On the 4d superconformal index near roots of unity: Bulk and Localized contributions

We study the expansion near roots of unity of the superconformal index of 4d $SU(N)$ $\mathcal{N}=4$ SYM. In such an expansion, middle-dimensional walls of non-analyticity are shown to emerge in the complex analytic extension of the integrand. These walls intersect the integration contour at infinitesimal vicinities and come from both, the vector and chiral multiplet contributions, and combinations thereof. We will call these intersections vector and chiral bits, and the complementary region bulk, and show that, in the corresponding limit, the integrals along the infinitesimal bits include, among other contributions, factorized products of either Chern-Simons and 3d topologically twisted partition functions. In particular, we find that the leading asymptotic contribution to the index, which comes from collecting all contributions coming from vector bits, reduces to an average over a set of $N$ copies of three-dimensional $SU(N)$ Chern-Simons partition functions in Lens spaces $L(m,1)$ with $m>1\,$, in the presence of background $\mathbb{Z}^{N-1}_m$ flat connections. The average is taken over the background connections, which are the positions of individual vector bits along the contour. We also find there are other subleading contributions, a finite number of them at finite $N$, which include averages over products of Chern-Simons and/or topologically $A$-twisted Chern-Simons-matter partition functions in three-dimensional manifolds. This shows how in certain limits the index of 4d $SU(N)$ $\mathcal{N}=4$ SYM organizes, via an unambiguously defined coarse graining procedure, into averages over a finite number of lower dimensional theories.Comment: 62pp. Significantly improved version, product of a deeper understanding of some of the results presented in v1. Two new sections added, one with a summary of results and discussion, another with further developments. Abstract changed accordingl

About the phase space of SL(3) Black Holes

In this note we address some issues of recent interest, related to the asymptotic symmetry algebra of higher spin black holes in $sl(3,\mathbb{R})\times sl(3,\mathbb{R})$ Chern Simons (CS) formulation. We compute the fixed time Dirac bracket algebra that acts on two different phase spaces. Both of these spaces contain black holes as zero modes. The result for one of these phase spaces is explicitly shown to be isomorphic to $W^{(2)}_3\times W^{(2)}_3$ in first order perturbations.Comment: improved presentatio

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

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 $\tau\to 0$.Comment: Version published in JHEP. New section added about contour deformation, and comments added about relations with Bethe-ansatz-type approac