  Exact expressions for certain integrated correlators of four half-BPS
operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group
$SU(N)$ have been recently obtained thanks to a beautiful interplay between
supersymmetric localisation and modular invariance. The large-$N$ expansion at
fixed Yang-Mills coupling of such integrated correlators produces an asymptotic
series of perturbative terms, holographically related to higher derivative
interactions in the low energy expansion of the type IIB effective action, as
well as exponentially suppressed corrections at large $N$, interpreted as
contributions from coincident $(p,q)$-string world-sheet instantons. In this
work we define a manifestly modular invariant Borel resummation of the
perturbative large-$N$ expansion of these integrated correlators, from which we
extract the exact non-perturbative large-$N$ sectors via resurgence analysis.
Furthermore, we show that in the 't Hooft limit such modular invariant
non-perturbative completions reduce to known resurgent genus expansions.
Finally, we clarify how the same non-perturbative data is encoded in the
decomposition of the integrated correlators based on $\rm{SL}(2,\mathbb{Z})$
spectral theory.
