The dual of a tensor in m dimensions is formed by using the completely antisymmetric tensor.

dual(t)=t~=e_ij...k * t_ij..n

A self dual tensor has dual(t)=t
An anti-self dual tensor has dual(t)=-t

if +=self dual, -=antiself dual

t+-= n (rank) even => t+-t~
n odd => t+-it~