Surface operators in = 2 4d gauge theories

Abstract$$ \mathcal{N} $$ = 2 four dimensional gauge theories admit interesting half BPS surface operators preserving a (2, 2) two dimensional SUSY algebra. Typical examples are (2, 2) 2d sigma models with a flavor symmetry which is coupled to the 4d gauge fields. Interesting features of such 2d sigma models, such as (twisted) chiral rings, and the tt* geometry, can be carried over to the surface operators, and are affected in surprising ways by the coupling to 4d degrees of freedom. We describe in detail a relation between the parameter space of twisted couplings of the surface operator and the Seiberg-Witten geometry of the bulk theory. We discuss a similar result about the tt* geometry of the surface operator. We predict the existence and general features of a wall-crossing formula for BPS particles bound to the surface operator.