This paper discusses the giant graviton expansion for the Schur index of $ \mathcal{N}=4 \, U(N) $ SYM with the insertion of Wilson lines of fundamental and anti-fundamental representations. The authors propose a double-sum giant graviton expansion and numerically confirm that it correctly reproduces the line-operator index. They also find that it reduces to a simple-sum expansion when the index is treated as a Taylor series with respect to a specific fugacity. The Schur index is defined as $ I_N = \mathrm{tr}[(-1)^F q^{J_1} x^{R_x} y^{R_y}] $, with $ q = x y $. The line-operator index is calculated using a localization formula involving characters of the adjoint, fundamental, and anti-fundamental representations. The authors analyze the large N limit of the line-operator index and find that it coincides with the supergravity multiplet index in $ AdS_5 \times S^5 $. They propose a giant graviton expansion for the line-operator index, which includes contributions from giant gravitons and fluctuations on the string worldsheet. The expansion is shown to reproduce the line-operator index up to certain orders. The authors also discuss the simple-sum expansion for the line-operator index and its relation to the Schur index. The paper concludes with a discussion of the implications of the results and potential future directions for research.This paper discusses the giant graviton expansion for the Schur index of $ \mathcal{N}=4 \, U(N) $ SYM with the insertion of Wilson lines of fundamental and anti-fundamental representations. The authors propose a double-sum giant graviton expansion and numerically confirm that it correctly reproduces the line-operator index. They also find that it reduces to a simple-sum expansion when the index is treated as a Taylor series with respect to a specific fugacity. The Schur index is defined as $ I_N = \mathrm{tr}[(-1)^F q^{J_1} x^{R_x} y^{R_y}] $, with $ q = x y $. The line-operator index is calculated using a localization formula involving characters of the adjoint, fundamental, and anti-fundamental representations. The authors analyze the large N limit of the line-operator index and find that it coincides with the supergravity multiplet index in $ AdS_5 \times S^5 $. They propose a giant graviton expansion for the line-operator index, which includes contributions from giant gravitons and fluctuations on the string worldsheet. The expansion is shown to reproduce the line-operator index up to certain orders. The authors also discuss the simple-sum expansion for the line-operator index and its relation to the Schur index. The paper concludes with a discussion of the implications of the results and potential future directions for research.