A New Supersymmetry Index for the D1-D5 CFT

TL;DR

The paper develops a Schur-Weyl-based reformulation of symmetric orbifold CFTs to define a refined supersymmetric index, the resolved elliptic genus (REG), for the D1-D5 CFT on $T^4$. REG decomposes the BPS spectrum into sectors labeled by the right-moving $ ilde{SU}(2)_2$ charge, providing enhanced, sector-resolved information beyond the standard modified elliptic genus (MEG). The authors derive REG from a Schur-Weyl decomposition, demonstrate exact matching with bulk supergravity below the black-hole threshold in several sectors, and show that above threshold REG distributes microstates across sectors invisible to MEG, highlighting deeper structure of black-hole microstates. They also discuss protection arguments, limitations away from the free orbifold point, and prospects for generalization to other holographic symmetric orbifolds and potential gravity-dual interpretations.

Abstract

We propose a new supersymmetry index for the D1-D5 CFT for $T^4$, relevant to the AdS$_3$/CFT$_2$ correspondence. In a novel formulation of symmetric orbifold CFTs based on the Schur-Weyl duality, we show how this index can be naturally described and its protection is argued based on the detailed nature of exactly marginal operators in these theories. This index is a one-parameter generalization of the standard index and gives more fine-grained information about the structure of microstates than previously available. We demonstrate precise matching of the new index between supergravity and CFT below the black-hole threshold, where the standard index -- the modified elliptic genus -- is trivial. Above the threshold, we uncover a decomposition of black-hole microstates into distinct sectors, invisible to the modified elliptic genus.

Figures (3)

• Figure 1: A single-hook Young diagram $\lambda\in H(1|1)$ with $n_\lambda$ boxes and $\rho_\lambda$ rows.
• Figure 2: Frobenius coordinates for Young diagrams in the case of (a) diagrams with $d_\lambda=1$, $\lambda=(a_1|a_2)\in H(1|1)$ and (b) diagrams with $d_\lambda=2$, $\lambda=(a_1,a_2|b_1,b_2)\in H(2|2)$ with $a_1>a_2$ and $b_1>b_2$.
• Figure 3: Plots of the REG logarithmic degeneracies $d^{\mathrm{CFT}}_{N=5,\tilde{\jmath}_2}$ in (a) and $d^{\mathrm{BH}}_{N=5,\tilde{\jmath}_2}$ in (b). For comparison, we show the analogous quantities obtained using the MEG, along with the universal Cardy growth. The first states contributing to the REG are at larger $h$ for sectors with larger $\tilde{\jmath}$.