  The giant graviton expansion of the line defect Schur index in four
dimensional $\mathcal N=4$ $U(N)$ SYM was recently proposed in arXiv:2403.11543
to be captured in the dual string theory by counting fluctuations states of two
half-infinite fundamental strings in $AdS_{5}\times S^{5}$ ending on the line
defect and D3 brane giant. However, agreement with the gauge theory data for
the defect line index at finite $N$ required the inclusion of ad hoc extra
contributions with unclear origin. We discuss the large $N$ leading order
contribution of the giant graviton expansion of the defect line index by a
direct analysis of semiclassical string partition function in a twisted
background. We discuss supersymmetric boundary conditions in the presence of
the D3 brane and evaluate the quadratic fluctuations effective action by
introducing a suitable projection of fluctuation modes. We show that the extra
contributions to the single giant graviton correction found in arXiv:2403.11543
correspond to a supersymmetric Casimir energy contribution.
