  Surface operators in gauge theory are analogous to Wilson and 't Hooft line
operators except that they are supported on a two-dimensional surface rather
than a one-dimensional curve. In a previous paper, we constructed a certain
class of half-BPS surface operators in N=4 super Yang-Mills theory, and
determined how they transform under S-duality. Those surface operators depend
on a relatively large number of freely adjustable parameters. In the present
paper, we consider the opposite case of half-BPS surface operators that are
``rigid'' in the sense that they do not depend on any parameters at all. We
present some simple constructions of rigid half-BPS surface operators and
attempt to determine how they transform under duality. This attempt is only
partially successful, suggesting that our constructions are not the whole
story. The partial match suggests interesting connections with quantization. We
discuss some possible refinements and some string theory constructions which
might lead to a more complete picture.
