Why exactly marginal deformations are common in N=1 d=4

Five authors, Daniel Green, Zohar Komargodski, Nathan Seiberg, Yuji Tachikawa, and Brian Wecht, wrote an interesting preprint

Exactly Marginal Deformations and Global Symmetries

In N=1 supersymmetric four-dimensional field theories, we used to think that the amount of supersymmetry seems rather small for the theory to preserve exactly marginal deformations. It has always seemed surprising why there were any theories with moduli spaces at all.

However, these authors extend the results of Leigh and Strassler (1995) and Barak Kol (2002) and explain that there's no real surprise here. For N=1 superconformal theories, it is actually very difficult to transform a superficially marginal operator to not-quite marginal one.




In fact, the only way to do so is to combine this operator with a conserved current. Consequently, one can also fully map the moduli space of possible, exactly marginal deformations: it is the quotient of all the deformations and the complexified gauge group. The understanding of the deformations becomes an exercise in group theory.

Similar arguments apply to N=2 d=3 theories, too.