Counting Giant Gravitons in AdS_3

TL;DR

The paper develops a comprehensive framework to count and characterize 1/4- and 1/2-BPS states in string theory on global AdS_3 × S^3 × T^4 by quantizing quarter-BPS brane probes. Recasting the problem via a Polyakov action reveals an SL(2,R) × SU(2) WZW structure, allowing a precise mapping between classical supersymmetric solutions and quantum states in discrete representations, with special-charge continua appearing at the bottom end. Exact analysis of the D-string through S-duality to an F-NS5 system yields explicit 1/4- and 1/2-BPS partition functions and an elliptic genus, showing agreement with symmetric-product results in many regimes while explaining missing chiral primaries and continuum subtleties. The work clarifies the moduli-space dependence of BPS spectra, explains protection of 1/2-BPS states as geodesics, and outlines how higher bound-state probes might be incorporated, highlighting both successes and open questions in matching bulk and boundary counts.

Abstract

We quantize the set of all quarter BPS brane probe solutions in global AdS_3 \times S^3 \times T^4/K3 found in arxiv:0709.1168 [hep-th]. We show that, generically, these solutions give rise to states in discrete representations of the SL(2,R) WZW model on AdS_3. Our procedure provides us with a detailed description of the low energy 1/4 and 1/2 BPS sectors of string theory on this background. The 1/4 BPS partition function jumps as we move off the point in moduli space where the bulk theta angle and NS-NS fields vanish. We show that generic 1/2 BPS states are protected because they correspond to geodesics rather than puffed up branes. By exactly quantizing the simplest of the probes above, we verify our description of 1/4 BPS states and find agreement with the known spectrum of 1/2 BPS states of the boundary theory. We also consider the contribution of these probes to the elliptic genus and discuss puzzles, and their possible resolutions, in reproducing the elliptic genus of the symmetric product.