In the previous section, we have seen that the structures of the gauge groups of the R1,11−2n low-energy theories from F-theory compactifications on elliptically fibered Calabi-Yau spaces Xn are primarily encoded in the codimension-one singularities of Xn. Such codimension-one
singularities were systematically analyzed by Kodaira on K3 and corresponding A-D-E gauge algebras in the effective theories, known as A-D-E classification. When the base Bn−1 is of
high dimension, monodromies around these codimension-one loci can lead to non-simply laced groups.
When going to higher codimensional loci, typically in the intersection of two irreducible divisors WI, one expect that the vanishing orders (f, g, ∆) would increases and hence the
singular types enhance. Indeed, as we will show momentarily, the matter would be generated at higher codimensional loci, and this has a counterpart in the effective fields on the 7-branes. Namely, these matter can be thought as arising from Higgsing of a higher rank gauge group located at the higher codimensional loci, known as the Vafa-Katz picture [105].
To be more precise, let’s first assume that two stacks of 7-branes wrapping divisor WI and WJ intersect on 2n − 6-cycles ΣIJ in the base Bn−1:
CIJ = WI
\
WJ, (2.114)
with the vanishing order of (ai, ∆) at WI and WJ being (niI, mI) and (niJ, mJ), respectively.
One can then read off the gauge algebra carried by the 7-brane wrapped on WI and WJ from the Kodaira-Tate table 2.3, denotes as gI and gJ, respectively. Then at the codimension-two loci ΣIJ the vanishing order typically would enhance to
(ai, ∆)|CIJ = (niI + niJ, mI+ mJ). (2.115)
One again can read off the gauge algebra gC from the Kodaira-Tate table 2.3, which contains the two gI,J as subalgebras.
However, this does not necessarily mean there are new gauge algebras in the effective theories but rather indicates new degrees of freedoms such as charged matter, which is more consistent with the picture of type IIB intersecting D-branes. How to show it? One can view the new charged matter arises from the decomposing of the adjoint representation of higher gauge algebra gC to its two subalgebras. In terms of branching, it reads
gC → gI× gJ
Ad(gC) → (Ad(gI), 1J) ⊕ (1I, Ad(gJ)) ⊕r[(RrI, RrJ) ⊕ c.c],
(2.116) where (RrI, RrJ) denotes the charged matter. Similar to the discriminant ∆, the codimension-two loci CIJ also split into several irreducible components based on the above branching, and we denote as
2.6. Chiral Matter and Codimension-two Singularities of Calabi-Yau
We will omit the superscript IJ occasionally if not confused, and denote the irreducible components CIJr as CRr as to each of component one can associate a certain representation R
r I.
Namely, one can view that the gauge theory on WI with corresponding gauge algebra gI as
the Higgsing of a parent gauge theory with gC by vevs < φI >s of the adjoint Higgs φ, where at the CIJ the vevs are zero and only at these loci the charged matter could be reflected at
the effective theory. In other words, if one move out from the codimension-two loci CIJ to the
codimension-one WI for example, only the adjoint representation gI appears in the effective theory of the 7-brane wrapping WI.
Now we would ask how are these new degrees of freedoms reflected from the geometry of Xn? Recall that we have said that Ad(gI) arise from the M2-branes wrapping the resolved
fiber components over WI, together with the C3 decomposition along the resolved fiber. Hence
one naturally expect that these new degrees of freedom come from the the fiber at the CIJ.
Let’s focus on the fibre EI =PiaiP1iI over any generic point p ∈ WI. To proceed, let us drop
the superscript I for conveniences. It turns out that the fiber typically becomes eE =P
keakP
1 k
when one move the point p inside CIJ, with more components P1 arising from the splitting
one (or more) component(s) P1i as P1i →
S
kP1k. In the language of representation theory, the
P1i correspond to a weight w adj
i , then the splitting of P1i →
S
kP1k amounts to a decomposition
of the adjoint weight wiadj →P
kwk, where wks are charges of the states with respect to the
Cartan subalgebra g. In other words, M2-branes wrapped on these new components P1k give
rise to states with weight βka in representation Ra of a corresponding gauge algebra g. Note that the representations Ra of g are typically not the adjoint representation ad(g) anymore. Indeed, [106, 107] showed that these new states generated by wrapped M2-branes and anti M2- branes lift to 4D N = 1 chiral multiplet and anti-chiral multiplets in F-theory compactification on a Calabi-Yau four-fold X4. And similar interpretation also applys to other dimensional Calabi-Yau manifolds.
Note that the above analysis also apply for the situations where the 7-brane divisor WI is singular, typically the single divisor WI has self-intersection, namely I = J in (2.114). However, such cases are subtle in many contexts. Nevertheless, the codimension-two loci arising from singular divisors will typically leads to exotic representation such as symmetric tensor representations or even more exotic representations [108,109]. While for the smooth divisors only leads to the fundamental, two-index antisymmetric ones for su(n) cases, as well as three-index antisymmetric ones for su(n), n = 6, 7, 8. Similar phenomenons also happen in the abelian gauge symmetries, which we will discuss in 2.9.2.
One should note that the possible singular types at codimension-two and higher loci, unlike the ones at codimension-one, are not completely classified yet. In [91], they classified cases with rank one enhancement with the assumption of the smooth 7-branes.
For our purposes in chapter 4, we would also introduce matter cycles for the localized charged matter in the Calabi-Yau five manifolds compactification. Namely, a fibration can be introduced by fibering the new fibers P1k, corresponding the weights βak, k = 1, ..., dim(Ra), over
the codimension-two irreducible loci CRI as
P1k−→SRk
↓ CR.
Here the fibration (complex ) 3-cycle SRk is called the matter 3-cycle 28.
2.6.1. Holomorphic couplings and higher codimensional singularities of Calabi-Yau manifolds
The parts of Yukawa couplings and higher codimensional singularities and their physical implications in F-theory are beyond the concrete applications in this thesis, thereby we will not give many details instead we will give a short summary and refer to the review [21] for more discussions.
From the codimension-two singularities, we have learned accompanying the enhancement of the singularities, there are new states arising from these singularities. Now as one approaches the codimension-three loci (apparently it needs n > 3 for Xn), Do we expect that are there
any new states? It was argued in [106, 107] though that there are no more new matter, instead codimension-three singulairities are responsible for hosting holomorphic Yukawa couplings at the perturbative level. For Calabi-Yau five-manifolds, there are also possible codimension-four singuliarites, and leads to E and J quartic interactions in 2D N = (0, 2) effective theories [64].
2.7. Non-flat Resolutions at High Codimensional Loci
So far we have sticked to the assumption that the singularities over the various codimensional loci in the base Bn−1 admit a flat, Calabi-Yau resolution. This typically requires that the vanishing
order of (f, g) are not at the same time exceed (4, 6) at the various codimensional loci. We call these Kadiaro singularities as the minimal singularities. However, even the minimal condition is satisfied, there are nevertheless no guarantees that a flat, Calabi-Yau (crepant) resolution of the singularities exists. Such types of singularities, typically occur in elliptic fibrations in higher codimensional loci than one, are so-called Q-factorial terminal singularities and will have some physical implications in the F/M-theory compactifications, which we will discuss in 2.8. In this section, let’s concentrate the situations with non-minimal singularities of Weierstrass models.
If the non-minimal singularities of Xn exist in codimension-one loci, one is still capable of carrying resolutions but such resolutions are not Calabi-Yau (crepant) resolution anymore and hence such a manifold cannot yield to a supersymmetric F-theory vaccum 29, as the dual M- theory compactification is not. Similar stories apply to the cases that codimension-two loci with the vanishing order (f, g, ∆)> (8, 12, 24) and codimesnion-three loci with (f, g, ∆) > (12, 18, 36). However, if there are Calabi-Yau Xnwith minimal codimension-one singularities but whose higher
codimensional singularities are non-minimal but wild, i.e., for those whose vanishing orders at codimension-two loci and codimension-three loci are (4, 6, 12)6 (f, g, ∆)|codim-2< (8, 12, 24) and (8, 12, 24) < (f, g, ∆)|codim-3< (12, 18, 36) respectively, some novel interesting physics comes to
play. Such situations have been worked out extensively and nicely in the elliptically fibered (local) Calabi-Yau three-folds X3, whose (f, g) vanishes to order (4, 6, 12)6 (f, g, ∆)|codim-2< (8, 12, 24) at some points p on the base B2 and have been dubbed conformal matter, which is a strongly
coupled sector [110]. To see this, let’s focus on an example with two colliding E8 7-branes. The
Weierstrass model can be written as
y2= x3+ αu2v2x + βu3v3, (u, v) ∈ C2, (2.119)
28
It is known as the matter surface in the four-folds bX4.