JHEP01(2019)215
may not fail: they fail for the spherical models we have analysed (all spherical boson stars)
but they are obeyed by some ultra-compact rotating boson stars. We also analyse the
swampland criteria for hairy black holes interpolating between rotating boson stars and
the Kerr solution and exhibit the region in parameter space where the criteria are violated,
which roughly correspond to the region where the scalar hair is the most compact around
the horizon. Another novel observation in this application is that the two criteria are not
independent. As a word of caution concerning all these results, the swampland criteria are
not defined as providing strict inequalities, but rather estimates. This should be considered
in interpreting the results herein.
This paper is organised as follows. In section 2 we discuss the swampland criteria. In
section 3 we present the compact objects we wish to consider and apply the swampland
criteria. In section 4 we discuss our results and their implications.
2 Swampland criteria
String theory provides a huge landscape of vacua. Such landscape is populated by consistent
low energy effective theories and is surrounded by a region where inconsistent semiclassical
effective theories live: the swampland [35, 38]. A significant portion of effective field theories
may fall into the swampland.
Two criteria have been proposed to test if a given configuration is safe from quantum
corrections within a consistent theory of quantum gravity [83, 84]. These are called the
swampland criteria. Following [41] and adapting to our case of interest these criteria are
the following.
Consider a theory of quantum gravity coupled to two scalar fields, φ
i
, i = 1, 2. The
effective Lagrangian, valid up to some cutoff scale, reads:
L =
R
2
−
1
2
g
µν
∂
µ
φ
1
∂
ν
φ
1
−
1
2
g
µν
∂
µ
φ
2
∂
ν
φ
2
− V (φ
1
, φ
2
) + . . . , (2.1)
where Planck units, M
p
=
√
8πG = 1, are used.
Swampland criterion 1. The range of the scalar field has an upper bound [38, 85, 86]:
∆φ
i
∼ O(1), (2.2)
where ∆ stands for the difference between the maximal and minimal value of the
scalar field.
Swampland criterion 2. There is a lower bound on the relative variation of the potential
in field space:
|∇
φ
V |
V
> c ∼ O(1) , (2.3)
where c is a positive constant of order 1 and one should take |∇V | =
q
δ
ij
∂
φ
i
V ∂
φ
j
V .
Criterion 1 signifies that the scalar field in the low energy effective field theory is only
valid over a field displacement [39] in Planck units, measured as a distance in the target
– 3 –
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