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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 theirmixtures compactified on two-dimensional tori with background magnetic fluxes,which preserve a partial ${\mathcal N}=1$ supersymmetry out of full ${\mathcalN}=2,3$ or $4$ in the original SYM theories. It is formulated in an $\mathcalN=1$ superspace respecting the unbroken supersymmetry, and the four-dimensionaleffective action is written in terms of superfields representing $\mathcal N=1$vector and chiral multiplets, those arise from the higher-dimensional SYMtheories. We also identify the dilaton and geometric moduli dependence ofmatter K\"ahler metrics and superpotential couplings as well as of gaugekinetic functions in the effective action. The results would be useful forvarious phenomenological/cosmological model buildings with SYM theories orD-branes wrapping magnetized tori, especially, with mixture configurations ofthem with different dimensionalities from each other.

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