Construction of the elliptic Calabi-Yau fourfold

5.3. Example 67 in the compact X. Here, we introduced the moduli bz_a corresponding to the charge vector ℓb^(a). Note that in our F-theory compactification of the next section we will not consider seven-branes naively wrapped on these divisors as we would in a compactification of the type IIB theory on CY orientifolds. Rather, we will construct a CY fourfold with seven-branes on its dis-criminant possessing additional moduli. These additional moduli correspond to eitherbz1orbz2

and allow deformations of the seven-brane constraint by the additional terms (5.91). Hence, zˆ_i can be interpreted as deformations of the seven-brane divisors inY, or as spectral cover moduli in the heterotic dual.

68 5. Lift to F-theory

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∆Yb ℓ⁽¹⁾_I ℓ⁽²⁾_I ℓ⁽³⁾_I ℓ⁽⁴⁾_I ℓ⁽¹⁾_{I I} ℓ⁽²⁾_{I I} ℓ⁽³⁾_{I I} ℓ⁽⁴⁾_{I I}

v0 0 0 0 0 0 0 −6 0 0 0 −6 0 0

v^b₁ 0 0 2 3 0 −2 1 −1 −1 −3 0 1 −2

v^b₂ 1 1 2 3 0 1 0 0 0 1 0 0 0

v^b₃ −1 0 2 3 0 0 0 1 −1 1 1 −1 0

v^b₄ 0 −1 2 3 0 1 0 0 0 1 0 0 0

v1 0 0 −1 0 0 0 2 0 0 0 2 0 0

v2 0 0 0 −1 0 0 3 0 0 0 3 0 0

vˆ₁ −1 0 2 3 −1 1 0 −1 1 0 −1 1 0

vˆ₂ 0 0 2 3 −1 −1 0 1 0 0 1 −1 1

vˆ3 0 0 2 3 1 0 0 0 1 0 0 0 1

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Table 5.4:Toric data for the main example

scribed by the polyhedron∆Ybas discussed in § 5.1.2. Its topological numbers are computed to be

χ=16848, h^3,1=2796, h^1,1=4, h^2,1=0, h^2,2=11244. (5.93) Note that∆Ybhas three triangulations corresponding to non-singular CY phases which are con-nected by flop transitions. In the following we will consider two of these phases in detail. These phases will match the two brane phases in Figure 3.1(b) in the local CY threefold geometry.

To summarize the topological data of the CY fourfold for the two phases of interest, we specify the generators of the Mori coneℓ⁽ⁱ⁾_I andℓ⁽ⁱ⁾_{I I} fori =1, . . . 4. These data are shown in Table 5.4. The charge vectors are best identified in phase II. The vectorsℓ⁽¹⁾_{I I} andℓ⁽²⁾_{I I} are the extensions of the threefold charge vectorsℓ⁽¹⁾andℓ⁽²⁾in Table 5.2 to the fourfold. The brane vectorℓb⁽¹⁾ is visible in phases II as a subvector ofℓ⁽³⁾_{I I}. The remaining vectorℓ⁽⁴⁾_{I I} arises since we complete the polyhedron such that it becomes reflexive implying thatYb is a CY manifold.

It represents the curve of theP¹basis ofYb. Phase I is related to phase II by a flop transition of the curve associated to ℓ⁽³⁾_I . Hence, in phase I the brane vector is identified with−ℓ⁽³⁾_I . Furthermore, after the flop transition we have to set

ℓ⁽³⁾_{I I} = −ℓ⁽³⁾_I , ℓ⁽¹⁾_{I I} =ℓ⁽¹⁾_I +ℓ⁽³⁾_I , ℓ⁽²⁾_{I I} =ℓ⁽²⁾_I +ℓ⁽³⁾_I , ℓ⁽⁴⁾_{I I} =ℓ⁽⁴⁾_I +ℓ⁽³⁾_I . (5.94) Note thatℓ⁽ⁱ⁾_I andℓ⁽ⁱ⁾_{I I} are chosen in such a way that they parameterize the Mori cone ofYb. The dual Kähler cone generators for phase I are then given by

J1=D2, J2=D1+2D2+D3+2D9, J3=D3+D9, J4=D9 (5.95) where as usual we writeDi={xi=0} for the toric divisors associated to the points∆Yb. In phase II we have the following

J1=D2, J2=D1+2D2+D3+2D9, J3=D1+3D2+2D9, J4=D9. (5.96)

5.3. Example 69 The J_i provide a distinguished integral basis ofH^1,1(Yb) since in the expansion of the Kähler form J in terms of theJ_i all coefficients will be positive parameterizing physical volumes of cycles inYb. TheJ_i are also canonically used as a basis in which we determine the topological data ofYb. The complete set of topological data ofYbincluding the intersection ring as well as the non-trivial Chern classes are summarized in appendix A.3.1.

Fibration structure

The polyhedron∆Ybhas only few Kähler classes making it possible to identify part of the fibra-tion structures from the intersecfibra-tion numbers. However, an analogous analysis is not possible for the mirror manifoldY since the dual polyhedron∆Y has more than two thousand Kähler classes. Therefore, we apply the methods reviewed in § 5.1.3 in analyzing bothYb andY. As already mentioned above,∆Ybintersected with the two hyperplanes

H₁=(0, 0,p₃,p₄, 0), H₂=(p₁,p₂,p₃,p₄, 0). (5.97) gives two reflexive polyhedra corresponding to the generic torus fiber and the generic CY three-fold fiberXb. The fibration structures ofY is studied by identifying appropriate projections to

∆^k_Fb⊂∆Yb. Three relevant projections are

P1(p)=(p3,p4), P2(p)=(p1,p2,p3,p4), P3(p)=(p3,p4,p5) (5.98) wherep=(p1, . . . ,p5) are the columns in the polyhedron∆Yb. Invoking the theorem of § 5.1.3, we see fromP1thatY is also elliptically fibered and since the polyhedron ofP²_1,2,3is self dual, the fibration is also of the same type. In addition, it is clear from P₂that Y is CY threefold fibered. The fiber threefold is X, the mirror CY threefold toXb. The fact, that the threefold fibers ofY andYb are mirror to each other is special to this example since the subpolyhedra obtained byH₂andP₂are identical. Finally, note thatY is also K3 fibered as inferred from the projectionP3. This ensures the existence of a heterotic dual theory by fiberwise applying the duality of F-theory on K3 to the heterotic strings onT². Replacing the K3 fiber by an elliptic fiber, we find the CY threefoldX. We will elaborate on this in § 6.

The hypersurface constraint forY depends on the four complex structure moduliz_i. This dependence is already captured by only introducing twelve out of the many (blow-up) coordi-nates needed to specify a non-singularY. This subset of points of∆Y is given in Table 5.5. In the table we have omitted the origin. Note that we have displayed in Table 5.5 the vertices of

∆Y and added the inner pointsv₁andv₂to list all points necessary to identify the polyhedron

∆X with vertices given in Table 5.3(a). The four-dimensional polyhedron∆X lies in the hyper-plane orthogonal to (0, 0, 0, 0, 1) and thus we have a CY threefold fibration with fiber X. The base of this fibration is given by the points labeled by the superscript (·)^b. Note that (0, 0, 1, 1, 0) is also needed to observe the elliptic fibration. The base of the elliptic fibration is obtained by performing the quotient∆base=∆Y/(P₁∆Yb)^∗which amounts to simply dropping the third and fourth entry in the points of∆Y.

70 5. Lift to F-theory

∆Y ⊃

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v₁ 0 0 1 1 0 z

v₂ −12 6 1 1 0 u₁ v₃ 6 −12 1 1 0 u₂

v4 6 6 1 1 0 u3

v5 0 0 −2 1 0 x

v₆ 0 0 1 −1 0 y

v^b₁ −12 6 1 1 −6 x₁ v^b₂ −12 6 1 1 6 x₂ v^b₃ 6 −12 1 1 −6 x3

v^b₄ 6 6 1 1 −6 x4

v^b₅ 0 −6 1 1 6 x₅ v^b₆ 0 6 1 1 6 x₆

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 Table 5.5:The relevant subset of points of the polyhedron∆Y

Weierstraß form and moduli

Additionally, we can also see the elliptic fibration directly on the defining polynomialµeofY which can be written in a Weierstraß form. Indeed, if we apply eq. (5.12) for the points in Table 5.5 and all points of∆Ybthat are not on codimension one faces, we obtain a hypersurface of the form¹⁴

e

µ=a6y²+a5x³+me1(x_j,u_i)x y z+me6(x_j,u_i)z⁶=0. (5.99) Here, x_j andu_i are the homogeneous coordinates on the base of the elliptic fibration while x,y, andz are the homogeneous coordinates of theP²_1,2,3fiber. The polynomialsme1andme6

are given by

me₁(x_j,u_i)=a₀u₁u₂u₃x₁x₂x₃x₄x₅x₆,

me₆(x_j,u_i)=u₁¹⁸¡a₇x²⁴₁ x¹²₂ x⁶₃x⁶₄+a₃x¹⁸₁ x¹⁸₂ x⁶₅x⁶₆¢

+a₄u¹⁸₂ x¹⁸₃ x₅¹²+a₂u¹⁸₃ x₄¹⁸x₆¹² +u⁶₁u⁶₂u⁶₃¡

a₁x₁⁶x₂⁶x₃⁶x₄⁶x₅⁶x₆⁶+a₉x₂¹²x₅¹²x₆¹²+a₈x₁¹²x₃¹²x¹²₄ ¢ (5.100) wherea_i denote coefficients encoding the complex structure deformations of Y. However, sinceh^3,1(Y)=h^1,1(Yb)=4, there are only four complex structure parameters rendering six of theairedundant. It is also straightforward to compareme1andme6for the fourfoldY with the corresponding threefold data given in eq. (5.83) and eq. (5.84).

For the different phases we can identify the complex structure moduli in the hypersurface constraint by using the charge vectorsℓ⁽ⁱ⁾_{I/I I} in Table 5.4. For phase I we find

z^I₁=a₂a₄a₇ a²₁a8

, z^I₂=a1a₅²a³₆

a₀⁶ , z₃^I =a₃a₈ a1a7

, z₄^I =a₇a₉ a1a3

. (5.101)

14The polynomialµecan be easily brought to the standard Weierstraß form by completing the square and the cube, i.e.ye=y+me₁xz/2 andxe=x−me²₁z²/12.

5.3. Example 71 For phase II we find in accord with the rules for flop transition (5.94) that

z₁^II=z^I₁z₃^I, z₂^II=z₂^Iz₃^I, z^II₃=(z₃^I)⁻¹, z₄^II=z₄^Iz₃^I. (5.102) In order to compare to the CY threefoldX, we choose the gaugeai=1,i =2, . . . , 6 anda8=1 such that

a⁶₀= 1

(z₁^II)^1/3z₂^IIz₃^II, a1= 1

(z^II₁)^1/3, a7=z₃^II(z₁^II)^1/3, a9= z₄^II

(z₁^II)^2/3. (5.103) It is straightforward to find the expression for phase I by inserting the flop transition (5.102) into this expression fora0,a1anda7,a9.

Having determined the defining equations for the CY fourfolds, we evaluate the discrimi-nant∆(Y) of the elliptic fibration. Using the formula for the discriminant (5.6), we find

∆(Y)= −me6(432me6+me⁶₁). (5.104)

We conclude that there will be seven-branes on the divisors {me6=0} and {432me6+me⁶₁=0} in the baseB_Y. The key observation is that in addition to a moduli independent partme⁰₆the full me6is shifted as

me6=me₆⁰+a1(u1u2u3x1x2x3x4x5x6)⁶+a7u₁¹⁸x₁²⁴x₂¹²x₃⁶x₄⁶+a9u⁶₁u⁶₂u⁶₃x¹²₂ x¹²₅ x¹²₆ . (5.105) The moduli dependent part is best interpreted in phase II witha₁,a₇anda₉given in eq. (5.103).

In fact, when setting the fourth modulus toz₄^II=0, we see that the deformation of the seven-brane locus {me6=0} is precisely parameterized by z₃^II. By setting x_i =1, we fix a point in the base ofY viewed as fibration with fiber X. We are then in the position to compare the shift in eq. (5.105) with the first constraint in eq. (5.91) finding agreement if one identifies bz1=z^II₃(z^II₁)^1/3. In the next section we will show that the open string BPS numbers of the lo-cal model with D5-branes of § 3.2 are recovered in thez₃^IIdirection. The shift of the naive open moduluszb1by the closed complex structure modulusz^II₁ fits nicely with a similar redefinition made for the local models in ref. [33]. This leaves us with the interpretation that indeedz₃^II de-forms the seven-brane locus and corresponds to an open string modulus in the local picture.

As we will show in the next section, az₃^IIdependent superpotential is induced upon switching on fluxes on the seven-brane. It can be computed explicitly and matched with the local re-sults for D5-branes for an appropriate choice of flux. A second interpretation of the shifts in eq. (5.105) by the monomials proportional toz^II₃,z₄^IIis via the heterotic dual theory on X and the spectral cover construction. This viewpoint will be treated in § 6.

As a side remark, let us again point out that the discriminant (5.104) with polynomials given in eq. (5.100) is not the full answer for the discriminant since we have set many of the blow-up coordinates to 1. However, we can use the toric methods of refs. [153, 159, 160] to determine the full gauge group in the absence of flux to be

G_Y =E₈²⁵×F₄⁶⁹×G₂¹⁸⁴×SU(2)²⁷⁶. (5.106)

Groups of such large rank are typical for elliptically fibered CY fourfolds with many Kähler moduli corresponding to blow-ups of singular fibers [160].

72 5. Lift to F-theory

Im Dokument String dualities and superpotential (Seite 79-84)