  We formulate the problem of counting 1/16 BPS states of N = 4 Yang Mills
theory as the enumeration of the local cohomology of an operator acting on
holomorphic fields on C^2. We study aspects of the enumeration of this
cohomology at finite N, especially for operators constructed only out of
products of covariant derivatives of scalar fields, and compare our results to
the states obtained from the quantization of giant gravitons and dual giants.
We physically interpret the holomorphic fields that enter our conditions for
supersymmetry semi-classically by deriving a set of Bogomolnyi equations for
1/16-BPS bosonic field configurations in N = 4 Yang Mills theory on R^4 with
reality properties and boundary conditions appropriate to radial quantization.
An arbitrary solution to these equations in the free theory is parameterized by
holomorphic data on C^2 and lifts to a nearby solution of the interacting
Bogomolnyi equations only when the constraints equivalent to Q cohomology are
obeyed.
