Constant Torque (CT) = Heavy Duty (HD) are the same type of ratings, simply different terms used in different industries/regions.
Variable Torque (VT) = Normal Duty (ND) = Quadratic Torque (QT) are all the same rating, simply different terms used in different industries/regions.
The difference between CT and VT has to do with the continuous output current ratings and overload output current ratings of VFD drives.
Constant Torque (CT) ratings will generally have lower continuous output current ratings but higher overload output current ratings.
Variable Torque (VT) ratings will generally provide higher continuous output current ratings but much lower overload output current ratings.
Some systems (like fans as an example) do not usually have increased loading once up to speed, nor significant disturbances to the load, so overload current is not a significant requirement and drives can be operated to higher continuous current levels (aka - VT/ND ratings) within the same thermal limitations of a particular drive design.
Other industrial systems require periodic overload output current due to changes in speed or loading, and because of this higher overload output current capability, the continuous current rating is reduced within the same thermal limitations of a particular drive design.