Ground state energy of twisted $AdS_{3}\times S^{3}\times T^{4}$ superstring and the TBA

We use the lightcone $AdS_{3}\times S^{3}\times T^{4}$ superstring sigma model with fermions and bosons subject to twisted boundary conditions to find the ground state energy in the semi-classical approximation where effective string tension $h$ and the light-cone momentum $L$ are sent to infinity in such a way that ${\cal J}\equiv L/h$ is kept fixed. We then analyse the ground state energy of the model by means of the mirror TBA equations for the $AdS_{3}\times S^{3}\times T^{4}$ superstring in the pure RR background. The calculation is performed for small twist $\mu$ with $L$ and $h$ fixed, for large $L$ with $\mu$ and $h$ fixed, and for small $h$ with $\mu$ and $L$ fixed. In these limits the contribution of the gapless worldsheet modes coming from the $T^4$ bosons and fermions can be computed exactly, and is shown to be proportional to $hL/(4L^2-1)$. Comparison with the semi-classical result shows that the TBA equations involve only one $Y_0$-function for massless excitations but not two as was conjectured before. Some of the results obtained are generalised to the mixed-flux $AdS_{3}\times S^{3}\times T^{4}$ superstring.