F- Theory on Manifolds without Section
8.3. THE EFFECTIVE ACTION 173 massless gauge field A e 0 that
˜k000=k000m3−3k001nm2+ 3k011n2m−k111n3 (8.2.12)
= m3
120(H−V −T −3) +1
4 X
R
H(R)X
w∈R
−m3lw2(lw+ 1)2
+ 2nm2qw1 lw(lw+ 1)(2lw+ 1) sign(w)
−n2m(q1w)2(1 + 6lw(lw+ 1)) + 2n3(qw1 )3(2lw+ 1) sign(w)
. (8.2.13)
Furthermore, one finds the one-loop contribution to ˜k0 to be k˜0=k0m−k1n
= m
6 (H−V + 5T+ 15)
+X
R
H(R)X
w∈R
mlw(lw+ 1)−nq1w(2lw+ 1) sign(w)
. (8.2.14)
8.3 The Effective Action
Having performed the M-theory reduction on a genus-one fibration without section and matched it to a fluxed circle reduction of a six-dimensional supergravity theory with an additional axion and an Abelian vector field ˆA1, we are finally in a position to summarize the low-energy effective F-theory action of this class of models.
Much of the data is the same as in section 7.1, including the condition thatkΛΘΣ must equal the intersection numberDΛ·DΘ·DΣ and the number of neutral hypermultiplets and tensors. The crucial addition is the presence of the massive vector field ˆA1 that we describe (using the St¨uckelberg mechanism) in terms of a massless Abelian vector field and an axion cthat is non-linearly charged under it.
As for the models with section, we do not have a closed formula to determine the number of charged hypermultiplets. Instead, they must be determined by making an ansatz for a spectrum such that the matching equations for the Chern-Simons terms are satisfied. There is, however, a relation between the charged matter spectra of the theory with a massive U(1) and the theory in which the U(1) is massless. For simplicity, let us concentrate on the case without a non-Abelian gauge group and denote byHU(1) the number of hypermultiplets charged under the massless U(1) vector field. In the transition to the multisection model, i.e. the one in which ˆA1 is massive, one of the hypermultiplets disappears and instead there is now an axion c that is charged non-linearly under ˆA1. To see how this works explicitly, let us denote the scalars in theHU(1)−1 linearly charged matter hypermultiplets by hs. In
174 CHAPTER 8. F-THEORY ON MANIFOLDS WITHOUT SECTION summary one then has2
Dcˆ =dc+mAˆ1, Dhˆ s=dhs+qsAˆ1hs, (8.3.1) whereqs is the charge of the statehs.
After gauge fixing theU(1) gauge symmetry, the kinetic term|Dc|ˆ 2 of the axioncbecomes a mass term for ˆA1, which is proportional to m2. Hence, the U(1) can become massive by
“eating” the axionc. In F-theory the shift gauging (8.3.1) can arise from a geometric St¨ uck-elberg mechanism [62]. More precisely, if the seven-brane action induces a six-dimensional coupling
SSt= Z
M5,1
m c4∧Fˆ1, (8.3.2)
then the four-formc4 can be dualized into the axion cto obtain the gauging (8.3.1).
For D7-branes at weak coupling the effective coupling (8.3.2) arises indeed from a non-trivial Chern-Simons couplingR
M8C6∧F, where C6 is the R-R six-form of Type IIB string theory, and M8 = M5,1× CD7 is the eight-dimensional subspace wrapped by the D7-brane and its orientifold image [211]. ComparingEquation 8.3.2with these Chern-Simons terms one findsm c4 =R
CD7C6, which determines mas an intersection number at weak string coupling.
Since the axion c is the dual of c4 in six dimensions, it arises in the expansion of the R-R two-formC2 as
C2 =cω ,˜ (8.3.3)
where ˜ω is a (1,1)-form on the Type IIB covering space that is negative under the orientifold involution and should be identified with the form in Equation 8.1.6. Since there is no flux involved in this mechanism, it was termed geometric St¨uckelberg mechanism in [62].
In fact, we can determinemfrom a purely geometric argument. Let us consider the fiber geometry of a two-section for a moment. By definition, a two-section cuts out two different points over a generic point in the base manifold. Let us call these pointsP and Q. Locally, the two-section is therefore indistinguishable from the sum of two separate sections cutting outP and Q, respectively. In a given patch, one could try to define divisorsV(P) andV(Q) and follow the usual procedure of applying the Shioda map [165,166] to obtain a suitable set of massless gauge fields. ChoosingV(P) as the zero section, one would thus obtain the two
“local divisors”
D0 =V(P), D1 =λ(V(Q)−V(P)) (8.3.4) up to some irrelevant vertical parts, whereλis an arbitrary normalization constant. However, since we have a two-section,globally the two points P and Qundergo monodromies and the
2Since the scalars c and hs remain scalars without redefinition when compactifying the theory to five dimensions, we have slightly abused notation and not put a hat on them to distinguish them from their five-dimensional counterparts.
8.3. THE EFFECTIVE ACTION 175 only well-defined quantity is the divisor V(P) +V(Q). Consequently, as the massless U(1) gauge field corresponds to the two-section, its associated divisor must satisfy
De0 ∼2λD0+D1, (8.3.5)
where the proportionality constant is just another normalization factor that we can choose arbitrarily. Comparing Equation 8.3.5 to the expression forAe0 in Equation 8.2.6, one hence finds
m= 2λ , n=−1. (8.3.6)
This geometric argument therefore implies that both the flux present in the circle reduction and the chargem of the axion under ˆA1 are in fact fixed uniquely up to physically irrelevant rescalings of the masslessU(1) gauge field.
For completeness, let us consider the effective theory at an energy scale below the mass of the U(1). In order to obtain this theory we have to integrate out the massive vector mul-tiplet containing ˆA1, which was obtained by a massless vector multiplet “eating” a massless hypermultiplet. In other words one finds
V → V −1, H → H−1, (8.3.7)
consistent with the cancelation of the gravitational anomaly. Furthermore, all hypermultiplets charged under the massiveU(1) are neutral in the effective theory and one has
Hcharged → 0, Hneutral→ Hneutral+HU(1)−1. (8.3.8) While this theory is a valid effective theory at the massless level, it cannot be used in order to perform the F-theory to M-theory duality.
In figure 8.2 we give a comprehensive summary of all the theories involved, including those in five dimensions, and give their matter spectra. While most of the discussion in this chapter has been abstract and focused on the six-dimensional theories, we will use the examples in section 9.3 to discuss the actual transitions from a massless U(1) to a massive one in more detail.
176 CHAPTER 8. F-THEORY ON MANIFOLDS WITHOUT SECTION
F-theory on Y Massless sector:
1 gauge field ˆA1 HU(1) charged hypers Hneutral neutral hypers
F-theory on Y Massless sector:
HU(1)−1 charged hypers Hneutral neutral hypers 1 massive gauge field ˆA1 St¨uckelberg mechanism
non-linear Higgsing
F-theory on Y×S1 Massless sector:
2 gauge fieldsAa Hneutral neutral hypers Massive sector:
HU(1) hypers charged underA1
+ KK towers of all fields
F-theory on Y ×S1 Massless sector:
1 gauge field Ae0
Hneutral+δ−1 hypers neutral underAe0 Massive sector:
1 gauge field Ae1 HU(1)−δ hypers charged
under Ae1 + KK towers of all fields compactifyonS1 compactifyonS 1withflux
M-theory on Y Massless sector:
2 gauge fieldsAa Hneutral neutral hypers
M-theory on Y Massless sector:
1 gauge field Ae0
Hneutral+δ−1 neutral hypers integrateoutmassivestates integrateoutmassivestates
Conifold transition δ states with charge (q0, q1) become light and Higgs gauge field
Conifold transition
Figure 8.2: A comprehensive summary of relations between the different theories and their spectra.
Chapter 9
Explicit Six-Dimensional F-Theory Models
Having derived the low-energy effective actions of six-dimensional F-theory models both with and without section in the preceding chapters, one might be tempted to argue that the discus-sion is complete — after all, the couplings and matter fields that we have been able to match are now determined in terms of general topological quantities of the compactification mani-fold. In practice, however, much is learned by nevertheless evaluating the general expressions for examples that one can explicit construct. Not only do these examples serve as a valuable additional check of the abstract calculations, but they also provide inspiration to reconsider and possibly weaken the assumptions that we make when deriving the effective actions. In the context of F-theory reductions, this led to studying models with non-holomorphic sections or genus-one fibrations without section, both of which had originally been neglected.
Constructing non-trivial F-theory backgrounds with the features one desires is another challenge in itself. While the low-energy effective action by itself a priori seems to impose few restrictions on the spectrum apart from anomaly freedom, there may well exist much stronger constraints from the geometry. One prominent such example is the rank of the Abelian gauge group. While one would hardly expect there to be stringent bounds from a purely field-theoretic argument, obtaining F-theory models with high Abelian rank is of considerable difficulty. No general bound has so far been proven, but it seems conceivable that one may exist, as the highest Abelian rank that has so far been explicitly constructed is only four.
In this chapter, we employ the toolkit developed in Part II of this thesis to construct genus-one fibered Calabi-Yau manifolds and use them as a test ground for the effective actions obtained inchapter 7 andchapter 8. We begin insection 9.1 with a general discussion about how to compute the matter spectrum using the loop-induced Chern-Simons terms derived in section 7.3. Next, we study three different F-theory compactifications with multiple sections
177
178 CHAPTER 9. EXPLICIT SIX-DIMENSIONAL F-THEORY MODELS insection 9.2 before we finally provide a whole class of genus-one fibrations without section in section 9.3 and study both the geometry and the physics of their transitions to elliptic Calabi-Yau manifolds with multiple sections.
9.1 Determining the Charged Spectrum from Chern-Simons Terms
One of the crucial observations ofchapter 7 was that the Chern-Simons terms in the circle-reduced theory contain much information about both the charged and the non-charged spec-trum of the theory, while they are given by intersection numbers determined purely in terms of the topology of the Calabi-Yau compactification manifold on the M-theory side. This in-sight was what allowed us to prove that under certain assumptions the gravitational and the mixed anomaly conditions are automatically fulfilled for any F-theory model. We however also noted that it was not possible to explicitly solve the matching equations obtained in the M-/F-theory duality for the F-theory spectrum — partially because of the sign-functions appearing in equations (7.3.10) and (7.3.11) that depend on the Mori cone of the compacti-fication manifold.
Despite the lack of a closed expression for the F-theory matter spectrum, one can still compute the matter multiplicities for given examples. To do so, one proceeds as follows:
• From the toric data of the compactification manifoldY one extracts the gauge group and thematter split using the methods discussed in section 4.4. Restricting the intersection numbers of the sections with the irreducible fiber components to a reasonable range of integers then allows one to make an ansatz for representations present in the matter spectrum. If the manifold is not toric, then one must determine this ansatz differently, for example by analyzing all possible degenerations of the fiber geometry independently of the base, such as in subsection 4.1.1.
• Keeping the multiplicities of the representations general, one next computes the induced Chern-Simons terms. The additional geometric input needed for this calculations is the sign-function for the weights of the matter representation as defined in Equation 7.3.5.
We explain in section C.2how it can be obtained for a toric Calabi-Yau manifold.
• Finally, one derives equations for the matter multiplicities by demanding that the Chern-Simons terms of the circle-reduced theory equal the intersection numbers of the Calabi-Yau geometry.
If one obtains multiple solutions or no solution at all, then the ansatz has been incor-rectly chosen. However, this has not happened for any of the examples we have studied so far. Otherwise, we have obtained the matter spectrum of our F-theory model.