![](https://csdnimg.cn/release/download_crawler_static/12276423/bg5.jpg)
singular elliptically fibered Calabi-Yau threefold X → B with base B. We reach an SCFT
when two-cycles of the base B simultaneously contract to zero size.
The data of the base of an F-theory compactification is conveniently summarized by a
writing down a diagram of P
1
’s with prescribed self-intersection numbers. These numbers
can range from 1 up to 12 (see [43] for details), and must obey the condition that minus
the intersection pairing defines a positive definite quadratic form (see [12]). Here is an
example of a base configuration of curves:
Example of a base: 2, 2 . . . 2, 2
| {z }
N
. (2.1)
When the elliptic fiber is trivial and all of the −2 curves of this configuration collapse
to zero size, we realize the A
N
(2, 0) SCFT. In addition to specifying the base, it often
happens that the elliptic fiber is non-trivial. As explained in [43], this occurs generically for
curves with self-intersection −n for 3 ≤ n ≤ 12. It can also happen for −1 and −2 curves
by further tuning the complex structure moduli of an F-theory model. We can incorporate
this additional data by the notation:
g
n
[N
f
=k]
, (2.2)
that is, we have a curve of self-intersection −n wrapped by a seven-brane with gauge
symmetry algebra g. Finally, the number of flavors [N
f
= k] which is often dictated by
anomaly cancellation considerations has also been indicated. Unless otherwise indicated,
a flavor will refer to a hypermultiplet in the fundamental representation (as this is by far
the most common situation). We shall denote such flavor contributions by square brackets.
Indeed, since we do not actually need the flavor symmetry group, we shall find it more
convenient to simply list all the flavors. Here is an example of a (1, 0) SCFT which exhibits
all of the rules mentioned above:
Example (1, 0) SCFT: [N
f
= k]
su
k
2 ,
su
k
2 . . .
su
k
2 ,
su
k
2
| {z }
N
[N
f
= k] (2.3)
The classification results of references [12–18] amount to a determination of all possible
bases and all possible fiber decorations which can occur. An important outcome of this
analysis is that all of these theories can be viewed as generalized quivers in which the
links (i.e. matter) between distinct gauge group can sometimes also be interacting fixed
points [14, 15]. Such links therefore have many interacting degrees of freedom. That means
Higgs or tensor branch flows involving conformal matter will likely proceed by larger jumps
in a candidate weak C-function and will therefore likely give weaker bounds.
Having given a brief review of some aspects of how to build 6D SCFTs, let us now
turn to RG flows for these theories. There are two qualitatively distinct kinds of RG flows
which can be seen as deformations of the F-theory geometry. First of all, there are tensor
branch flows. Geometrically these correspond to resolving some of the singular curves of
the base B. In the 6D effective field theory, this corresponds to giving a vev to the scalar
of a tensor multiplet. In this flow, the UV R-symmetry remains unbroken (as the scalar is
neutral under SU(2)
R
). Second of all, there are Higgs branch flows. Geometrically these
– 4 –