Tensionless hybrid strings in $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{S}}^3\times {\mathrm{S}}^1$: Free field realisation

We discuss a Wakimoto-like free field realisation of $\mathfrak{d}(2,1;\alpha {)}_1$, whose $\mathfrak{s}\mathfrak{l}(2,\mathbb{R})$ subalgebra has level $k=1$, that requires no gauging, i.e., realises the current algebra exactly. We then compute the partition function of the theory and show that, by combining this with the ghost contribution, the full, on-shell projected string partition function reproduces precisely the single-particle partition function of the ${\mathrm{S}\mathrm{y}\mathrm{m}}^N({{\mathcal{S}}^{\mathcal{\prime }}}_0^2)$ theory, i.e. the symmetric orbifold theory of 8 free fermions, 1 compact free boson, and 1 non-compact free boson. We also discuss other aspects such as DDF operators and BRST and physical state conditions.