What is the difference between $d$ and $d_k$? I have the explanation: Quadratic Program subproblem formulated as \ \ \begin{equation*} \begin{aligned} & \underset{d}{\text{minimize}} & & \frac{1}{2} d^T H_k d+\nabla f(x_k)^T d \\ & \text{subject to} & & \nabla g_i(x_k)^Td+g_i(x_k)\leq 0 \; \;i = 1, \ldots, m\\ & & & \nabla h_j(x_k)^Td+h_j(x_k)=0 \; \;j = 1, \ldots, n\\ %&&& x_k^{low}\leq x\leq x_k^{high} \end{aligned} \end{equation*} where the Hessian is positive definite, to ensure a minimum is found. Every iteration $k$ computes a new Hessian $H_k$ and solve the corresponding subproblem. The search direction $d_k$ is used to chose the next point, which is closer to the local minima, as follows $$x_{k+1}=x_k+\alpha_k d_k.$$
Why does the $d$ in the formulation not have the index $d_k$? What is the difference between $d$ and $d_k$?
It could have been stated as you suggest, however, the author in question has chosen to do otherwise.
It is clear that in this context, the search direction $d_k$ is the argmin, i.e., the optimal value of $d$, in the Quadratic Program subproblem. It's as simple as that.