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 IN,k, the Schur index of 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 IN,k we propose a quadruple-sum expansion. Objects causing finite k and finite N corrections are disk D3-branes ending on the D5-brane.