Brane expansions for anti-symmetric line operator index

A bstract 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 $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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.