This is more of a particle physics question than maths. Since $\operatorname{SO}(6)$ and $\operatorname{SU}(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ Super Yang Mills in $4d$) transforming under 6 dimensional vector representation of $\operatorname{SO}(6)$ related to the fields transforming under antisymmetric 6 of $\operatorname{SU}(4)$?
Relation between $SU(4)$ and $SO(6)$
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group-theory
representation-theory
lie-groups
mathematical-physics
quantum-field-theory