chiral ring just described does indeed take place.

The detailed map between the electric theory and the magnetic theory ,  satisfies a number of non-trivial constraints:

[1.]{} The electric theory has a complicated vacuum structure which depends on the details of the superpotential . The magnetic theory should have the same vacuum structure.

[2.]{} The expectation values of the operators $\Tr X^j$ can be calculated in all the vacua of the electric theory. They should agree with the corresponding calculation done using the magnetic variables and the map of operators .

[3.]{} As various perturbations are turned on, some fields become massive and can be integrated out both in the electric and in the magnetic theories. The two resulting low energy theories are calculable. They should be dual to one another. In particular, powerful constraints arise from requiring consistency of the operator map implied by duality with various deformations.

Additional constraints on duality follow from the requirement that certain scale matching relations are consistent with all deformations. These relations are discussed in the next subsection.

It is highly non-trivial and surprising that there exists a duality transformation which satisfies such a large number of consistency checks.

An important chiral operator 