Black Holes and (0,4) SCFTs from Type IIB on K3

TL;DR

The study computes central charges and current algebra levels for 2d $\mathcal{N}=(0,4)$ SCFTs living on strings from D3-branes wrapped on curves in K3 with ALE/ALF transverse spaces, whose near-horizon geometry is $\text{AdS}_3\times S^3/\Gamma$. By reducing Type IIB on K3 to 6d $\mathcal{N}=(2,0)$ supergravity, the authors obtain classical contributions to CS terms and, for ALF, one-loop KK corrections; they then perform a microscopic analysis via ADE quivers for ALE and a dual MSW/M5-brane setup for ALF-A-series. The results show a perfect match between macroscopic and microscopic computations up to $\mathcal{O}(1)$ in the charges, with one-loop hair contributions essential for ALF, and pave the way for extensions to D/E-series and F-theory setups. Overall, the work reinforces the holographic relation between 6d supergravity data and 2d $(0,4)$ SCFT data in string compactifications with intricate transverse geometries.

Abstract

We study the central charges and levels of 2d $\mathcal{N}=(0,4)$ superconformal field theories that are dual to four- and five-dimensional BPS black holes in compactifications of type IIB string theory on a K3 surface. They arise from wrapping a D3-brane on a curve $C$ inside K3 and have transverse space either an ALE or ALF space. These D3-branes have an AdS$_3 \times \mathrm{S}^3/ Γ$ near horizon geometry where $Γ$ is a discrete subgroup of $SU(2)$. We compute the central charges and levels of the 2d SCFTs both in the microscopic picture and from six-dimensional $\mathcal{N}=(2,0)$ supergravity. These quantities determine the black hole entropy via Cardy's formula. We find agreement between the microscopic and macroscopic computations. The contributions from one-loop quantum corrections to the macroscopic result are crucial for this matching.