  In a recent paper [1], the semiclassical quantization of a string, winding
once around the compact Euclidean time circle, on a supergravity background
dual to the deep infrared regime of a confining finite temperature gauge
theory, was carried out. The string mass-shell condition and, by extrapolation,
the Hagedorn temperature to leading order in the holographic limit was deduced.
In this work, we improve on those results in three ways. First, we fix some
missing details of the related light-cone quantization analysis. Second, we
reconsider the problem under the lens of a background-covariant geometrical
formalism. This allows us to put the semiclassical mass-shell condition on more
solid grounds. Finally, going beyond the semiclassical regime, we compute the
Hagedorn temperature at next-to-leading order in the holographic limit. The
sub-leading correction turns out to arise entirely from the contribution of the
zero modes of the massive worldsheet scalar fields. Our result matches that of
a recent analysis in the literature based on the Horowitz-Polchinski stringy
star effective model.
