BPS spectrum on AdS$_3\times $S$^3 \times $S$^3 \times $S$^1$

The BPS spectrum of string theory on AdS$_3\times {\rm S}^3 \times {\rm S}^3 \times {\rm S}^1$ is determined using a world-sheet description in terms of WZW models. It is found that the theory only has BPS states with $j^+ = j^-$ where $j^{\pm}$ refer to the spins of the $\mathfrak{su}(2)$ algebras of the large ${\cal N}=4$ superconformal algebra. We then re-examine the BPS spectrum of the corresponding supergravity and find that, in contradistinction to previous claims in the literature, also in supergravity only the states with $j^+=j^-$ are BPS. This resolves a number of long-standing puzzles regarding the BPS spectrum of string theory and supergravity in this background.