Construction of 4d SYM compactified on open Riemann surfaces by the superfield formalism

By compactifying gauge theories on a lower dimensional manifold, we often find many interesting relationships between geometry and supersymmetric quantum field theories. In this paper we consider conformal field theories obtained from twisted compactification on a Riemann surface with a boundary. Various kinds of supersymmetric boundary conditions are exchanged under S-duality. To consider these transformations one need to take into account boundary degrees of freedom. So we study how these degrees of freedom can be added at the boundary of the Riemann surface. For these the boundary fields to be added it is convenient to rewrite the theory by means of superfields. Therefore, I show in this paper that the 4d SYM action can be surely expressed as 2d superfields.