Superfield description of 10D SYM theory with magnetized extra dimensions

We present a four-dimensional (4D) ${cal N}=1$ superfield description of supersymmetric Yang-Mills (SYM) theory in ten-dimensional (10D) spacetime with certain magnetic fluxes in compactified extra dimensions preserving partial ${cal N}=1$ supersymmetry out of full ${cal N}=4$. We derive a 4D effective action in ${cal N}=1$ superspace directly from the 10D superfield action via dimensional reduction, and identify its dependence on dilaton and geometric moduli superfields. A concrete model for three generations of quark and lepton superfields are also shown. Our formulation would be useful for building various phenomenological models based on magnetized SYM theories or D-branes.