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.