From multi-gravitons to Black holes: The role of complex saddles

TL;DR

This work exploits the ABBV equivariant localization to transform 4d superconformal index matrix integrals into a controlled fixed-point sum, enabling exact finite-$N$ and large-$N$ analyses. By deforming the action with a parameter $\\lambda$ and employing a double-periodic extension of the integrand, the authors derive an exact ABBV formula for the index as a sum over fixed points, with a Bethe-Ansatz-like structure for half of the fixed-point conditions. In the large-$N$ regime, they show that for 4d $SU(N)$ ${\\mathcal N}=4$ SYM counting large operators (charges $\\sim N^2$) is governed by two complex saddles, $(1,0)$ and $(1,1)$, whose contributions are of the same exponential order and whose phases interfere to reproduce oscillations around the Bekenstein-Hawking entropy; saddles with other $(m,n)$ are exponentially suppressed in this regime. For small charges, the suppression weakens and more saddles contribute, signaling a crossover toward a multi-graviton/different sector regime. The combination of ABBV with Picard-Lefschetz theory thus provides a nonperturbative, analytic framework connecting microcanonical operator counting to black hole entropy in AdS$_5$, while clarifying the role and hierarchy of complex saddles across charge scales.

Abstract

By applying the Atiyah-Bott-Berline-Vergne equivariant integration formula upon double dimensional integrals, we find a way to compute the matrix integral representations of $4d$ $\mathcal{N}=1$ superconformal indices. The final formula allows us to easily extract analytic results in the large-rank expansion of certain theories. As an example, we compute the leading one-loop corrections to the effective action of the known complex saddles in those theories. For a superconformal index of $SU(N)$ $\mathcal{N}=4$ SYM, we use the equivariant integration formula and the Picard-Lefschetz method to show that at large enough values of $N$, only two, among the known complex saddles, dominate the counting of large operators i.e. of operators with charges of order $N^2$. Contributions from other known complex saddles are present, but we show they are exponentially suppressed in that range of charges; the smaller the charges the less suppressed they are, and eventually, to count small operators i.e. operators with charges smaller than $N^{2/3}$, like multi-gravitons, they can not be neglected.

Figures (7)

• Figure 1: A first trial to the problem: One starts with integrals of a family of exponentials of moment maps which are smooth on the torus $A\times B$. These integrals are parameterized by a real number $\lambda$. As $\lambda\,\to\,1^-$ the integrand develops a discontinuity at an $A$-period and it is now well-defined and smooth in the cylinder (not in the torus). If the limit integrand is holomorphic and the height of cylinder is one, the integral over the cylinder collapses to an integral over an $A$-period. Every $A$-period gives the same answer.
• Figure 2: The profile of $O_\lambda(u)$ that follows from the specific choice of function $\{\cdot\}_\lambda$ that we have chosen to work with. The plot shows three different values of $\lambda$. $O_\lambda(u)$ is positive for any value of $0\leq\lambda<1$. $O_{\lambda}(u)\to 1$ for small enough values of $\lambda$. $O_{\lambda}(u)$ tends to the periodic Dirac-delta function centered at integer values of $u$ for $\lambda\,\to\,1^-$. In fact, the specific details of the function $\{\cdot\}_\lambda$, the one that defines the dependence on $\lambda$ in both the action $S_\lambda$, and the two-form $\omega_\lambda$ (through $O_{\lambda}$), are irrelevant as long as $\{\cdot\}_\lambda$ is periodic, smooth and it preserves the conditions \ref{['OLambdaCOnd']}, \ref{['initialCondSForm']} and \ref{['measureLocalise']}.
• Figure 3: In the figure in the top we have plotted values of the two-saddles approximation to the graded number of states $d(\mathfrak{q})$ for $N=100$, and the Bekenstein-Hawking entropy of the dual black holes. In this plot $q\,=\,\mathfrak{q}$. In this scale, both plots are superposed and both are the same. In this scale of charges and entropy, the two saddles oscillations are negligible. For smaller values of $N$ oscillation become noticeable as shown "experimentally" in Agarwal:2020zwm( Murthy:2020rbd) and as analytically predicted by the two-saddles approximation. Something interesting happens at the point $q=0$. This point encodes information about small operators, i.e roughly speaking, operators with charges of order $N^{\frac{2}{3}}$ or less. If we zoom in close enough into the point $q=0$, oscillations start to enhance, first caused by the interference among contributions coming from $(1,0)$ and $(1,1)$, and at some point contributions from generic $(m,n)$ saddles start to become important and can not be neglected. In principle, that flow evolution can be exactly probed (at least the one driven by the competition among $(m,n)$ fixed points) with the help of the ABBV formula, but implementing that with numerical precision lies beyond the scope of the present paper and it is left for future work.
• Figure 4: The contour $C_\eta$ in $q^{\frac{1}{3}}$ and $\tau$-planes. The original contour in $q$-plane is a circle inside the unit disk $|q|<1$. The grey dashed circle (segment) represents $|q|=1$ ($\text{Im}(\tau)=0$). The integration along the two vertical contours in $\tau$-plane cancel each other due to the periodicity properties of the index. Due to reasons explained in \ref{['par:HolomorphicExtension']} they must cross the real axis across irrational values. Thus, the integral along $C_\eta$ equals the integral along the original contour $C$.
• Figure 5: The plots of the analytic solutions (of the relevant algebraic equations) obtained for the Lefschetz thimbles and ascent paths associated to the $(1,1)$ (left) and $(1,0)$ (right) fixed points. The solution for generic $(m,n)$ has the same form as one of the two plots above. Each critical point of the entropy functional $\mathcal{E}_{(m,n)}$ in \ref{['Epsilon']}, has two thimbles and two ascent paths associated. In the figure we have only indicated the direction of the flows corresponding to the thimbles that appear (have non-zero intersection number) in the decomposition of the original integration contour $C_{\eta}$ which we have denoted as $I$, $II$, $III$, $IV$, $V$. The different $H$'s in the plots denote the values of the Morse function at the closest critical point and also at $\tau_2\approx\pm \infty$ where $H=\pm T$, respectively, with $T$ a very large positive number.