Freckled instantons in two and four dimensions

Field theory with instantons can be partially regularized by adding degrees of freedom at some scale. These extra degrees of freedom lead to the appearance of new topological defects. These defects which we call freckles have some characteristic size depending on the scale at which the extra degrees of freedom revive. The examples of a two-dimensional sigma model, four-dimensional gauge theory are studied. The compactification of the four-dimensional supersymmetric gauge theory down to two dimensions is also considered and new phenomena are found.