The double copy of maximal supersymmetry in D = 4

We realize off-shell, local and gauge invariant \( \mathcal{N} \) = 8 supergravity in D = 4, to cubic order in fields, as the double copy of \( \mathcal{N} \) = 4 super Yang-Mills theory (SYM). Employing the homotopy algebra approach, we show that, thanks to a redundant formulation for the fermionic fields, the kinematic algebra \( \mathcal{K} \) of \( \mathcal{N} \) = 4 SYM is compatible with an action of the global supersymmetry algebra. The double copy space is then a subspace of \( \mathcal{K} \) ⨂ \( \overset{\sim }{\mathcal{K}} \) that inherits an L_∞ algebra on which the two copies of the \( \mathcal{N} \) = 4 action combine into an action of the \( \mathcal{N} \) = 8 supersymmetry algebra, with a corresponding enhancement of the R-symmetry group to SU(8).