The large-N limit of 4d superconformal indices for general BPS charges

We study the expansion near roots of unity of the superconformal index of 4d SU(N) 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 $$ {\mathbb{Z}}_m^{N-1} $$ ℤ m N − 1 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) 𝒩 = 4 SYM organizes, via an unambiguously defined coarse graining procedure, into averages over a finite number of lower dimensional theories.

On the 4d superconformal index near roots of unity: bulk and localized contributions

Cabo-Bizet

2023

S-duality in the Cardy-like limit of the superconformal index

Amariti,

Zanetti

2024

We evaluate the superconformal index of 4d $$ \mathcal{N} $$ N = 4 SYM with gauge algebra so(2Nc + 1) in the Cardy-like limit. We then study the relation with the results obtained for the S-dual usp(2Nc), discussing the fate of S-duality in different regions of charges. We find that S-duality is preserved thanks to a non-trivial integral identity that relates the three sphere partition functions of pure 3d Chern-Simons gauge theories.

Exact QFT duals of AdS black holes

Choi,

Jeong,

Kim

et al. 2023

We construct large N saddle points of the matrix model for the $$ \mathcal{N} $$ N = 4 Yang- Mills index dual to the BPS black holes in AdS5 × S5, in two different setups. When the two complex chemical potentials for the angular momenta are collinear, we find linear eigenvalue distributions which solve the large N saddle point equation. When the chemical potentials are not collinear, we find novel solutions given by areal eigenvalue distributions after slightly reformulating the saddle point problem. We also construct a class of multi-cut saddle points, showing that they sometimes admit nontrivial filling fractions. As a byproduct, we find that the Bethe ansatz equation emerges from our saddle point equation.