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 $\mathcal{N}$ \( \mathcal{N} \) = 4 U(N) SYM decorated by line operators in the anti-symmetric representation of rank k. For the large N index I_∞,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.