Covariant quantization of the superstring in $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$ with mixed flux

A quantizable and manifestly $\text{PSU}(1,1\mathrm{\mid }2)\times \text{PSU}(1,1\mathrm{\mid }2)$-invariant action for the superstring in $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times {\mathrm{S}}^3\times {\mathrm{T}}^4$ with mixed NS-NS and R-R self-dual three-form flux is constructed, which is the analogue of the $\mathrm{A}\mathrm{d}{\mathrm{S}}_5\times {\mathrm{S}}^5$ pure spinor action for $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times {\mathrm{S}}^3$. The model is then quantized and proven to be conformal invariant at the one-loop level. We conclude by showing how one can relate the supersymmetric description with the Berkovits-Vafa-Witten $\mathrm{A}\mathrm{d}{\mathrm{S}}_3\times {\mathrm{S}}^3$ worldsheet action with mixed flux.