Finite N Index and Angular Momentum Bound from Gravity

TL;DR

This work identifies a maximal angular momentum limit in the finite $N$ index of $\ ext{N}=4$ SYM, defined by $t\to 0$, $y\to\infty$ with $t^{3}y$ fixed, motivated by a gravity dual. Using a matrix-model approach, it demonstrates that the index is free of finite-$N$ corrections and is governed solely by the free $U(1)$ sector, while the free BPS partition function exhibits finite-$N$ structure; weakly interacting dynamics render non-Abelian BPS operators non-BPS, aligning with a gravity picture where only decoupled degrees contribute. The maximal angular momentum limit is supported by supergravity reasoning, where KK modes are bounded by spin-two and the limit is universal across Sasaki–Einstein compactifications; AdS black holes do not contribute in this limit, reinforcing the finite-$N$ independence of the index and interacting BPS partition function. Overall, the results illuminate how finite-$N$ enumeration programs in AdS/CFT can isolate a universal, gravity-compatible sector and motivate further geometric and nonperturbative explorations of BPS spectra.

Abstract

We exactly compute the finite N index and BPS partition functions for N=4 SYM theory in a newly proposed maximal angular momentum limit. The new limit is not predicted from the superconformal algebra, but naturally arises from the supergravity dual. We show that the index does not receive any finite N corrections while the free BPS partition function does.