Modified supersymmetric indices in AdS$_3$/CFT$_2$

TL;DR

This paper studies modified supersymmetric indices in AdS$_3$/CFT$_2$ dualities, revealing a fractional match with black brane entropy, highlighting unique features of the duality and index calculations.

Contribution

It introduces and analyzes modified supersymmetric indices in the context of AdS$_3$/CFT$_2$ dualities with orbifold geometries, uncovering fractional entropy matches.

Findings

Saddle-point analysis reproduces fractional parts of black brane entropy.

Modified indices vanish due to fermionic zero modes, requiring special treatment.

Fractional entropy matches depend on the orbifold parameter k.

Abstract

We consider the $\mathcal{N}=(2,2)$ AdS$_3$/CFT$_2$ dualities proposed by Eberhardt, where the bulk geometry is AdS$_3\times(S^3\times T^4)/\mathbb{Z}_k$, and the CFT is a deformation of the symmetric orbifold of the supersymmetric sigma model $T^4/\mathbb{Z}_k$ (with $k=2,\ 3,\ 4,\ 6$). The elliptic genera of the two sides vanish due to fermionic zero modes, so for microstate counting applications one must consider modified supersymmetric indices. In an analysis similar to that of Maldacena, Moore, and Strominger for the standard $\mathcal{N}=(4,4)$ case of $T^4$, we study the appropriate helicity-trace index of the boundary CFTs. We encounter a strange phenomenon where a saddle-point analysis of our indices reproduces only a fraction (respectively $\frac{1}{2},\ \frac{2}{3},\ \frac{3}{4},\ \frac{5}{6}$) of the Bekenstein-Hawking entropy of the associated macroscopic black branes.