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.