Comments on 1/16 BPS Quantum States and Classical Configurations

TL;DR

The authors address finite-$N$ counting of $1/16$ BPS states in ${\cal N}=4$ SYM on $S^3$ by recasting the problem as the local cohomology of a special supercharge acting on holomorphic generating fields, linking the spectrum to giant graviton constructions. They show that below energy $E$ of order $N$, the cohomology is captured by a Fock space built from single-trace cohomology, and they obtain exact partition functions in several bosonic subsectors while deriving an upper bound that grows too slowly to account for black-hole entropy. The paper also develops a thorough classical analysis of ${1\over16}$-BPS configurations via Bogomolnyi equations on $S^3\times\mathbb{R}$ and their reformulation on ${\mathbb R}^4$, followed by a radial quantization that maps classical constraints to the quantum $Q$-cohomology problem. It is shown that naive dual giant graviton quantization reproduces the upper bound only in limited regimes, indicating the need for a refined bulk description that accounts for brane intersections. Overall, the results suggest that purely bosonic $1/16$-BPS states are insufficient to reproduce known black-hole entropy and that fermionic degrees of freedom likely play a crucial role at high energies.

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 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.