• abelian with matter is the trouble

Compare with pure abelian gauge theory, the Coulomb branch of general cases needs two direction modifications, with matter representations, and with non-abelian gauge group. BFN did a combination of both.

Why is it hard? Is it hard to write down the result, or is it hard to prove that the answer is right?

In the end, it turns out, we need an affine variety that maps over the base $T$ with some conic singular fiber and with some affine blow-up. Then, we just need to take Weyl cover. If you don't do the affine blow-up just do the abelian gauge theory with matter, then you almost get it right, except you need to blow-up near the root hyperplane. The question is maybe, how do you do the affine blow-up?

It is always like: first make the ring bigger, then make the ring smaller. Take affine blow-up makes the ring bigger, then passing to