Conformal $\alpha$-attractor Inflation with Weyl Gauge Field
  Conformal scaling invariance should play an important role for understanding
the origin and evolution of universe. During inflation period, it appears to be
an approximate symmetry, but how it is broken remains uncertain. The appealing
$\alpha$-attractor inflation implements the spontaneous breaking of conformal
symmetry and a mysterious SO(1,1) global symmetry. To better understand the
SO(1,1) symmetry, here we present a systematic treatment of the inflation
models with local conformal symmetry in a more general formalism. We find SO(2)
is the other possible symmetry in the presence of Weyl gauge field. We also
obtain all the analytic solutions that relate the inflation fields between
Jordan frame and Einstein frame. We illustrate a class of inflation models with
the approximate SO(2) symmetry and trigonometric potential, and find that it
can fit the current observations and will be probed by future CMB experiments.
