4.2 F-Theory Compactifications
4.2.2 Elliptic Calabi-Yau Manifolds and Seven-Branes in F-theory
As we have seen in an F-theory compactification on a Calabi-Yau n-fold Xn to (11−2n, 1)-dimensional Minkowski space the axio-dilatonτ is described as the complex structure modulus of an elliptic curve fibered over the Type IIB target manifold Bn−1 that is a K¨ahler manifold with positive curvature,
F-theory onXn = Type IIB onBn−1 . (4.29) The Calabi-Yau Xn geometrizes non-perturbative seven-branes by non-trivial monodromies of τ around degeneration loci of the elliptic curve, which includes D7-branes and O7-branes as special cases. Here we systematically discuss compactifications of F-theory with a focus on four dimensional vacua and on the consistency condition imposed by tadpoles inherited from the M-theory description of F-theory [29].
Let us study the F-theory geometry of an elliptically fibered Calabi-Yaun-foldXn→Bn−1 with a section. This section can be used to express Xn as an analytic hypersurface in the projective bundleW =P(OBn−1⊕ L2⊕ L3) with coordinates (z, x, y), which is a fiber-bundle over Bn−1 with generic fiber P2(1,2,3). The hypersurface constraint can be brought to the Weierstrass form
y2=x3+g2(u)xz4+g3(u)z6 . (4.30) Here L = KB−n−11 for Xn being Calabi-Yau and g2(u), g3(u) are sections of L4 and L6 for (4.30) to be a well-defined constraint equation. Locally on the base Bn−1 they are functions in local coordinatesu on Bn−1. We refer to appendix B.1 for details.
F-theory defined on Xn automatically takes care of a consistent inclusion of spacetime-filling seven-branes, as we have seen in section 4.2.1. These are supported on the in general reducible divisor ∆in the base Bn−1 determined by the degeneration loci of (4.30) given by the discriminant
∆={∆ := 27g22+ 4g33 = 0}. (4.31) The degeneration type of the fibration specified by the order of vanishing of g2, g3 and ∆ along the irreducible components ∆i of the discriminant have an ADE–type classification that physically specifies the four-dimensional gauge group G. It can be determined explicitly
using generalizations of the Tate formalism [101]. For the order of vanishing of ∆ at most one the Calabi-Yau manifoldXnis smooth corresponding to a single seven-brane on ∆. This is the I1 locus of the elliptic fibration [101]. However, this is not the generic and interesting situation in F-theory in general and in this work. Indeed in all our examples we will consider a singular Xn with a rich gauge symmetry generated.
The weak string coupling limit of F-theory is given by Imτ → ∞and yields a consistent orientifold setup with D7-branes andO7-planes on a Calabi-Yau manifold [75,76]. In general, as the axio-dilaton of Type IIB string theory τ corresponds to the complex structure of the elliptic fiber, it can be specified by the value of the classical SL(2,Z)-modular invariant j-function which is expressed through the functionsg2 and g3 in (4.30) as
j(τ) = 4(24g2)3
∆ , ∆ = 27g22+ 4g33 . (4.32)
The function j(τ) admits a large Imτ expansion j(τ) =e−2πiτ + 744 +O(e2πiτ) from which we can directly read off the monodromy (4.28) of τ around a D7-brane, for example.
There are further building blocks necessary to specify a consistent F-theory setup. This is due to the fact that a four-dimensional compactification generically has a tadpole of the form [77, 78, 208]
χ(X4)
24 =n3+ 1 2
Z
X4
G4∧G4, (4.33)
which can be deduced by considering the dual M-theory formulation. There a tadpole of the the three-form C3 is induced due to the Green-Schwarz term R
C3∧X8, the coupling to the M2-brane and the Chern-Simons termR
C3∧(dC3)2. In the case that the Euler characteristic χ(X4) of X4 is non-zero a given numbern3 of spacetime-filling three-branes on points in B3 and a specific amount of quantized four-form flux G4 have to be added in order to fulfill (4.33). In addition, non-trivial fluxes on the seven-brane worldvolume contribute as [102]
χ(X4)
24 =n3+ 1 2
Z
X4
G4∧G4+X
i
Z
∆i
c2(Ei), (4.34) whereEi denotes the corresponding gauge bundle17localized on the discriminant component
∆i. The cancellation of tadpoles in F-theory compactifications on Calabi-Yau fourfolds thus is restrictive for global model building since the total amount of allowed flux is bounded by the Euler characteristic of X4 that is, in known explicit constructions, at most 1820448 [95].
The choice of G4 is further constrained by the F-theory consistency conditions on allowed background fluxes. The first constraint comes from the quantization condition for G4, which depends on the second Chern class of X4 as [209]
G4+c2(X4)
2 ∈H4(X4,Z) (4.35)
17It has been argued in [29] that also the brane fluxes should be describable by transcendental fluxG4.
4.2. F-THEORY COMPACTIFICATIONS 81 and has been deduced from anomaly freedom of the M2-brane theory. More restrictive is the condition that G4 has to be primitive, i.e. orthogonal to the K¨ahler form of X4. In the F-theory limit of vanishing elliptic fiber this yields the constraints
Z
X4
G4∧Ji∧Jj = 0 . (4.36)
for every generator Ji, i= 1, . . . , h(1,1)(X4) of the K¨ahler cone. There is one caveat in order when working in the K¨ahler sector of X4 and when evaluating topological constraints like (4.33), (4.35) and (4.36). In the case that X4 is singular, which happens for enhanced gauge symmetry, it is not possible to directly work with the singular space since the topological quantities such as the Euler characteristic and intersection numbers are not well-defined.
Thus the above constraints can be naively evaluated only in the case of a smooth X4, which corresponds to the physically simplest situation with single D7-branes18. To remedy the problems when working with singular X4 we systematically blow up the singularities19 to obtain a smooth geometry [101] for which the constraints (4.33), (4.35) and (4.36) are valid.
The resulting smooth geometry still contains the information about the gauge-groups on the seven-branes and allows to analyze the compactification in detail.
Let us comment on the effect of three-branes and fluxes on the F-theory gauge group.
For a generic setup with three-branes and fluxes, the four-dimensional gauge symmetry as determined by the seven-brane content is not affected. However, if the three-brane happens to collide with a seven-brane, it can dissolve, by a similar transition as discussed in section 4.1.2, into a finite-size instanton on the seven-brane worldvolume that breaks the four-dimensional gauge groupG. During this transition the numbern3 of three-branes jumps and a bundleEi over the seven-brane worldvolume is generated describing the gauge instanton [29]. However, we will not encounter this since we restrict our discussion to the case that the gauge bundle on the seven-branes is trivial and no three-branes sit on top of their worldvolumes.
We conclude by a discussion of the complex structure moduli space of elliptic fourfolds X4, that is of central importance in this work, and the interpretation in terms of the Type IIB moduli. Consider a smooth elliptic Calabi-Yau fourfold X4 with a number ofh(3,1)(X4) complex structure moduli, that may be obtained from a singular fourfold by multiple blow-ups20. In order to compare to the Type IIB weak coupling picture the complex structure moduli can be split into three classes [29]:
(1) One complex modulus that physically corresponds to the complex axio-dilaton τ and that geometrically parametrizes the complex structure of the elliptic fiber.
18Our examples in chapter 6 are significantly more complicated and admit seven-branes with rather large gauge groups. Technically this is a consequence of working with fourfolds X4 with few complex structure moduli, which typically have in the order of thousand elements ofH(1,1)(X4) of which many correspond to blow-ups of singular elliptic fibres signaling the presence of enhanced gauge groups.
19In the cases considered in this work this is done using the methods of toric geometry [101, 210, 211].
20This affects only the number of K¨ahler moduli, which we will not discuss in the following.
(2) A number ofP
ih(2,0)(∆i) complex structure moduli corresponding to the deformations of the seven-branes wrapped on the discriminant loci inB3.
(3) h(2,1)(Z3) complex structure moduli corresponding to the deformations of the basis and its double covering Calabi-Yau threefold Z3 obtained in the orientifold limit.
For a more detailed analysis of this and the organization in terms of the low-energy effective action of F-theory we refer to [212].