Brane expansions for anti-symmetric line operator index

Based on the D5-brane realization of Wilson line operators in anti-symmetric representations, we propose brane expansion formulas for $I_{N,k}$, the Schur index of ${\cal N}=4$ $U(N)$ SYM decorated by line operators in the anti-symmetric representation of rank $k$. For the large $N$ index $I_{\infty,k}$ we propose a double-sum expansion, and for finite $N$ index $I_{N,k}$ we propose a quadruple-sum expansion. Objects causing finite $k$ and finite $N$ corrections are disk D3-branes ending on the D5-brane.