  The mixed equation, defined as a combination of the anti-self-duality
equation in gauge theory and Cauchy-Riemann equation in symplectic geometry, is
studied. In particular, regularity and Fredholm properties are established for
the solutions of this equation, and it is shown that the moduli spaces of
solutions to the mixed equation satisfy a compactness property which combines
Uhlenbeck and Gormov compactness theorems. The results of this paper are used
in a sequel to study the Atiyah-Floer conjecture.
