :=η(2)δγCabγ0 (1,1,2), Cabc0 0 =η(1)edK0abcd that are related to the classical three-point function Cabγ0,(1,1,2) and the intersection numbers K0abcd, cf. (5.68). These couplings can be calculated in the classical cohomology ring of ˜X4 in the basis (5.55) via the integrals (5.56) and (5.58). In particular, the grading ({k}) = (0,1,2,3,4) in powers of ta corresponds to a grading of γa ∈ H4(X4) which matches the grading of the dual cohomology groupHH4(X4,Z) in the fixed complex structure given by the point z. We note that the periods (5.102) contain instanton corrections that we suppressed for convenience, that are however crucial for the A-model.
For applications of fourfolds for example to F-theory the instanton corrections in particular help to identify the physical meaning of the periods, like e.g. the interpretation in terms of the
flux or brane superpotential of the underlying Type IIB theory in the limit (4.41). Indeed the comparison of the enumerative data of the double logarithmic periodsF0(γ) with the Ooguri-Vafa double-covering (5.97) will allow us in section 6.1 to identify periods corresponding to Wbrane for specific flux choicesG4.
Chapter 6
Constructions and Calculations in String Dualities
In this chapter we present explicit calculations of the effective superpotentials in F-theory and in heterotic/F-theory dual setups, where we mainly follow [79] and [81]. First in sec-tion 6.1 we explicitly calculate the F-theory flux superpotential for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds and extract the Type IIB flux and seven-brane superpotential along the lines of the discussion in section 4.2.3. Before delving into the details of the calculations we first present a general strategy to obtain elliptic Calabi-Yau fourfolds X4 with a little number of complex structure moduli. This is necessary in order to work with a technically controllable complex structure moduli space ofX4, which we will mainly study using Picard-Fuchs equations and cross-checks from mirror symmetry. We put particular emphasis on the toric realization of examples with few moduli and the toric means to analyze possible fibration structures. Then we start with the construction of a concrete Calabi-Yau fourfold, that we realize as a toric Calabi-Yau hypersurface. We focus on one main example and refer to appendix C.2 for two further examples. For this example we perform a detailed analysis of the seven-brane dynamics as encoded by the discriminant of the elliptic fibration, comment on the Calabi-Yau threefold geometry of its heterotic dual and finally calculate, using fourfold mirror symmetry, the Type IIB flux and seven-brane superpotential.
This is technically achieved by specifying appropriate four-flux G4, that singles out linear combination of fourfold periods, that are then identified with the Type IIB Calabi-Yau three-fold periods upon matching the classical terms and the instanton corrections on the fourthree-fold with the Calabi-Yau threefold results. The Type IIB seven-brane superpotential is similarly identified by a matching of the brane disk instanton invariants with the fourfold instanton invariants.
Then in section 6.2 we compute the heterotic superpotential for a Calabi-Yau threefold compactification on Z3 from its F-theory dual setup. Before dealing with concrete examples we comment on the general matching strategy of the F-theory flux superpotential with the heterotic superpotential. We emphasize the duality map for heterotic five-branes and their superpotential to the F-theory side. Then we study in detail two explicit examples of heterotic string compactifications. Here we essentially take the same threefold geometries that we used in section 6.1.2 in order to obtain Calabi-Yau fourfolds X4 with a small number of complex
structure moduli. This surprising coincidence goes back to the rich fibration structure ofX4 and its mirror ˜X4, which is, on the one hand, a consequence of the requirement of a little number of moduli but, on the other hand, also natural for heterotic/F-theory dual setups, see the schematic diagram (6.1) and [103].
The first example of a heterotic threefold we consider, denoted by ˜Z3, has a small number of K¨ahler moduli, which allows to explicitly calculate intersection numbers and topological indices as necessary to e.g. concretely construct heterotic bundlesEvia the methods of section 4.1.4. We equip ˜Z3with anE8×E8bundle and explicitly construct the F-theory dual geometry X4, once in the absence and once in the presence of horizontal heterotic five-branes. Indeed we are able to check the duality map (4.62) between F-theory and heterotic moduli explicitly for both cases. The corresponding heterotic superpotentials can however not be computed due to a huge amount of complex structure moduli of ˜Z3 and X4. The converse situation applies in the second example, which is basically a heterotic compactification on the mirror threefold Z3 of ˜Z3. For this example we are not able to explicitly demonstrate the matching of moduli, but directly study the complex geometry ofZ3 and its F-theory dual fourfold X4. This allows both to extract the heterotic bundle and five-brane moduli and to calculate the heterotic flux and five-brane superpotential explicitly. Finally, we specify the corresponding flux G4 that, together with the knowledge of the periods from section 6.1, determines the dual heterotic superpotential completely.
Before we start let us for future reference summarize the fibration structures of the Calabi-Yau three- and fourfolds we consider in the following, both in sections 6.1 and 6.2:
Z3tt
M S
**
πZ
E o ? _
Het/F
// K3
oo /X4
πX
oo M S // ˜X4
πX˜
Z˜3 o ? _
B2Z B2Z P1
. (6.1)
Here we used the abbreviationsM SandHet/F for the action of mirror symmetry respectively heterotic/F-theory duality. In words, starting from a heterotic string compactification on an elliptic threefold πZ : Z3 → B2Z with generic elliptic fiber E we obtain the F-theory dual elliptic K3-fibered fourfoldπX : X4→B2Z as the fourfold mirror to the Calabi-Yau threefold fibered fourfold πX˜ : ˜X4 →P1 with generic fiber ˜Z3, which in turn is the mirror of Z3.
6.1 F-theory, Mirror Symmetry and Superpotentials
In this section we explicitly perform the computation of the F-theory flux superpotential (5.98). The class of Calabi-Yau fourfoldsX4 that we consider here have to have, for technical reasons, a low number of complex structure moduli. We outline in section 6.1.1 a strategy,
6.1. F-THEORY, MIRROR SYMMETRY AND SUPERPOTENTIALS 135 cf. [94, 95], to construct such examples of fourfolds X4 as in (6.1) as the mirror dual to a fourfold ˜X4 with a small number of K¨ahler moduli, that itself is realized as a Calabi-Yau threefold fibration ˜Z3over aP1-base. In addition we discuss toric means to identify interesting fibration structures like an elliptic or K3-fibration, which is of particular importance both for constructions of F-theory and heterotic/F-theory dual geometries.
Then in section 6.1.2 we fix a concrete Calabi-Yau fourfold ˜X4 by specifying the threefold fiber ˜Z3, that is given for the example at hand as an elliptic fibration overP2. This guarantees a small number of only four complex structure moduli in the mirrorX4. We emphasize that Z˜3 can be viewed as a compactification of the local geometry KP2 → P2 which was studied in [107] in the context of mirror symmetry with D5-branes on the local mirror geometry given by a Riemann surface Σ. We exploit this fact in our analysis of the seven-brane content of the F-theory compactification on X4, where the local brane geometry of Σ can be made visible as an additional deformation modulus of the discriminant of the elliptic fibration of X4. Finally in section 6.1.3 we determine the solutions of the Picard-Fuchs system for X4 and obtain the linear combination of solutions F0(γ), which depends on this distinguished deformation modulus and which we thus identify as the Type IIB seven-brane superpotential.
In addition, we check this assertion further by mirror symmetry, namely a comparison of the fourfold instanton invariants of F0(γ) with the disk instantons1 in the limit of the local brane geometry considered in [107]. Analogously we determine the Type IIB flux superpotential by a matching of the classical terms and the world-sheet instanton corrections from the fourfold periods. This explicitly demonstrates the split of the F-theory flux superpotential into the Type IIB flux and brane superpotential as required in (4.41) and thus confirms the unified description of Type IIB open-closed deformations and obstructions in F-theory. We conclude with an independent check via the heterotic dual on Z3, compare to the diagram in (6.1).