Conformal inflation in the metric-affine geometry
  Systematic understanding for classes of inflationary models is investigated
from the viewpoint of the local conformal symmetry and the slightly broken
global symmetry in the framework of the metric-affine geometry. In the
metric-affine geometry, which is a generalisation of the Riemannian one adopted
in the ordinary General Relativity, the affine connection is an independent
variable of the metric rather than given e.g. by the Levi-Civita connection as
its function. Thanks to this independency, the metric-affine geometry can
preserve the local conformal symmetry in each term of the Lagrangian contrary
to the Riemannian geometry, and then the local conformal invariance can be
compatible with much more kinds of global symmetries. As simple examples, we
consider the two-scalar models with the broken $\mathrm{SO}(1,1)$ or
$\mathrm{O}(2)$, leading to the well-known $\alpha$-attractor or natural
inflation, respectively. The inflaton can be understood as their pseudo
Nambu-Goldstone boson.
