On the brane expansion of the Schur index

We consider the Schur index of $\mathcal N=4$ $U(N)$ SYM theory in 4d and its holographic giant graviton-type expansion at finite $N$. We compute the world-volume brane superconformal index by a recently proposed definition of the gauge holonomy integral as a multivariate residue. This is evaluated by a novel deformation algorithm that avoids Gr\"obner basis methods. Various terms of the brane expansion are computed and shown to be free of wall-crossing singularities to the order we explored. The relation between the brane expansion and previous giant graviton-type represenations of the Schur index is clarified.