Fortuity and R-charge concentration in the D1-D5 CFT

TL;DR

This work probes finite-$N$ BPS cohomology in the D1--D5 CFT, focusing on fortuitous classes within the $1/4$-BPS sector through explicit $Q$-cohomology analysis at $N=2$ and $N=3$. By refining cochains with ${\rm SU}(2)_a\times{\rm SU}(2)_b$ and right-moving ${\rm SU}(2)_{\rm outer}$, it proposes that, at fixed holomorphic weight, states in the largest representations are fortuitous and demonstrates this via explicit constructions and lifting arguments. The study provides evidence for R-charge concentration, showing fortuitous states reside in central, maximal-dimension spaces of the longest $Q$-cochain complex for both $N=2$ and $N=3$, with implications for their holographic interpretation as black-hole microstates. It outlines a program to extend these results to larger $N$ and other cochain complexes, aiming to sharpen the boundary-bulk dictionary and refine microstate counting in the D1--D5 system.

Abstract

We investigate finite-$N$ BPS cohomology in the D1--D5 CFT, focusing on the sector of fortuitous classes. Analyzing the supercharge cochain complexes in the $N=2$ and $N=3$ theories, we construct several explicit fortuitous classes. We study the decomposition of these cohomology classes into ${\rm SU}(2)_a\times {\rm SU}(2)_b$ representations and conjecture that, at fixed holomorphic weight, those transforming in the largest representation are necessarily fortuitous. Our results also provide strong evidence that the $R$-charge concentration phenomenon extends to the D1--D5 CFT.

Theorems & Definitions (1)

• Conjecture 1