6.3 Examples
6.3.1 Five-branes in the elliptic fibration over P 2
We begin the discussion of our first example of the heterotic/F-theory duality by defining the geometric setup on the heterotic side. Following § 6.1, the heterotic theory is specified by
6.3. Examples 95 an elliptic CY threefold Z with a stable holomorphic vector bundleE =E1⊕E2obeying the anomaly constraint (2.33). We choose the threefold Z as the elliptic fibration over the base BZ =P2 with generic torus fiber given by a degree 6 hypersurface in P21,2,3. This is the CY threefold of § 5.3.1. It is given as the hypersurface {P =0} in the toric variety whose reflexive polyhedron we repeat here for convenience
∆Z=
-1 0 0 0 3B+9H
0 -1 0 0 2B+6H
3 2 0 0 B
3 2 1 1 H
3 2 -1 0 H
3 2 0 -1 H
(6.50)
with the class of the hypersurfaceZ given by [Z]=X
Di=6B+18H (6.51)
as explained in § 5.1.2. Here, we denote the two linearly independent toric divisorsDi byH andB: The pull-back of the hyperplane class of the baseP2 and the class of the base itself, respectively. From the toric data the basic topological numbers ofZ are obtained as
χ(Z)=540, h1,1(Z)=2, h2,1(Z)=272. (6.52)
The second Chern-class ofZ is given in eq. (5.7) which we repeat here
c2(Z)=12c1(BZ)·σ+11c1(BZ)2+c2(BZ) (6.53) whereσ:BZ →Z is the section of the elliptic fibration. Here, we haveσ=Band thus obtain
c2(Z)=36H·B+102H2. (6.54)
To fulfill the anomaly formula (2.33), we have to construct the vector bundleE1⊕E2and determine the characteristic classesλ(Ei). Now, we first need to specify the classesη1,η2∈ H2(BZ,Z) essential in the spectral cover construction. We furthermore restrictE1⊕E2to be an E8×E8bundle overZ and choose both classes asη1=η2=6c1(BZ). Then, we use the formula for the second Chern class ofE8bundles (6.22) to obtain
λ(E1)=λ(E2)=18H·B−360H2. (6.55)
The anomaly constraint then leads to conditions on the coefficients of the independent classes inH4(Z). The class of five-branesCwill have the following general form
[C]=σ·H2(BZ,Z)+ ···. (6.56) The first part represents horizontal five-branes, i.e. a curve in the baseBZ and the rest vertical five-branes wrapping the elliptic fiber. This implies that no horizontal five-branes are present
96 6. Heterotic/F-theory duality and five-brane superpotential [180]. For the classH·Bthis is trivially satisfied by the choice ofλ(Ei). For the class of the fiber Fthe anomaly forces the inclusion of vertical five-branes in the class
[C]=c2(BZ)+91c1(BZ)2=822H2=nFF. (6.57) SinceFis dual to the baseBZ, the number of vertical five-branes is determined by integrating C over the base
nF= Z
P2
C=822. (6.58)
To conclude the heterotic side we compute the indexI(Ei) since it appears in the identification of the moduli (6.28) and thus is crucial for the analysis of the heterotic/F-theory duality. ForZ we use the index formula (6.29) to obtain
I(E1)=I(E2)=8+4·360+18·3=1502. (6.59) Next, we include horizontal five-branes to the setup by shifting the classesηi appropri-ately. We achieve this by puttingη2=6c1(B)−H. The class of the five-braneCcan then be de-termined analogous to the above discussion by evaluating the characteristic class of the bundle and imposing the anomaly condition. It takes the following form
[C]=91c1(BZ)2+c2(BZ)−45c1(BZ)·H+15H2+H·B=702H2+H·B. (6.60) As discussed above, this means that we have to include five-branes in the base on a curveC in the classH of the hyperplane ofP2. Additionally, the number of five-branes on the fiber F is changed tonF =702. Accordingly, the shift ofη2 changes the second index toI2=1019 whereasI1=1502 remains unchanged.
Let us now turn to the dual F-theory description. We first construct the fourfoldY dual to the heterotic setup with no five-branes. In this case the baseBY of the elliptic fourfold isBY =P1×P2. This can be seen from the form of ηi given in eq. (6.27) and the fibration structure ofBY forE8bundles. Since both classes equal 6c1(Z), we have t=0 and thus the bundleT =OP2 as well as the projective bundleBY =P(OP2⊕OP2). Then, the CY fourfoldY is constructed as the elliptic fibration overBY with generic fiber given byP21,2,3[6]. Again,Y is described as a hypersurface in the toric varietyVY as described by the toric data in Table 6.1 if one drops the point (3, 2,−1, 0, 1) and sets the divisorDto zero. The class ofY is then given by
[Y]=X
i
Di=6B+18H+12K (6.61)
where the independent divisors are the baseBY denoted byB, the pull-back of the hyperplane HinP2and of the hyperplaneK inP1. Then, the basic topological data reads
χ(Y)=19728, h1,1(Y)=3, h3,1(Y)=3277, h2,1(Y)=0. (6.62) Now, we have everything at hand to discuss the heterotic/F-theory duality along the lines of
§ 6.2.2, in particular the map of the moduli formula (6.28). As discussed there, the complex
6.3. Examples 97
∆(Ye)=
−1 0 0 0 0 3D+3B+9H+6K D1
0 −1 0 0 0 2D+2B+6H+4K D2
3 2 0 0 0 B D3
3 2 1 1 0 H D4
3 2 −1 0 0 H−D D5
3 2 0 −1 0 H D5
3 2 0 0 1 K D7
3 2 0 0 −1 K+D D8
3 2 −1 0 1 D D9
Table 6.1:Toric data of the CY fourfoldYeblown up in the base
structure moduli of the F-theory fourfold are expected to contain the complex structure mod-uli ofZas well as the bundle and brane moduli of possible horizontal five-branes. Indeed, we obtain a complete matching by adding up all contributions
h3,1(Y)=3277=272+1502+1502+1 (6.63)
where it is crucial that no horizontal five-branes with possible brane moduli are present.
To obtain the F-theory dual of the heterotic theory with horizontal five-branes, we have to apply the recipe discussed in § 6.2.2. We have to perform the described geometric transition.
Firstly, by tuning the complex structure ofY, the fourfold becomes singular over the curveC which we then blow up into a divisorD. This way we obtain a new smooth CY fourfold denoted by Ye. The toric data of this fourfold are given in Table 6.1 where we included the last point (3, 2,−1, 0, 1) and a corresponding divisorD9=Dto perform the blow-up along the curveCas follows: Since the curveCon the heterotic theory is in the classHwe have to blow-up over the hyperplane class ofP2inBY. Firstly, we project the polyhedron∆(Y) to the baseBY which is done just by omitting the first and second column in Table 6.1. Then, the last point maps to the point (−1, 0, 1) that subdivides the two-dimensional cone spanned by (−1, 0, 0) and (0, 0, 1) in the polyhedron ofBY. Thus, upon adding this point the curveC=HinBZcorresponding to this cone is removed fromBY and replaced by the divisorD corresponding to the new point.
We see that the toric data in Table 6.1 contain this blown-up baseBYein the last three columns.
The CY fourfold is then realized as a generic constraint {P=0} in the class
[Ye]=6B+18H+12K+6D. (6.64)
Note that this fourfold has now three different triangulations which correspond to the various five-brane phases on the dual heterotic side. The topological data for the new fourfoldYe are given by
χ(Ye)=16848, h1,1(Ye)=4, h3,1(Ye)=2796, h2,1(Ye)=0 (6.65) where the number of complex structure moduli has reduced in the transition as expected. If we now analyze the map of the moduli (6.28) in the heterotic/F-theory duality, we observe that we have to puth0(C,NC/Z)=2 in order to obtain a matching. This implies, from the point of view of the heterotic/F-theory duality, that the horizontal five-brane wrapped onChas to have two deformation moduli. Indeed, this precisely matches the fact that the hyperplane
98 6. Heterotic/F-theory duality and five-brane superpotential class ofP2has two deformations since a general hyperplane is given by the linear constraint a1x1+a2x2+a3x3=0 in the three homogeneous coordinatesxi ofP2. Discarding the overall scaling, it thus has two moduli parameterized byP2with homogeneous coordinatesai. In this way, we have found an explicit construction of an F-theory fourfold with complex structure moduli encoding the dynamics of heterotic five-branes.
In § 6.3.2 we provide further evidence for this identification by showing that we can also constructYe as a complete intersection starting with a heterotic non-CY threefold. Unfortu-nately, it will be very hard to compute the complete superpotential for the fourfoldYesince it admits a large number of complex structure deformations. It would be interesting, however, to extract the superpotential for a subsector of the moduli including the two brane deforma-tions.6Later on, we will take a different route and consider examples with only a few complex structure moduli which are constructed by using mirror symmetry.