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We show that the spontaneous compactification of the Abelian and non-Abeliantwo-form gauge field theories from $D=4+1$ to $D=3+1$ leads to the sametheories plus the Maxwell and Yang-Mills ones, respectively. The vectorpotential comes from the zero mode of the fifth component of the tensor gaugefield in D=5. Concerning to the non-Abelian case, it is necessary to make amore refined definition of the three-form stress tensor in order to becompatible, after the compactification, with the two-form stress tensor of theYang-Mills theory.

Spontaneous compactification of D=5 two-form gauge fields and the obtainment of Maxwell and Yang–Mills theories