Small data global existence and decay for relativistic Chern-Simons equations

We establish a general small data global existence and decay theorem for Chern-Simons theories with a general gauge group, coupled with a massive relativistic field of spin 0 or 1/2. Our result applies to a wide range of relativistic Chern-Simons theories considered in the literature, including the abelian/non-abelian self-dual Chern-Simons-Higgs equation and the Chern-Simons-Dirac equation. A key idea is to develop and employ a gauge invariant vector field method for relativistic Chern-Simons theories, which allows us to avoid the long range effect of charge.