Giant graviton integrated correlators at finite coupling and all orders in $1/N$

We study the giant graviton integrated correlator in SU$(N)$ $\mathcal{N}=4$ super Yang-Mills at finite complexified coupling $\tau$. Despite the formidable complexity arising from the heavy nature of the operators considered, the large-$N$ expansion simplifies dramatically and exhibits manifest modular invariance. At each order in $1/N$, the expansion coefficients are linear combinations of non-holomorphic Eisenstein series thus capturing the full spectrum of perturbative and non-perturbative effects in the Yang-Mills coupling. Furthermore, we find additional contributions which are modular functions exponentially suppressed in $N$. In the 't Hooft limit, this yields an all-orders result in the $1/N$ expansion at arbitrary coupling $\lambda$, extending beyond prior results of leading orders. For the U$(N)$ theory, we obtain a closed-form expression valid for all $N$ and $\tau$, and show that the coupling-dependent sector of the large-$N$ expansion is universal between SU$(N)$ and U$(N)$ to all orders. Crucially, we exploit the integrated correlator constraints and determine the giant graviton correlator itself to two-loop order at finite $N$, previously only accessible in the planar limit.