3.5 Compactifications for a General Base: U(1)×U(1)
3.5.3 Anomaly Cancellation
With the hypermultiplet representations and their multiplicities at hand, six-dimensional anomaly cancellation can be verified. For the convenience of the reader we summarize below the expressions for the respective mixed Abelian-gravitational, pure Abelian and pure gravitational anomaly cancellation:
KB·bmn = −1 6∑qxqm,qnqmqn, bmn·bkl+bmk·bnl+bml·bnk =
∑
q xqm,qn,qk,qlqmqnqkql, 273=H−V+29T, KB·KB=9−T. (3.121) Here xqm,qn and xqm,qn,qk,ql denote the number of matter hypermultiplets with charges(qm,qn)and(qm,qn,qk,ql)under U(1)m×U(1)n, respectively, U(1)m×U(1)n×U(1)k×
U(1)l. In addition, H, V and T denote the total number of hyper-, vector- and tensor-
multiplets, respectively. Thebmn denote curves in the baseBdefined as
bmn=−π(σ(sˆm)·σ(sˆn)) = −2[KB] S9− S7−[KB] S9− S7−[KB] 2(S9−[KB]) mn . (3.122) Here we have collectively denoted rational sections ˆsm= (sˆQ,sˆR)mand their divisor classes
Sm= (SQ,SR)m, with m=1,2 as before. The Shioda map of rational sections ˆsmto corre- sponding divisor classesσ(sˆm)gives:
σ(sˆQ) =SQ−SP−[KB−1], σ(sˆR) =SR−SP−[KB−1]− S9. (3.123) In (3.122) we have further used (3.116).
cellation equations (3.121), we see that all equations are satisfied and thus all anomalies are cancelled. This verifies the six-dimensional anomaly cancellation for the general F-theory compactification with U(1)×U(1) gauge symmetry over an arbitrary baseB.14
14The baseBhas to admit a generic elliptic fibration of the form (3.47), i.e. all line bundles in (3.114) have to have generic sectionssi. See the discussion in Section 2 of [30].
Chapter
4
Engineering U(1)×
U(1) in 4D -
Adding
G4
Flux
Encouraged by the results of the previous chapter, we push toward more realistic scenarios and construct U(1)×U(1) F-theory compactifications in four dimensions. To achieve this goal, we need to construct four-complex dimensional geometries (fourfolds). The appear- ance of Yukawa couplings at codimension three is expected. Finally, in order to obtain chiral matter in 4D,G4flux have to be engineered on top of the geometry.
In this chapter, most of the content have been taken from [30], where the author of this dissertation is a co-author.
The content of this chapter is organized as follows. First, we leverage the knowledge from the previous chapter and construct the four-complex dimensional fibration using the dP2ambient space for the fiber. As in the previous chapter, we proceed to find the spectrum of the theory. We realize that the charges of the hypermultiplets under U(1)×U(1) remain unchanged. The number of chiral matter, however, requires the specification of the flux and the matter surfaces. As advertised, we find the Yukawa couplings at codimension three in the base, located at the intersection of three matter curves.
We continue with the construction of theG4flux. Given that this flux is an element of HV(2,2), we require the cohomology ring of the fourfold. The calculation of chiralities can then in principle be done with the integrating the flux over the matter surfaces. However, one problem appears, we do not have the explicit description of all matter surfaces. In order to achieve our goal we make use of the 4D/3D F- M-theory duality. The trick is to read the chirality from the Chern-Simons terms of the 3D theory. There is an interesting twist in the story here given that the section is not holomorphic. Finally we check anomaly cancellation of the spectrum.
Let us point to some results beyond the construction of the geometry. In order to calcu- late the Yukawa points we again made use of algebraic techniques, in particular, we had to describe the varieties as the zero locus of ideals. In order to obtain the varieties associated to each charged curve, we have to decompose the ideals in its prime ideals using primary decomposition. This techique has been adopted by the community and it has been dubbed as the ‘ideal technique’. Additionally, the existence of a non-holomorphic zero section had non-trivial implications in the F-theory M-theory duality. This was a very important result and this is why it is explained at length in a full section.
In the original article [30], almost half of the paper is dedicated to the SU(5)×U(1)×U(1) case. During the full study of thais non-abelian example we found a non-flat fibration. The non-flat fiber can be wrapped by a M5 brane giving rise to stringy excitations in the low energy theory. It was an interesting discovery, however, we decided not to include this topic in this dissertation. We point the curious reader to the article for more information.
4.1
The Elliptic Curve in
dP
2And Its Fibrations
In this section we review the construction of the elliptic curveE indP2and its Calabi-Yau elliptic fibrations over a generalB. These Calabi-Yau manifolds have a rank two Mordell- Weil group, that gives rise to U(1)×U(1) gauge symmetry in F-theory.
In section 4.1.1 we construct resolved elliptically fibered Calabi-Yau manifolds ˆπ: ˆX→ Xover an arbitrary baseBwith this elliptic curveEas the general fiber. The singular Calabi- Yau manifold is denoted byX We show that these Calabi-Yau manifolds ˆX are classified by the choice of two divisors S7, S9 in the baseB. In particular, we work out all the line bundles that are relevant to formulate the Calabi-Yau constraint of ˆX, which is the analog of the Tate model for elliptic fibrations withdP2-elliptic fiber.
The content of section 4.1.1 is a direct extension of the discussion in [34], where the possibility of a full classification of all Calabi-Yau elliptic fibrations with general fiber E was pointed out, but demonstrated explicitly only forB=P2.