One-loop determinants in AdS$_3$ supergravity with extended supersymmetry

We re-examine the computation of the one-loop partition function of Type II supergravity theory compactified on $AdS_3 \times \mathbf{S}^3 \times X$, where $X$ can be $K_3$, $T^4$ and $\mathbf{S}^3 \times \mathbf{S}^1$. These backgrounds preserve eight supercharges (four left and four right moving) and are known to be holographically dual to either small or large $N=(4,4)$ superconformal field theories. By extending well-established heat kernel techniques, we evaluate the one-loop determinants associated to these supergravity backgrounds. The resulting expressions are then used to reconstruct the characters of short multiplets of the corresponding dual boundary theories. A distinctive aspect of our analysis is the emergence of contributions from multiple gravitational saddle points in the path integral, reflecting the spectral flow structure present in the boundary theory.