  We propose a hydridizable discontinuous Galerkin (HDG) method for solving the
Cahn-Hilliard equation. The temporal discretization can be based on either the
backward Euler method or the convex-splitting method. We show that the fully
discrete scheme admits a unique solution, and we establish optimal convergence
rates for all variables in the $L^2$ norm for arbitrary polynomial orders. In
terms of the globally coupled degrees of freedom, the scalar variables are
superconvergent. Another theoretical contribution of this work is a novel HDG
Sobolev inequality that is useful for HDG error analysis of nonlinear problems.
Numerical results are reported to confirm the theoretical convergence rates.
