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 flux G4 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 integers N(k) athe flux quanta. Both periods and flux quanta are summarized
in vectors N, Π, where Σ denotes a h4H× h4
H-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 flux G4 is restricted by the two conditions (4.35) and (4.36). The
latter condition implies that G4 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
flux G4 in an integral basis in the form
G4= 4 X k=0 X pk Npk(k)R(k) pkΩ4 z=0 . (5.101)
28Recall that in contrast to H3(Z
3, Z) of Calabi-Yau threefolds the fourth cohomology group of X4does not
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 flux G4 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 of WG4 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 of HH
4 (X4, Z). Moreover, an intrinsic
definition of the integral basis γa(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 point z = 0 several cycles γa(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) = Rγ
0Ω4 corresponding to the fundamental period, h
(3,1)(X
4) logarithmic periods
Xa(z) = Rγ
aΩ4 ∼ X
0(z) log(z
a), 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, 12Cab0 δtatb, 3!1Cbcd0 atbtctd, 4!1Cabcd0 tatbtctd . (5.102)
Here we have introduced the constant coefficients C0 δ
ab := η(2) δγC 0 (1,1,2)
abγ , Cabc0 0 = η(1) edK0abcd
that are related to the classical three-point function Cabγ0 ,(1,1,2) and the intersection numbers K0
abcd, 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 γ
a ∈ H4(X4) which matches the
grading of the dual cohomology group HH4(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 periods F0(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 choices G4.
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 of X4, 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 fourfold 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 of X4
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 bundles E via the methods of section 4.1.4. We equip ˜Z3with an E8×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 of Z3 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:
Z3 tt M S ** πZ E ? _ o Het/F // K3 oo /X4 πX oo M S //X˜4 πX˜ ˜ Z3 ? _ o B2Z B2Z P1 . (6.1)
Here we used the abbreviations M S and Het/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 fourfold πX : 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 fourfolds X4 that we consider here have to have, for technical
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 a P1-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 over P2. This guarantees
a small number of only four complex structure moduli in the mirror X4. We emphasize that
˜
Z3 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).