Microstate Counting via Bethe Ansätze in the 4d ${\cal N}=1$ Superconformal Index

TL;DR

The paper extends the Benini–Milan Bethe Ansatz method to compute the four-dimensional ${\cal N}=1$ superconformal index for toric quiver gauge theories at large $N$, enabling entropy-type analyses without invoking a Cardy-like limit. By proposing a class of holonomies with $u_{ij}^{ab}=\frac{\tau}{N}(i_a-j_b)$, the authors evaluate the index for several models (conifold, SSP, ${\rm dP}_1$, and $Y^{p,q}$) and show that, in a region of chemical potentials, the leading result reproduces the familiar cubic entropy structure $S_E\propto C_{IJK}\Delta_I\Delta_J\Delta_K$ after a suitable shift and constraint. They demonstrate that outside this region, the index receives additional $\tau$-dependent corrections (captured by $S_\tau$) that modify the Legendre transform and hence the black hole entropy, with explicit treatment provided for the conifold. These results illustrate a non-Cardy route to black hole microstate counting in AdS$_5$ and motivate further study of $1/N$ corrections and explicit gravity duals for rotating, electrically charged AdS$_5$ black holes.

Abstract

We study the superconfomal index of four-dimensional toric quiver gauge theories using a Bethe Ansatz approach recently applied by Benini and Milan. Relying on a particular set of solutions to the corresponding Bethe Ansatz equations we evaluate the superconformal index in the large $N$ limit, thus avoiding to take any Cardy-like limit. We present explicit results for theories arising as a stack of $N$ D3 branes at the tip of toric Calabi-Yau cones: the conifold theory, the suspended pinch point gauge theory, the first del Pezzo theory and $Y^{p,q}$ quiver gauge theories. For a suitable choice of the chemical potentials of the theory we find agreement with predictions made for the same theories in the Cardy-like limit. However, for other regions of the domain of chemical potentials the superconformal index is modified and consequently the associated black hole entropy receives corrections. We work out explicitly the simple case of the conifold theory.

Figures (2)

• Figure 1: The figure shows the complex plane of chemical potentials for a generic $\Delta$ where the region specified by (\ref{['const']}) is shown in grey.
• Figure 2: The figure shows the complex plane of chemical potentials for generic $\Delta$ including also the region for the corresponding $X_I$ inside the dashed strip. notice that the grey and the dashed region overlap in a zone where the identification $\left(\left[\Delta_I\right]_{\tau} + \frac{1}{2}\right) \Longleftrightarrow X_I$ is valid.