4.6 Picard-Fuchs equations
4.6.2 Picard-Fuchs equations of complete intersections
As we have seen in the previous section, the blow-up geometry can be represented as a com-plete intersection. There is a well-defined algorithm, GD method, to determine the PF system if the geometry is given as a hypersurface or as a complete intersection. In this section we describe the algorithm for complete intersections.
LetM be a threefold withh3,0(M)=1. So, M can be a CY threefold or its blow-up. Fur-thermore, let us assume for simplicity thatMis given by two equationsf1and f2inP5. Let us denote {fi=0} byDi. The generalization to general ambient toric variety is straight forward.
The only difference is that there are more homogeneous coordinates with possibly different weights. Using the Griffiths residuum formula, we can represent the holomorphic three-form ΩM as follows
ΩM = Z
T(f1,f2)
1
f1f2∆=ResM µ ∆
f1f2
¶
with ∆=X
j
(−)jxjd x1∧ ··· ∧d xdj∧ ··· ∧d x5 (4.59)
where(b·) denote the omission of the argument andT(f1,f2) is the union of two tubular neigh-borhoods aroundDi, i.e.S1-bundles overDiinNDi/P5. Generally, an elementηofH3(M) can be represented by [137]
η=ResM p
f1if2n−i∆ (4.60)
andpis a homogeneous polynomial of appropriate degree. For notational brevity, we will omit ResM from now on. To apply the GD method of reduction of pole order, we use the following two-form
ω= 1 f1af2b
X
i<j
(−)i+j(xjhi−xihj)Λi j (4.61)
whereΛi j=d x1∧ ··· ∧d xdi∧ ··· ∧d xdj∧ ··· ∧d x5. Sincedωis exact, we obtain aP
ihiJ1i f1a+1f2b +bP
ihiJ2i f1af2b+1 =
Pi∂hi/∂xi
f1af2b with Jij= ∂fi
∂xj (4.62)
4.6. Picard-Fuchs equations 43 up to exact forms. Let us now assume thatηdepends on a parameterψ, then
∂ψη= − i p∂ψf1
f1i+1f2n−i −(n−i)p∂ψf2
f1if2n−i+1 + ∂ψp
f1if2n−i. (4.63)
Thus, if the numerators of the first two terms of∂ψηare elements of the Jacobian ideal of f1
andf2, i.e. ideal ofC[x1,···,x5] spanned by partial derivatives offi, then we can reduce the pole order by using eq. (4.62). The strategy to determine PF equations is as follows: We differentiate ΩM w.r.t. the parameterψ. After takingkderivatives, we have an expression of the form
ηk= p
f1af2b∆ with a+b=k. (4.64)
The pole order isk. Using the algorithm described above, we reduce the pole order of this ex-pression by 1. From the resulting exex-pressionαk−1we separate the part proportional to∂kψ−1ΩM
and denote the rest byηk−1. Here, proportionality means
β=gβ˜ with g∈C[ψ,ψ−1]. (4.65)
Then, we recursively apply the algorithm toηk−1till we are left with an expression proportional toΩM. The result of this procedure is a PF equation
∂kψΩM=
kX−1 i=0
qi∂iψΩM. (4.66)
If we apply the algorithm in practice, we proceed as follows. Thek-th derivative ofΩM
with respect to the complex structure parameterψcontains all possible pole order forfiof the total pole orderk+2. We then construct a vector with entries being the numerator of each pole order, i.e.
PkT =QTk =³
pk+1,1 pk,2 ··· p2,k p1,k+1´
(4.67) wherepa,bdenotes the numerator (polynomial) of the term of pole order (a,b) in (f1,f2). For the reduction of the pole order, we use
Kk=
k J1
J2 (k−1)J1
2J2 . .. f1·1(k+1)×(k+1) f2·1(k+1)×(k+1)
. .. J1
k J2
(4.68)
where Ji denote (Ji1,···,Jmi ) with m being the number of homogeneous coordinates. This means that the matrixKkis a (k+1)×(km) matrix. We then solve the matrix equation
PkT =Kk·A with A=³
A1 ··· Ak Ak+1 Ak+2 ´
(4.69)
44 4. D5-branes, mixed Hodge structure and blow-up whereAi≤k=(A1i,···,Ami ),Ak+r=(A1k
+r,···Akk+1
+r) andAij∈C[x1,···,xm]. The entries ofA cor-respond tohiin eq. (4.62). Thus, fromA, we build the following vector
Qek−1=
∇A1+A1k+1
∇A2+A2k
+1+A2k
+2
...
∇Ak−1+Akk+1+Akk+2
∇Ak+Akk++12
. (4.70)
As discussed above, we separate the part which is proportional (in the sense of eq. (4.65)) to Pk−1and denote the rest byQk−1. We then apply the same reduction algorithm toQk−1until we reachQe0which necessarily is proportional toΩM. The algorithm can terminate earlier if
e
Qk−Qkvanishes for somek. We described the case for one-parameter model.
For multi-parameter models, several different derivatives can produce terms of the same order of poles. We have to include all possible terms to determine theQekpart. For brane geom-etry we also have to take also the exact pieces into account since we are integrating over chains instead of cycles. The technical difficulty arises since we have to perform multi-variable poly-nomial division. This is not possible inMathematica 7. The algebra systemMacaulay 2[138]
can accomplish this, but not with rational functions in the parameter as coefficient. However, this can be easily circumvented. It would be interesting to determine the PF systems for exam-ples and solve the system. This would enable us to check the blow-up proposal would allow for computations for large class of brane geometries, not restricted to branes given torically by charge vectors.
5
Lift to F-theory
Everything popular is wrong.
O. Wilde,
The Importance of Being Earnest
F-theory is a non-perturbative description of the type IIB theory with D7-branes [31]. For a nice “derivation” of F-theory from M-theory, see ref. [79]. It allows for holomorphically vary-ing axio-dilatonτ=C0+i e−φand theSL(2,Z) symmetry of the type IIB theory is built right into the geometry. The main advantage of F-theory over the type IIB theory is that it ge-ometrizes the axio-dilaton and D7-branes or their generalizations, (p,q) seven-branes, into a twelve-dimensional manifold, i.e. four complex dimensional internal manifoldY. The in-ternal manifold has to have an extra structure, namely the elliptic fibration. If we want to embed D7-branes in a type IIB compactification, we have to include O7 orientifolds to cancel the tadpole. This means also that we have to find a consistent orientifold involution of the CY threefold which is a very difficult task in general. F-theory does this automatically once we have a suitable CY fourfold with desired seven-brane content. Sen provided an elegant way of obtaining consistent orientifold configurations from F-theory [139, 140, 141].
Recently, there have been lots of activities for F-theory GUT model building starting with refs. [142, 143, 144, 145]. These refs. construct non-compact, i.e. local, models. For compact examples, cf. for example refs. [146, 147, 148, 149, 150, 151].
Since D7-branes can have non-trivial worldvolume flux inducing D5-brane charge, we will argue and show in this chapter that we can lift the setting with a D5-brane to a F-theory setting.
This means that we will construct an F-theory compactification with a CY fourfold which has the appropriate singularity and flux on the seven-brane worldvolume. Using this embedding and employing mirror symmetry for CY fourfolds, we will be able to compute the superpo-tentialWB. Explicit checks will be made using the BPS numbers computed for non-compact geometries.
46 5. Lift to F-theory This chapter is therefore organized as follows: Firstly, we will review the elliptic fibration and seven-branes in F-theory. Since we will be extensively using toric varieties and CY hyper-surfaces in them, we quickly review the construction of elliptic CY threefolds and fourfolds.
Our examples will have many CY fibration structures. Therefore, we will study how we can determine and construct such CY fibration structures in great detail. Then, we will describe mirror symmetry for CY fourfolds which differ from mirror symmetry for CY threefolds con-siderably. We will discuss the states and operators of A- and B-model and their underlying algebra structure, the Frobenius algebra. After having set all the required techniques, we will compute the superpotential for one main example. Results for further examples are relegated to appendix. This chapter is based on ref. [53].
5.1 F-theory and elliptic Calabi-Yau fourfolds
The computation of the F-theory flux superpotential (2.24) will be done for a class of CY four-foldsY which we will introduce in this section. Our basic strategy in constructing a fourfoldY with a low number of complex structure moduli is first to construct its mirrorYbas a CY three-fold fibrationXbover aP1base. The threefoldsXbwe are interested in are themselves elliptically fibered and admit a local limit yielding the non-compact geometriesKBXb studied in ref. [33].
This fact will be exploited when we analyze the seven-brane content of the F-theory compacti-ficationY and later on determine the F-theory flux superpotential which we split into flux and brane superpotential as in § 2.2.
Due to the importance of the involved geometries we will introduce the geometrical pre-requisites here. We will first review the elliptic fibration and seven-branes in F-theory. Then, we will discuss the hypersurface description of CY manifolds in toric varieties since our exam-ples will be of this type. Our example geometries admit rich CY fibration structure and thus, we will study in great detail how these structures arise and how we can determine them. We will see how we can construct CY fourfolds of desired type.