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.

Figures (7)

• Figure 1: The form of a Young diagram $\lambda$ satisfying the $(b|f)$-hook condition in \ref{['eq.bfHook']}.
• Figure 2: A single-hook Young diagram $\lambda\in H(1|1)$ is uniquely defined from its number of boxes $n_\lambda$ and number of rows $\rho_\lambda$.
• Figure 3: $su(2)_R'\oplus \widetilde{su}(2)_2'$ representations and $\mathscr{A}$-algebra representations
• Figure 4: Examples of diamond diagrams for various $\mathscr{A}$-representations. We have omitted the the dots at the vertices representing states.
• Figure 5: Examples of garnet diagrams. (a) A garnet, which represents the product of ${\tilde{\psi}}$- and ${\cal G}$-quartets. This can also be interpreted as the garnet diagram for ${\cal R}^{\mathscr{G}}_{0,0,0}$. (b) The garnet diagram for ${\cal R}^{\mathscr{G}}_{\frac{1}{2},0,0}$. Two garnets are arrayed along the $\tilde{m}$ direction. Two vertical diamonds (bluish) centered at the origin overlap, and two horizontal diamonds (yellowish) centered at the origin overlap.