Supersymmetric phases of 4d N=4 SYM at large N

TL;DR

This work analyzes the large-N matrix model for the N=4 SYM superconformal index on $S^3\times S^1$ with a single chemical potential $\tau$ and uncovers an infinite family of complex saddles labeled by lattice points $(m,n)$ on $\Lambda_\tau$. The action is computed via a torus extension using the Bloch-Wigner elliptic dilogarithm and real-analytic Eisenstein sums, yielding explicit expressions that reproduce the pure AdS$_5$ and the supersymmetric AdS$_5$ black hole, with the black hole saddle dominating near $\tau\to0$ and new saddles emerging near rational points. The results unify multiple strands of prior work, including Cardy-like limits and Bethe-Ansatz approaches, and establish a detailed phase structure with anti-Stokes lines separating dominant saddles. The analysis highlights modular and elliptic structures as central to holographic degeneracies, and provides exact and asymptotic entropy formulas for general $(m,n)$ saddles, offering a path to broader applications in supersymmetric holography and beyond.

Abstract

We find a family of complex saddle-points at large N of the matrix model for the superconformal index of SU(N) N=4 super Yang-Mills theory on $S^3 \times S^1$ with one chemical potential $τ$. The saddle-point configurations are labelled by points $(m,n)$ on the lattice $Λ_τ= \mathbb{Z} τ+\mathbb{Z}$ with $\text{gcd}(m,n)=1$. The eigenvalues at a given saddle are uniformly distributed along a string winding $(m,n)$ times along the $(A,B)$ cycles of the torus $\mathbb{C}/Λ_τ$. The action of the matrix model extended to the torus is closely related to the Bloch-Wigner elliptic dilogarithm, and the related Bloch formula allows us to calculate the action at the saddle-points in terms of real-analytic Eisenstein series. The actions of $(0,1)$ and $(1,0)$ agree with that of pure AdS$_5$ and the supersymmetric AdS$_5$ black hole, respectively. The black hole saddle dominates the canonical ensemble when $τ$ is close to the origin, and there are new saddles that dominate when $τ$ approaches rational points. The extension of the action in terms of modular forms leads to a simple treatment of the Cardy-like limit $τ\to 0$.

Figures (11)

• Figure 1: The saddle-point configurations of eigenvalues in the fundamental domain and on the torus for $\tau=\frac{1}{2}(1+{\rm i})$. The saddles shown are $(0,1)$ (red), $(1,0)$ (blue), and $(1,1)$ (green). The original definition of the matrix model is the interval $[0,1)$ on the left which maps to the outer circle of unit radius on the right.
• Figure 2: The new contour of integration that passes through the saddle-point is the sum of the contour along $[0,1]$ and the closed contour $D_k$ in each coordinate $u_k$. The figure shows the $u_k$ plane for one $k$ with the axes denoting the two components of $u_k$ (decomposed in our notation $z=z_1+\tau z_2$) running from 0 to 1 so as to cover the fundamental parallelogram.
• Figure 3: The orange curves are plots of extremal curves $\tau^*$. The blue curves are plots of their entropy as a function of the worldline parameter $\tau_1$. These curves agree with the respective gravitational curves with the same labels. In the gravitational theory the extremal curves for $n_0=-1$ (solid) and for $n_0=+1$ (dashed) have the same metric and gauge field configurations.
• Figure 4: The blue curve denotes the extremal black hole-locus with the horizon radius $r_+=\sqrt{2 a+a^2}$. The red curves correspond to different values of $r_+$ (where the solution is complex). Notice that the envelope of these complex solutions in $\mathbb{H}$ is bounded by the semi-circle on the right.
• Figure 5: The orange lines are extremal curves in the $\tau$-plane for the $(m,n)$ saddle (where $J$, $Q$, and $\mathcal{S}_{(m,n)}$ are all real). The blue line plots the normalized entropy $\frac{\mathcal{S}_{(m,n)}}{N^2}$ along the corresponding extremal curve. For every $\frac{n}{m}$, with $m>0$, $\text{gcd}(m,n)=1$, and such that $(m+n) \,\text{mod}\,3 \neq 0$, the normalized entropy grows to $+\infty$ when the extremal curve approaches $-\frac{n}{m}$ from the left ($(m+n)\,\text{mod}\,3 = 1$), or from the right ($(m+n) \,\text{mod}\,3 = -1$).