The Resolved Elliptic Genus and the D1-D5 CFT

Abstract

This paper is a follow-up to the short paper arXiv:2509.19425, greatly expanding the discussion with examples and providing derivations and justifications of results presented there. We introduce a new supersymmetry index for the D1-D5 CFT on $T^4$, which we call the resolved elliptic genus (REG). It is a one-parameter generalisation of the standard supersymmetry index, the modified elliptic genus (MEG), and arises naturally in the free symmetric orbifold description of the theory within a new formalism, based on Schur-Weyl duality, that we develop. In this formalism, the Hilbert space of the symmetric orbifold CFT is decomposed into symmetry sectors in which the structure of the states contributing to the MEG is transparent. By examining the action of the supercharge deformed by an exactly marginal operator on the relevant symmetry algebra, we propose a superselection rule governing the lifting process of BPS states, and use it to construct the REG by summing only over those symmetry sectors that can mix according to this rule. The REG exhibits detailed agreement between the CFT and supergravity below the black-hole threshold, a regime in which the MEG is essentially trivial. Above the threshold, the REG is dominated by black-hole microstates, which are now distributed amongst distinct sectors that are invisible to the MEG. We expect both the new formalism and the REG to provide useful new tools for studying the structure of black-hole microstates. In particular, we comment on their possible relevance to the fortuity program for understanding black-hole microstates within CFT.

45 pages + appendices. V2: References added and minor typos corrected; V3: Typos corrected, matches published version

References in corpus (35)

• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Microscopic Formulation of Black Holes in String Theory
• Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS${}_5$ black holes
• New Connections Between 4D and 5D Black Holes
• Split States, Entropy Enigmas, Holes and Halos
• Supersymmetric Index from Black Hole Entropy
• Arithmetic of Quantum Entropy Function
• A holographic dual for string theory on ${\mathrm{AdS}}_3\times {\mathrm{S}}^3\times {\mathrm{S}}^3\times {\mathrm{S}}^1$
• Proving the PP-Wave/CFT_2 Duality
• The chiral ring of AdS3/CFT2 and the attractor mechanism
• Words to describe a black hole
• 1/16 BPS States in N=4 SYM
• Effect of the deformation operator in the D1D5 CFT
• Moulting Black Holes
• Emission from the D1D5 CFT: Higher Twists
• Superstrata
• Lifting of D1-D5-P states
• No entropy enigmas for N=4 dyons
• Lifting at higher levels in the D1D5 CFT
• Lifting 1/4-BPS States in $Ad{S}_3\times S^3\times T^4$
• Beyond the Tensionless Limit: Integrability in the Symmetric Orbifold
• Lifting of level-1 states in the D1D5 CFT
• Decoding stringy near-supersymmetric black holes
• BPS Chaos
• Universal lifting in the D1-D5 CFT
• Lifting of states in 2-dimensional $N=4$ supersymmetric CFTs
• The Holographic Landscape of Symmetric Product Orbifolds
• Black Hole Berry Phase
• Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems
• Dynamical evolution in the D1D5 CFT
• $S_N$ Orbifolds and String Interactions
• Symmetric Product Orbifold Universality and the Mirage of an Emergent Spacetime
• Fortuity and Supergravity
• Perturbed symmetric-product orbifold: first-order mixing and puzzles for integrability
• A New Supersymmetry Index for the D1-D5 CFT