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

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 $\tau$. The saddle-point configurations are labelled by points $(m,n)$ on the lattice ${\Lambda }_{\tau }=\mathbb{Z}\tau +\mathbb{Z}$ 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 $\mathbb{C}/{\Lambda }_{\tau }$. 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 $\tau$ is close to the origin, and there are new saddles that dominate when $\tau$ approaches rational points. The extension of the action in terms of modular forms leads to a simple treatment of the Cardy-like limit $\tau \to 0$.

arXiv commentVersion published in JHEP. New section added about contour deformation, and comments added about relations with Bethe-ansatz-type approach

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