  Hot Jupiters (HJ) are defined as Jupiter-mass exoplanets orbiting around
their host star with an orbital period < 10 days. It is assumed that HJ do not
form in-situ but ex-situ. Recent discoveries show that star clusters contribute
to the formation of HJ. We present direct $N$-body simulations of planetary
systems in star clusters and analyze the formation of HJ in them. We combine
two direct $N$-body codes: NBODY6++GPU for the dynamics of dense star clusters
with 32 000 and 64 000 stellar members and LonelyPlanets used to follow 200
identical planetary systems around solar mass stars in those star clusters. We
use different sets with 3, 4, or 5 planets and with the innermost planet at a
semi-major axis of 5 au or 1 au and follow them for 100 Myr in our simulations.
The results indicate that HJs are generated with high efficiency in dense star
clusters if the innermost planet is already close to the host star at a
semi-major axis of 1 au. If the innermost planet is initially beyond a
semi-major axis of 5 au, the probability of a potential HJ ranges between
$1.5-4.5$ percent. Very dense stellar neighborhoods tend to eject planets
rather than forming HJs. A correlation between HJ formation and angular
momentum deficit (AMD) is not witnessed. Young Hot Jupiters ($t_{\rm age} <
100$ Myrs) have only been found, in our simulations, in planetary systems with
the innermost planet at a semi-major axis of 1 au.
