5.3 Example
6.1.1 Spectral cover
80 6. Heterotic/F-theory duality and five-brane superpotential duality. Then, we will investigate how the blow-up geometry enters the heterotic setting and how it is related to the horizontal five-branes and the blow-up of the base of the CY fourfold of F-theory. We will also argue that there is a map between heterotic and the F-theory flux superpotentials since both blown-up threefolds and the fourfolds can be given as complete intersections. In the last section, we will treat explicit examples of heterotic/F-theory pairs checking the moduli map. We will also construct the CY fourfold for F-theory explicitly from the blown-up CY threefold of the heterotic theory. As the last example, we will re-investigate the main example of § 5 in the light of this duality. This chapter is based on ref. [54].
6.1 Heterotic/F-theory duality
In this section we will describe the crucial ingredients for the heterotic/F-theory duality: The spectral cover construction and the identification of the moduli. The spectral cover construc-tion is the only known method for general construcconstruc-tion of stable vector bundles on elliptic CY manifolds. In addition, it plays a very important role in the heterotic/F-theory duality. There-fore, its importance cannot be overstated. Since the beginning of the duality many mappings of the moduli of both sides have been uncovered. We will describe some of the most important mappings which we will need later in our computations.
6.1. Heterotic/F-theory duality 81 For a class of manifolds however, namely for elliptically fibered manifolds, there are construc-tive methods to obtain stable bundles. The seminal works [84, 186] use the method of spectral cover, del Pezzo surfaces, and the parabolics to construct stable bundles. We will review the spectral cover method of ref. [84] which works forSU(n) andSp(n) bundles. We will concen-trate on theSU(n) case.
Stable bundles on elliptic curves
LetZ be an elliptically fibered manifold with a sectionσ, meaning T2 //Z πZ //BZ.
σ
bb (6.3)
The basic strategy of the spectral cover method is to use stable bundles over the elliptic curve E. We first construct stableSU(n) bundles overEand pull it back toZ. LetV be a stableSU(n) bundle onE. It is obvious from eq. (6.1) thatV is flat. The fact thatV is a flatSU(n) bundle means the following
V = Mn i=1
Li, On i=1
Li=1 (6.4)
whereLiare line bundles and1denotes the trivial line bundle. It is a well-know fact that we can define a group law on points ofE with the identity element being the distinguished point ponE. This group can be obtained by the degree zero Picard group whose elements are of the formLQ=OE(Q)⊗OE(p)−1. It is clear thatLQis of degree zero and thus flat. This means that there is an one-to-one correspondence between points ofE and flat line bundles onE. The first equation in eq. (6.4) translates to
Xn ′ i=1
Qi=p (6.5)
where the primed sum denotes the sum under the group law just discussed. Now, it is easy to construct a stableSU(n) bundleV onE: Choose a set ofn points {Qi}⊂E andV is given by the direct sum of theLQi. Consequently, the moduli spaceMfor SU(n) stable bundles on E is isomorphic to P(H0(E,OE(np))∼=Pn−1. This can be explained since an element of H0(E,OE(np)) has a pole of ordern atp andn zeroes corresponding to theQi. Usually, the sections ofH0(E,OE(np)) have zeroes of ordernatp. However, we have the following short exact sequence for a divisorDinM, cf. for example ref. [187, p. 84]
0 //O(−D) α //OM //OD //0 (6.6)
where the mapαis given by multiplication with the (non-unique) non-trivial sectionswhose zero locus isD. The dual map
OM //O(D) (6.7)
is then given by division bys. Thus, we have to consider sections with pole of appropriate order since we are going to work with the coordinates of the ambient space. An elementwinMhas
82 6. Heterotic/F-theory duality and five-brane superpotential the following explicit form in the affine coordinatesxandy
w=a0+a2x+a3y+a4x2+a5x2y+ ··· +
anxn/2 forneven, anx(x−3)/2y fornodd
(6.8) withai being the homogeneous coordinates ofM. The pointpcorresponds to the infinity in xandy.
For elliptically fibered manifolds
We are now in place to start to construct a stable bundle over an elliptically fibered manifold.
In § 5.1 we have seen that we need a line bundleLto specify the elliptic fibration. We want to fiberMover the base manifoldBto obtain the moduli space of stable bundles onZ. Now, the coefficientaibecomes holomorphic sections ofK⊗L−i, i.e.ai ∈H0(B,K⊗L−i). We will momentarily explain the role of the line bundleK. EachPnb−1withb∈Bfits to aPn−1bundle overBdenoted byW, thus
W=P(K⊗(OB⊕L−2⊕L−3⊕ ··· ⊕L−n)). (6.9) Before we proceed further, let us summarize what we have learned: If we have a stable bundle V on Z, then it uniquely determines a sections∈W. Reversely, if we choose a sections∈ W, then it determines a stable bundle, but not uniquely. The section s and the Weierstraß equation determine a hypersurfaceCinBwhich is an-fold cover ofBsinceshasnsolutions corresponding to thenpoints. This hypersurfaceCis called thespectral cover. The line bundle Kdetermines the class of the spectral cover inZ as [C]=nσ+Kwhere we write againKfor the associated divisor to the line bundleK. We rephrase the correspondence between bundles and the sections ofWas follows
bundles
unique
++spectral cover.
not unique
jj (6.10)
Let us assume that we are given a spectral coverC and want to construct the stable bundle from it. To do this, we need the fiber product, the Poincaré line bundle, and the push-forward of a vector bundle. Let us go through them in steps.
Fibert product
Thefiber productis a central and general construction1in algebraic geometry. Here, we discuss the construction adapted to our need, namely only for elliptically fibered manifolds. ForπZ: Z→Bwe define the fiber productZ×BZ as follows
Z×BZ={(z1,z2)∈Z×Z |πZ(z1)=πZ(z2)}. (6.11) There is a naturally defined divisor∆inZ×BZ, the diagonal
∆={(z1,z2)∈Z×BZ |z1=z2}. (6.12)
1Cf. for example [65, § III.3].
6.1. Heterotic/F-theory duality 83 We can define three natural projectionsπ1/2, ˜π
Z×BZ
π1
π2
//
˜
Pπ
PP PP
''P
PP PP PP
Z
πZ
Z π
Z //B
(6.13)
where ˜π=πZ◦π1=πZ◦π2.
Poincaré line bundle
Having defined the fiber product, we now define the Poincaré line bundleP for the elliptic curveEand then construct it forZ using the fiber product. Here, we considerEas the elliptic fibration over a point. We have seen above that the degree zero line bundles ofEare param-eterized by points ofE, i.e. byE itself. Therefore, we want to construct a line bundlePE, the Poincaré line bundle, on the direct product2E×Ewith the following property
PE¯¯Q
×E∼=OE(Q)⊗OE(p)−1 ∀Q∈E. (6.14)
The line bundlePE is the universal bundle for degree zero line bundles onE. If we setPE= OE×E(DE) whereDE=∆E−E×p−p×Ewith∆Ebeing the diagonal ofE×E, the above property is fulfilled. To obtain the Poincaré line bundleP over Z×BZ which restricted toEb×Eb is isomorphic toPEand trivial restricted toσ×BZ, we set
P=OZ×BZ(D)⊗π˜∗L−1 (6.15)
whereD=∆−σ×BZ−Z×Bσ. The second factor inPis needed sinceOZ×BZ(D)¯¯σ
×BX∼=π˜∗L. Push-forward
Now, we discuss the last point of our list: thepush-forwardof a vector bundle. LetV →M a vector bundle on M and f :M→N a map fromM toN. The push-forward bundle f∗V is a vector bundle onNand is defined as follows3
(f∗V)(U)=V(f−1(U)) , U⊂Nopen (6.16)
The relevant caseV being a line bundle andMann-fold cover ofNis illustrated in Figure 6.1.
For a line bundle on an n-fold coverM the resulting vector bundle onN is a rankn vector bundle.4
2This can be seen as the trivial fiber product over a point.
3The push-forward of a vector bundle might not be a vector bundle anymore iffis not surjective. For our case f will be a projection and thus surjective. For sheaves, containing the vector bundles as a subclass, this is not a problem and the push-forward operation is called thedirect image, cf. ref. [65, § II.1].
4There is a Higgs bundle interpretation of the spectral cover, cf. for example refs. [188, 189].
84 6. Heterotic/F-theory duality and five-brane superpotential C
C
C
f∗L(f−1(U)), f−1(U)⊂M L(U),U∈N f−1 C
f−1 f−1
Figure 6.1:Push-forward of a line bundle
The bundle, finally
We are now in place to give the stable vector bundle corresponding to the spectral coverC. The data given are with the notation of eq. (6.13)
Z //B, C //B, P //Z×BZ, N //C (6.17)
whereN is an a-priori arbitrary line bundle onC called thetwisting line bundle. From these data we obtain the following diagram
N
P⊗π∗1N
π2∗(P⊗π∗1N)
C C×BZπ
1
oo π2
// Z
(6.18)
and setV =π2∗(P⊗π∗1N). In recent literatures this construction is also called the Fourier-Mukai transform. The non-uniqueness in eq. (6.10) comes from the line bundleN. Since a line bundle can always be locally trivialized, during the push-forward,N contributes only by tensoring a factor ofC.
Characteristic classes
Later when we apply the heterotic/F-theory duality, the second Chern class of the bundles will play an important role. Also, on the way to the second Chern class, we can constrain the line bundleN by fixing its characteristic class. The computation of the characteristic classes involves the application of (GRR, A.1.6), for e.g. the projectionsπ2andπZ|C, and is quite elab-orate. We refrain from the lengthy derivation, but collect the most important results of the computation. The spectral coverC is given by a sections∈Wand is ann-fold cover ofB. This means that its class inZ isnσ+ηwhereη=c1(K). Thus,
OZ(C)=OZ(σ)n⊗K with η=c1(K). (6.19)
The characteristic class ofN is given by πZ∗c1(N)= −1
2πZ∗(c1(C)−π∗Zc1(B)) ⇒ c1(N)= −1
2(c1(C)−π∗Zc1(B))+γ (6.20)
6.1. Heterotic/F-theory duality 85 whereγis a class in the kernel ofπZ∗. Finally, the characteristic class ofV is given by, writing alsoσforc1(OZ(σ)),
λ(V)=c2(V)=ησ−c1(L)2(n3−n)
24 −nη(η−nc1(L))
8 −πZ∗(γ2)
2 . (6.21)
ForE8bundles, we have to use the del Pezzo surfaces or the method of parabolics. We refer to ref. [84] for those methods and only quote the corresponding formula for the characteristic class for anE8bundleV
λ(V)=c2(V)
60 =ησ−15η2+135ηc1(L)−310c1(L)2. (6.22) We see that the classηis essential in the general construction of the spectral cover. In the next section we will see that, in the context of the heterotic/F-theory duality,ηcan be constructed from the dual F-theory manifold.