Black hole microstate counting in Type IIB from 5d SCFTs

TL;DR

This work tests the $AdS_{6}$/CFT$_{5}$ duality by matching microstates of magnetically charged AdS$_6$ black holes in Type IIB to topologically twisted indices of 5d SCFTs engineered by 5-brane webs. It develops and deploys matrix-model and Bethe-ansatz techniques to compute the twisted indices for theories such as $T_N$, $[N,M]$, and $T_{N,K,j}$ in the large-$N$ limit, and confirms the universal relation $\log Z_{\Sigma_{\mathfrak{g}_1}\times \Sigma_{\mathfrak{g}_2}\times S^1} = -\frac{8}{9}(1-\mathfrak{g}_1)(1-\mathfrak{g}_2)F_{S^{5}}$ by matching to Bekenstein-Hawking entropies. On the gravity side, it uplifts AdS$_6$ solutions in Type IIB to AdS$_2$ near-horizon geometries and computes the corresponding black-hole entropies, including backgrounds with monodromy. The results provide strong evidence for a universal large-$N$ structure relating five-dimensional sphere partition functions and twisted indices across a broad class of holographic duals and illustrate how detailed brane-construction data encode BH microstates via field-theory indices.

Abstract

We use recently established AdS$_6$/CFT$_5$ dualities to count the microstates of magnetically charged AdS$_6 \times S^2 \times Σ$ black holes in Type IIB. The near-horizon limit is described by solutions with AdS$_2 \times Σ_{\mathfrak{g}_1} \times Σ_{\mathfrak{g}_2} \times S^2 \times Σ$ geometry, where $Σ_{\mathfrak{g}_i}$ are Riemann surfaces of constant curvature and $Σ$ is a further Riemann surface over which the geometry is warped. Our results show that the topologically twisted indices of the proposed dual superconformal field theories precisely reproduce the Bekenstein-Hawking entropy of this class of black holes. This provides further support for a prescription to compute five-dimensional topologically twisted indices put forth recently, and for the proposed dualities. We confirm the $N^4$ scaling found in the sphere partition functions and extend previous matches of sphere partition functions to AdS$_6$ solutions with monodromy.

Figures (4)

• Figure 1: 5-brane junctions: (a) $\#_{N,M}$ theory and (b) unconstrained $T_N$ theories.
• Figure 2: $T_{N,K,j}$ junctions: (a) $N=4$, $K=j=2$ and (b) $N=5$, $j=1$, $K=3$.
• Figure 3: Ratio of the topologically twisted index and the five-sphere partition function, evaluated numerically as described in section \ref{['sec:numerics']}. On the left hand side for the $T_N$ (red), $T_{N,N/2,1}$ (blue), $T_{N,N/2,2}$ (green) and $T_{N,N/4,3}$ (black) theories. On the right hand side for $\#_{N,M}$ as functions of $M$. From top to bottom the curves are $N\in\lbrace 5,10,15,20,25\rbrace$. As function of $N$ the ratio exhibits similar convergence properties.
• Figure 4: Plots of $\mathcal{S}^{(1)}$ (upper curve) and $\mathcal{S}^{(2)}$ (lower curve) as functions of $k\equiv K/N$.