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Superfield description of (4+2n)-dimensional SYM theories and their mixtures on magnetized tori

We provide a systematic way of dimensional reduction for $(4+2n)$-dimensional $U(N)$ supersymmetric Yang-Mills (SYM) theories ($n=0,1,2,3$) and their mixtures compactified on two-dimensional tori with background magnetic fluxes, which preserve a partial ${\mathcal N}=1$ supersymmetry out of full ${\mathcal N}=2,3$ or $4$ in the original SYM theories. It is formulated in an $\mathcal N=1$ superspace respecting the unbroken supersymmetry, and the four-dimensional effective action is written in terms of superfields representing $\mathcal N=1$ vector and chiral multiplets, those arise from the higher-dimensional SYM theories. We also identify the dilaton and geometric moduli dependence of matter Kähler metrics and superpotential couplings as well as of gauge kinetic functions in the effective action. The results would be useful for various phenomenological/cosmological model buildings with SYM theories or D-branes wrapping magnetized tori, especially, with mixture configurations of them with different dimensionalities from each other.