Question

Does target space have the same structure when we compactify from 3D to 2D? I mean when we compactify from 4D to 3D, the moduli space is essentially a torus fibration over the Coulomb branch of $\mathcal{N}=2$ theory and we know that the Coulomb branch has a rigid special Kähler structure. So when we reduce to 2D, why does  the same Hyperkähler target space (moduli space) of 3D theory arise?

Answer 1

Moduli spaces of supersymmetric vacua of supersymmetric field theories are parametrized by expectation values of massless scalar fields. When we compactify a theory the massless scalars of the original theory are still present in the compactified theory and so the moduli space of the original theory is certainly naturally inside the moduli space of the compactified theory. The moduli space of the compactified theory can be bigger if there are new massless scalar fields in the compactified theory which were not present in the original one. At a generic point of the moduli space, the only way it can happen is by dimensional reduction of the gauge vectors of the original theory along the compact directions: if you have a gauge vector $A_\mu$ in the original theory and if you compactify some of the directions corresponding to some indices $i$ then the corresponding components $A_i$ become massless scalar fields in the compactified theory. This is why when one goes from $6d$ to $5d$, from $5d$ to $4d$, from $4d$ to $3d$ the moduli spaces are in general different. Now the key point in that in spacetime dimension $d$ a gauge vector has only $d-2$ physical components. In particular in $3d$  an abelian (which is the case at a generic point of the moduli space) gauge field can be dualized in a scalar field. If you include this scalar in the description of the moduli space of the $3d$ theory, what is usually done, then there is no longer any non-trivial gauge vector in $3d$. In particular going from $3d$ to $2d$ there are no non-trivial gauge vectors to reduce along the compactified direction to give new massless scalar fields and so the moduli space do not change.

Comments

@40227, Thank you very much! I have other questions that I asked here: Coulomb branch of $\mathcal{N}=2$ field theories