Comments on 1/16 BPS quantum states and classical configurations

Abstract

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 C². 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 R⁴ 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 C² and lifts to a nearby solution of the interacting Bogomolnyi equations only when the constraints equivalent to Q cohomology are obeyed.