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Non-Abelian Electric-Magnetic Duality with Supersymmetry in 4D and 10D

We present electric-magnetic (Hodge) duality formulation for non-Abelian gauge groups with N=1 supersymmetry in 3+1 (4D) dimensions. Our system consists of three multiplets: (i) A super-Yang-Mills vector multiplet (YMVM) $(A_μ^I, λ^I)$, (ii) a dual vector multiplet (DVM) $(B_μ^I, χ^I)$, and (iii) an unphysical tensor multiplet (TM) $(C_{μν}{}^I, ρ^I, φ^I)$, with the index I for adjoint representation. The multiplets YMVM and DVM are dual to each other like: $G_{μν}{}^I = (1/2) ε_{μν}{}^{ρσ} F_{ρσ}{}^I$. The TM is unphysical, but still plays an important role for establishing the total consistency of the system, based on recently-developed tensor-hierarchy formulation. We also apply this technique to non-Abelian electric-magnetic duality in 9+1 (10D) dimensions. The extra bosonic auxiliary field $K_{μ_1\cdotsμ_6}$ in 10D is shown to play an important role for the closure of supersymmetry on fields.