soft wall model. However, a complete analysis of the thermal spectral function for vector states of bottomonium and charmonium like the one performed in this article is not present in the literature.

Dual space and Hawking Page transition
--------------------------------------

Gauge string duality at finite temperature was discussed originally in refs. [@Witten:1998qj; @Witten:1998zw]. Considering Euclidean signature and a compactified time coordinate, the geometry dual to a gauge theory at finite temperature is one of the two solutions of Einstein equations with constant negative curvature. One of these solutions is the AdS black hole space that in Euclidean signature reads $$ds^2 \,\,= \,\, \frac{R^2}{z^2}  \,  \Big(    f(z) dt^2 + \frac{dz^2}{f(z) }  + d\vec{x}\cdot d\vec{x}  \Big)   \,,
 \label{EuclideanMetric}$$ where the Schwarschild factor is $ f (z) = 1 - z^ 4/ z_h^ 4  $ and $z_h$ is the horizon position. The other solution is the thermal AdS space, that is just AdS space corresponding to $ f(z) = 1 $, with a compactified time.

Following the work by Hawking and Page [@Hawking:1982dh], one uses the semiclassical argument that there is “competition” between the two solutions and the one with smaller Einstein Hilbert action will be stable at a given temperature. For the conformal 