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We study supersymmetric open strings in type IIB $AdS_3 \times S^3 \times S^3\times S^1$ with mixed R-R and NS-NS fields. We focus on strings ending along astraight line at the boundary of $AdS_3$, which can be interpreted as lineoperators in a dual CFT$_2$. We study both classical configurations andquadratic fluctuations around them. We find that strings sitting at a fixedpoint in $S^3 \times S^3 \times S^1$, i.e. satisfying Dirichlet boundaryconditions, are 1/2 BPS. We also show that strings sitting at different pointsof certain submanifolds of $S^3 \times S^3 \times S^1$ can still share somefraction of the supersymmetry. This allows to define supersymmetric smearedconfigurations by the superposition of them, which range from 1/2 BPS to 1/8BPS. In addition to the smeared configurations, there are as well 1/4 BPS and1/8 BPS strings satisfying Neumann boundary conditions. All thesesupersymmetric strings are shown to be connected by a network of interpolatingBPS boundary conditions. Our study reveals the existence of a rich moduli ofsupersymmetric open string configurations, for which the appearance of masslessfermionic fields in the spectrum of quadratic fluctuations is crucial.

On the abundance of supersymmetric strings in $AdS_3\times S^3\times S^3\times S^1$ describing BPS line operators