Let us first discuss the flux superpotential Wflux as it occurs in Type IIB and heterotic
compactifications, cf. sections 3.3.2 respectively 4.1.3, and its meaning for mirror symmetry. As noted before it takes in both setups the same form and can be evaluated in terms of the periods (XK,FK) of the holomorphic three-form Ω as
Wflux= ˆNKXK(z)− ˆMKFK(z) , XK = Z AK Ω , FK = Z BK Ω , (5.94) where ( ˆMK, ˆNK) = (MK − τ ˜MK, NK − τ ˜NK) are complex numbers in an N = 2 Type
IIB theory formed from the flux quantum numbers (MK, NK) of F3 and ( ˜MK, ˜NK) of H3.
Whereas in the O3/O7-orientifold setup both F3 and H3 fluxes contribute to Wflux, in the
O5/O9-orientifold setup of section 3.3.2 there are only fluxes F3 due to the orientifold pro-
jection O. Similarly in the heterotic setup of 4.1.3 only NS–NS fluxes are present. As before we have introduced the symplectic basis (AK, BK) of three-cycles in H3(Z3, Z).
As we have reviewed in section 5.2 the complete complex structure moduli dependence of Wflux is determined, once the flux is specified, from the periods obeying the Picard-Fuchs
differential system. The use of the Picard-Fuchs system makes it even possible to evaluate the flux superpotential deep inside the complex structure moduli space, where conventional supergravity breaks down due to strong curvature effects for example from singularities in Z3 like the conifold. Thus, Wflux in general inherits the characteristic properties of special
geometry, in particular the existence of the prepotential. In other words, even in N = 1 effective actions, the flux superpotential enjoys remnants of an underlying N = 2 structure.
In the context of mirror symmetry and the enumerative interpretation of the A-model a few further general observations can be made. Although the concrete form of the flux superpotential Wflux, will highly depend on the point at which it is evaluated on the complex
structure moduli space, we can make statements about its structure at particular points in th moduli space. One particularly distinguished point is the large complex structure point which by mirror symmetry corresponds to a large volume compactification of Type IIA string theory. As we have seen in section 5.2 the solutions at this point have a characteristic grading by powers of log(z). Mirror symmetry then maps the logarithmic terms to classical large-volume contributions on the Type IIA side while the regular terms in the periods (X,F) encode the closed string world-sheet instantons corrections26. Inserting the form of the prepotential (5.38) into the flux superpotential (5.94) one finds the characteristic structure
Wflux= X0 ˆN0+ ˆM0K0− ˆMiKi+ ( ˆNi− ˆMjKij+ ˆM0Ki)ti−12MˆiKijktjtk+3!1Mˆ0Kijktitjtk
+ ( ˆMi− ˆM0ti)X β din0βLi2(qβ) + 2 ˆM0 X β n0βLi3(qβ). (5.95)
26Recall that the mirror map takes the form zi
= e2πiti+ . . ., where ti= Xi/X0 on the IIB side which is identified with the world-sheet volume complexified with the NS-NS B-field on the Type IIA side.
5.4. BASICS OF ENUMERATIVE GEOMETRY 129 This expression directly shows that in addition to a cubic polynomial of classical terms, also instanton correction terms proportional to Lik(z) = P∞n=1 z
n
nk for k = 2, 3 are induced by
non-vanishing fluxes ˆMi, ˆM0. Thus, we can directly read off the Gromov-Witten invariants n0
β of (5.90) from the flux superpotential Wflux for particular flux choices. Conversely, given
a few invariants ngβ, we can determine the flux numbers for any given superpotential Wfluxor
compare to superpotentials obtained from different setups, like e.g. Calabi-Yau fourfolds, or different string theories using string dualities. Indeed, this will be our strategy to explicitly relate Calabi-Yau fourfold superpotentials to dual Type IIB or heterotic superpotentials in sections 6.1 and 6.2.
Let us now turn to the superpotential for the open string sector that is given, for a curve Σu, by the chain integral
Wbrane=
Z
Γ(u)
Ω(z) , ∂Γ(u)⊃ Σu, (5.96)
which we encountered in the Type IIB context in section 3.3.2 as a D5-brane superpotential, in section 4.1.3 in the heterotic context as a small instanton/five-brane superpotential and in F-theory setups in section 4.2.3 as a seven-brane superpotential.
Ideally one would like to explicitly compute the functional dependence of Wbrane on the
brane deformations u of the curve Σu and the complex structure moduli z e.g. by evaluating,
as in the closed string case, a set of open-closed Picard-Fuchs equations. Indeed, one way to achieve this is to lift the setup to an F-theory compactification on a Calabi-Yau fourfold, as we will demonstrate in section 6.1. Another, more direct and mathematically canonical procedure is discussed in part III of this work [60, 100]. There a constructive method is used to directly compute the superpotential Wbrane for five-branes on a curve Σu on the complex
geometry side by mapping the deformation problem of the curve Σu in Z3 to the deformation
problem of complex structures on a non-Calabi-Yau threefold ˆZ3, that is canonically obtained
by blowing up Z3 along the curve Σu.
Before performing any calculations we infer some crucial properties of Wbrane by applying
mirror symmetry at the large complex structure/large volume point. Recall that under mirror symmetry, a Type IIB compactification with D5- or D7-branes is mapped to a Type IIA compactification with D6-branes wrapping special Lagrangian cycles L in the mirror Calabi- Yau space ˜Z3, cf. section 5.1. Thus, in order for mirror symmetry with branes to hold [96]
the superpotentials have to agree on both sides. However, on the A-model side the moduli of L are counted by elements in H1(L, Z) and are generically unobstructed [170]. In contrast,
the deformations of the curve Σu are in general obstructed, which is a basic fact in classical
geometry [259], and reflected in physics by the non-trivial superpotential Wbrane [66, 67].
Consequently, the superpotential in the A-model has to be induced entirely by quantum corrections27, which are string world-sheet discs ending on L. As was reviewed in detail in
27In mathematical terms this equivalence can be formulated as a matching of classical obstruction theory of
section 5.4.2 this superpotential induced by the open string world-sheets reads Wbrane= Citit + Cˆ ijtitj+ Cˆt2+ X β, m nmβ Li2(qβQm) , Qm = e2πimiu i . (5.97) with constants C, Ci, Cij and the Gromov-Witten invariants nmβ determined by the brane
geometry and ˜Z3, as well as the flux F2 in the seven-brane context.