Leading Giant graviton expansion of Schur correlators in large representations

We consider 4d $\mathcal N=4$ $U(N)$ SYM and the leading giant graviton correction to the Schur defect 2-point functions of $\frac{1}{2}$-BPS Wilson lines in rank-$k$ symmetric and antisymmetric representations. We study in particular the large $k$ limit for the symmetric case and the regime $1\ll k \ll N$ in the antisymmetric one. We present exact results for the correction in agreement with matrix model evaluation at finite $N,k$. The Wilson lines in symmetric/antisymmetric representations admit a description in terms of D3$_{k}$ and D5$_{k}$ brane probes representing a collection of $k$ fundamental strings. In this picture, giant graviton corrections come from fluctuations of brane probes in presence of a wrapped D3 brane giant graviton. In particular, for the antisymmetric case, our leading correction matches the half-index of the 3d $\mathcal N=4$ Maxwell theory living on the 3d disk which is a part of the giant graviton divided out by the D5$_{k}$ probe, as recently proposed in arXiv:2404.08302. For the symmetric case at large $k$, we derive an explicit exact residue formula for the leading correction.