Finite-$N$ superconformal index via the AdS/CFT correspondence

TL;DR

This work develops a finite-$N$ framework for the ${ m olinebreak N}=4$ U$(N)$ superconformal index using AdS/CFT, incorporating D3-branes wrapped on topologically trivial three-cycles of $oldsymbol{S}^5$ and summing over wrapping numbers $(n_1,n_2,n_3)$. The authors provide a detailed integral formula ${ m I}_{U(N)}={ m I}_{ m KK}\sum_{n_i}igl({ m I}_{(n_1,n_2,n_3)}igr)$ with a classical factor ${ m I}_{ m cl}$ and explicit vector and hypermultiplet contributions, addressing contour choices and tachyonic mode issues via a modified single-particle index and pole-cancellation conditions. They numerically confirm that single-, double-, and triple-wrapping sectors reproduce known finite-$N$ corrections to the gauge index up to substantial order in $q$, and they propose a general leading form for higher wraps, supported by pole-cancellation arguments. The results illuminate how finite-$N$ effects in the boundary theory emerge from wrapped brane sectors in the gravity dual and hint at connections to black-hole entropy via refined Fugacities.

Abstract

We propose a prescription to calculate the superconformal index of the ${\cal N}=4$ $U(N)$ supersymmetric Yang-Mills theory with finite $N$ on the AdS side. The finite $N$ corrections are included as contributions of D3-branes wrapped around three-cycles in $\boldsymbol{S}^5$, which are calculated as the index of the gauge theories realized on the wrapped branes. The single-wrapping contribution has been studied in a previous work, and we further confirm that the inclusion of multiple-wrapping contributions correctly reproduces the higher order terms as far as we have checked numerically.

Figures (7)

• Figure 1: The quiver diagram of the theory realized on the D3-brane system with the wrapping numbers $(n_1,n_2,n_3)$.
• Figure 2: The pole structure of the integrand in (\ref{['s1u2']}) is shown.
• Figure 3: The pole structure of the function $\Theta$ on the $x$-plane.
• Figure 4: Terms in the polynomial $\Delta_n$ for $n=3$ are shown.
• Figure 5: Rough behavior of the $q$-expansion of the partition function in the thermodynamic limit.