Black hole entropy function for toric theories via Bethe Ansatz

TL;DR

The paper develops a Bethe Ansatz framework to evaluate the large-N limit of superconformal indices for toric quiver gauge theories and derives entropy functions for dual AdS5 black holes. By solving BAEs with a universal basic solution (and its SL(2,Z) transforms), it obtains a compact Θ-based exponent that reproduces Cardy-like entropy predictions and reveals Stokes-line structure in the complex chemical-potential plane. The results are worked out in detail for the conifold and extended to infinite toric families Y^{pq}, X^{pq}, L^{pqr}, as well as several del Pezzo variants, with the Θ-coefficients tied to U(1) triangle anomalies C_IJK from toric data. In the equal-charge sector, the analysis yields tractable extremization and BH entropy relations, mirroring the known N=4 SYM case and supporting the proposed toric entropy function S ∝ −π i N^2 C_IJK X_I X_J X_K /(ω1 ω2). The work clarifies the large-N index structure, clarifies Stokes phenomena, and lays groundwork for extending entropy-function extremization beyond the equal-Δ regime.

Abstract

We evaluate the large-$N$ behavior of the superconformal indices of toric quiver gauge theories, and use it to find the entropy functions of the dual electrically charged rotating $\mathrm{AdS}_5$ black holes. To this end, we employ the recently proposed Bethe Ansatz method, and find a certain set of solutions to the Bethe Ansatz Equations of toric theories. This, in turn, allows us to compute the large-$N$ behavior of the index for these theories, including the infinite families $Y^{pq}$, $X^{pq}$ and $L^{pqr}$ of quiver gauge theories. Our results are in perfect agreement with the predictions made recently using the Cardy-like limit of the superconformal index. We also explore the index structure in the space of chemical potentials and describe the pattern of Stokes lines arising in the conifold theory case.

Figures (9)

• Figure 1: The definition of $\left[\Delta\right]_{\tau}$ divides the complex $\Delta$-plane into domains with boundaries which are the elements of the set $\gamma+\mathbb{Z}$, where $\gamma$ is the line passing through $0$ and $\tau$. The green filled area is the fundamental domain, corresponding to $\textrm{Im}\left(-\frac{1}{\tau}\right)>\textrm{Im}\left(\frac{\Delta}{\tau}\right)>0$, where the function $\left[\Delta\right]_{\tau}$ takes its values.
• Figure 2: Toric digram of $T^{1,1}$ conifold.
• Figure 3: Quiver diagram of the conifold theory.
• Figure 4: Toric digram of $Y^{pq}$. We show $p=3,q=2$ on this picture.
• Figure 5: Quiver diagram of the $Y^{32}$ theory.