A related post might be What are marginal fields in CFT? where Qmechanic♦ pointed to Ginsparg secion 8.6.
However, I heard about two argument.
Claim 1:In a $D$ dimension CFT, the marginal operator must satisfy $h+\bar h=D$.
Though, the Gingsparg's paper provided the definition of the relevant and the irreverent operator, it did not rule out the possibility such as $(h,\bar h)=(2,0)$
Claim 2: In 2D, the marginal operator must satisfy $(h,\bar h)=(1,1)$, not other combination?
How to prove the claim 1 and the claim 2, that in 2D CFT the marginal operator must have $(h,\bar h)=(1,1)$?