4D effective action from non-Abelian DBI action with magnetic flux background

We derive four dimensional $mathcal{N}=1$ supersymmetric effective theory from ten dimensional non-Abelian Dirac-Born-Infeld action compactified on a six dimensional torus with magnetic fluxes on the D-branes. For the ten dimensional action, we use a symmetrized trace prescription and focus on the bosonic part up to $mathcal{O}(F^4)$. In the presence of the supersymmetry, four dimensional chiral fermions can be obtained via index theorem. The matter K{a}hler metric depends on closed string moduli and the fluxes but is independent of flavor, and will be always positive definite if an induced RR charge of the D-branes on which matters are living are positive. We read the superpotential from an F-term scalar quartic interaction derived from the ten dimensional action and the contribution of the matter K{a}hler metric to the scalar potential which we derive turns out to be consistent with the supergravity formulation.