testable in the future CMB B-mode experiments [5, 6]. Such an interesting moduli space
geometry has several symmetries, one of which is the shift symmetry of the inflaton.
Most of the α-attractor models studied so far are constructed in such a way that the
inflaton shift symmetry is only slightly broken by its potential, whereas other symmetries
are completely broken. Therefore only the inflaton field is light during inflation. However,
it turns out that, even if the superpartner axion is light, it effectively freezes during in-
flation and is not harmful for successful inflation [7]. Such light particles can result in a
phenomenologically rich structure in the low energy physics without affecting the successful
inflationary predictions of the single-field α-attractors.
Only a very narrow class of α-attractors with such properties was known until now:
T-models with α = 1/3. In this paper, we will construct a much more broad class of α-
attractor models with global (nonlinear) U(1) symmetries in supergravity, including models
with α 6= 1/3. We assume that the U(1) symmetry is slightly broken, e.g. by instanton
effects. There appears a light axion as a pseudo Nambu-Goldstone mode. We show how to
realize (approximately) U(1) symmetric model in supergravity, and extend the construction
to the multi-field case.
We also discuss the dual description of the U(1) symmetric α-attractor models by
using a two-form superfield, which gives us different perspectives of the models in terms of
the dual gauge (form) superfields. From such a dual viewpoint, the shift symmetry of the
axion can be seen as a consequence of the gauge symmetry of the dual two-form field. As
we will show, the Poincar´e dual of the U(1) symmetric α-attractor can be identified as the
system with two- and three-form superfields. Gauging the two-form under the three-form
gauge symmetry leads to a mass term of the axion in the dual scalar system as applied to
natural chaotic inflation model [8–10]. Such a mass term is naturally small and as a result
the light axion would appear.
This paper is organized as follows. First, in section 2 we briefly review the symmetry of
the hyperbolic geometry and find that there is a U(1) symmetry corresponding to an axion
shift symmetry. We show concrete examples of U(1) symmetric models with a single field
as well as multiple moduli fields. We discuss the dual formulation of the U(1) symmetric
model in section 3. Also we show that the deformation of the dual system naturally yields
the mass of the axion. Finally, we give a summary of this paper in section 4 and briefly
discuss possible applications of the U(1) symmetric models. In appendix A, we discuss the
possible deformation of the potential by adding a non-perturbative superpotential.
2 U(1) symmetric α-attractors
2.1 U(1) symmetry in hyperbolic geometry
The α-attractor models are known to be described by the hyperbolic geometry of moduli
space [3–6]. The metric of the moduli space is given by
ds
2
= 3α
dT d
¯
T
(T +
¯
T )
2
, (2.1)
– 2 –
