I have a question about sigma models in 3D.

If we have $\mathcal{N}=2$ field theory on $\mathbb{R}^4$ and compactify it on $\mathbb{R}^3 \times S^1_R$ (in which $S^1_R$ is a circle of radius $R$) we get a 3D effective field theory whose Lagrangian is dependent on $R$. If we change variables of Lagrangian in suitable way and impose the preservation of SUSY (8 real supercharges), then we get $\mathcal{N}=4$ sigma model whose target space is a Hyperkähler manifold. My question is:

How we can prove this rigorously or using theorems of supersymmetry? Is there any reference other than Gauge Dynamics & Compactification To Three Dimensions that explains this more carefully?

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