600 E. Palti / Nuclear Physics B 907 (2016) 597–616
[14,15]. When considering all the matter representations in the spectrum the restrictions on the
flux values read
C
i
10
Q
i
10
N
i
10
+
C
j
5
Q
j
5
N
j
5
= 0 , (2.3)
C
i
10
N
i
10
=
C
j
5
N
j
5
= 0 , (2.4)
3
C
i
10
Q
i
10
2
N
i
10
+
C
j
5
Q
j
5
2
N
j
5
= 0 . (2.5)
The interpretation in terms of anomalies is that the U(1) symmetry can be anomalous but only
in a GUT universal way which means that the introduction of the hypercharge flux should not
modify the anomalies [14,15]. This is because the global triviality of the hypercharge flux means
it can not participate in the Green–Schw
arz (GS) anomaly cancellation mechanism of string the-
ory.
2
In essence this particular restriction on the spectrum, between full anomaly cancellation as
usually considered in field theory and no restriction from Abelian anomalies as usually expected
in string theory (due to GS closed-string anomaly cancellation), leads to the signature charges of
the matter which we explore in this note.
In [13] it wa
s shown that (2.3) implies that if the Higgs up and down states of the MSSM
spectrum have different charges under the U(1) symmetry then the MSSM spectrum must be
extended by additional states which are vector-like under the SM gauge symmetries but not
under the U(1). This prediction of necessary additional v
ector-like charged fields beyond the
MSSM spectrum, and singlets which couple to them, forms one of the key motivations for this
note in light of the possible 750 GeV excess.
The pre
vious discussion amounts to general statements about possible F-theory models, let
us summarise the status of actual explicit examples manifesting these properties. There are two
important aspects for different F-theory models relevant to the analysis in this note: the U(1)
charges of the spectrum and the restriction of the hypercharge flux to the matter loci. Before
considering h
ypercharge flux there are two types of U(1) charges we will study, the first type of
models are those for which we have a complete well-understood global F-theory geometry, and in
particular know the associated smooth resolved geometry in which the charges can be calculated,
we will call these global smooth models. The second type of models we will consider are ones
presented in [24]. These models are natural from an F-theory perspecti
ve though we do not know
a smooth resolved geometry associated to them. Nonetheless they can be reasonably expected
to arise from F-theory in the sense that in [24] a Higgsing chain to reach them from certain
so-called f
actorised Tate models was presented, and the singular elliptic fibrations associated to
these factorised Tate models were presented in [23]. They are also consistent with, and form a
subset of, the general analysis of possible U(1) char
ges in global F-theory geometries [19–21].
With re
gards to hypercharge flux, the constraints (2.3)–(2.5) which we will consider amount
to a distribution of hypercharge flux over different U(1) charged matter loci. As yet we only have
a local understanding of this possibility but a global implementation is still missing. Note that
the difficult part is implementing a net h
ypercharge flux restriction to a matter state with a U(1)
charge, while a global implementation of hypercharge flux which has no net restriction to a U(1)
2
For an F-theory study of the GS mechanism see [17].