$\mathcal N=4$ SYM line defect Schur index and semiclassical string

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.