Rules as written, can monks use a Sun Blade as a monk weapon?
Weapon (longsword), rare (requires attunement)
Properties: Versatile, Finesse
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Monks are proficient with shortswords:
Weapons: Simple weapons, shortswords
therefore they are proficient with the Sun Blade as well, because the Sun Blade’s description says that anyone proficient with a short sword or long sword is proficient with it. However, the Sun Blade is a magic longsword and therefore normally not a monk weapon
[...] monk weapons, which are shortswords and any simple melee weapons that don’t have the two-handed or heavy property
TCoE introduces the Dedicated Weapon optional class feature:
[...] you can touch one weapon, focus your ki on it, and then count that weapon as a monk weapon until you use this feature again.
The chosen weapon must meet these criteria:
- The weapon must be a simple or martial weapon.
- You must be proficient with it.
- It must lack the heavy and special properties
The Sun Blade meets all of the criteria for monks, so it is possible to use it as monk weapon.
Slaves_of_the_coast explains how this can work for any monk if you have the Dedicated Weapon optional class feature. But this is an optional rule from Tasha's Cauldron of Everything; not all groups may have that book, and even those who do might not be using it (it is an optional feature after all).
However, the Kensei monk from Xanathar's Guide to Everything can also make this work, even without that feature. At third level, they gain the Kensei Weapons subclass feature:
Choose two types of weapons to be your kensei weapons: one melee weapon and one ranged weapon. Each of these weapons can be any simple or martial weapon that lacks the heavy and special properties. The longbow is also a valid choice. You gain proficiency with these weapons if you don't already have it. Weapons of the chosen types are monk weapons for you.
A longsword, in particular, is a martial weapon that lacks the heavy and special properties. If a Kensei monk chooses longswords as their melee kensei weapon, the Sun Blade can be a monk weapon for them.