't Hooft expansion of $SO(N)$ and $Sp(N)$ $\mathcal{N}=4$ SYM revisited

TL;DR

This paper revisits the 't Hooft expansion of 4D N=4 SYM theories with SO(N) and Sp(N) gauge groups, revealing a more string-like structure in the 1/N_5 expansion related to RR flux shifts.

Contribution

It demonstrates that the 1/N_5 expansion in SO(N) and Sp(N) N=4 SYM theories is more 'closed string like' than the traditional 1/N expansion, considering RR flux shifts.

Findings

The 1/N_5 expansion aligns more with string theory structures.

The shift in N_5 due to RR charge affects the expansion.

The expansion applies to free energy and Wilson loops.

Abstract

We study the 't Hooft expansion of $d=4$ $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory with the gauge group $SO(N)$ or $Sp(N)$. We consider the $1/N_5$ expansion with fixed $g_sN_5$, where $g_s$ denotes the string coupling of bulk type IIB string theory on $AdS_5\times\mathbb{RP}^5$ and $N_5$ refers to the RR 5-form flux through $\mathbb{RP}^5$. $N_5$ differs from $N$ due to a shift coming from the RR charge of O3-plane. As an example, we consider the $1/N_5$ expansion of the free energy of $\mathcal{N}=4$ SYM on $S^4$ and the $1/2$ BPS circular Wilson loops in the fundamental representation of $SO(N)$ or $Sp(N)$. We find that the $1/N_5$ expansion is more "closed string like" than the ordinary $1/N$ expansion.