Entropy function from toric geometry

TL;DR

This work shows that the Cardy-like limit of the superconformal index for 4d toric quiver gauge theories yields an entropy function $S_E$ that can be read directly from toric data, with $S_E = -rac{i \, ext{π} N^2}{6 oldsymbol{ au}_1 oldsymbol{ au}_2} C_{IJK} oldsymbol{ riangle}_I oldsymbol{ riangle}_J oldsymbol{ riangle}_K$ under a single constraint on fugacities. For a wide range of toric geometries, including the conifold, SPP, $ ext{F}_0$, and several del Pezzo and infinite families ($Y^{pq}$, $X^{pq}$, $L^{pqr}$), explicit entropy functions are computed and shown to align with the toric data via the cubic coefficients $C_{IJK}$. The Legendre transform of these entropy functions yields expressions for the black hole entropy in terms of charges and angular momenta, providing multi-charge AdS$_5$ black hole entropy formulas that extend known results from $ ext{N}=4$ SYM to broad classes of holographic quivers. The results support a geometric interpretation of black hole microstate counting in AdS/CFT and open avenues for exploring universality and non-toric generalizations in this Cardy-like framework.

Abstract

It has recently been claimed that a Cardy-like limit of the superconformal index of 4d $\mathcal{N}=4$ SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole. Here we study this Cardy-like limit for $\mathcal{N}=1$ toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data. Furthermore, for some families of models, we compute the Legendre transform of the entropy function, comparing with similar results recently discussed in the literature.

Figures (3)

• Figure 3: $V_2$ for $SU(3)$ conifold with cyclically identified holonomies: on the left we fixed $\Delta_{F_1}=0.12,\Delta_{F_2}=0.15, \Delta_B=0.22$ and we can observe that the function enjoys a local maximum at the origin; on the right we fixed $\Delta_{F_1}=0.05,\Delta_{F_2}=0.1, \Delta_B=0.3$ and we can observe that $V_2$ only possesses plateaux.
• Figure 4: $V_2$ in the $a^{(1)}_1-a^{(1)}_2$ plane for $SU(4)$ conifold with cyclically identified holonomies and $\Delta_{F_1}=0.1,\Delta_{F_2}=0.15, \Delta_{B}=0.2$; from the top left in clockwise sense we fixed $a^{(1)}_3=0$, $a^{(1)}_3=0.08$ and $a^{(1)}_3=0.16$. We notice a minimum whose lowest value is reached for $a^{(1)}_3=0$.
• Figure 5: $V_2$ in the $a^{(1)}_1-a^{(1)}_2$ plane for $SU(4)$ conifold with cyclically identified holonomies and $\Delta_{F_1}=0.05,\Delta_{F_2}=0.1, \Delta_{B}=0.3\,, a^{(1)}_3\,=\,0$; only plateaux are present.