The double copy of maximal supersymmetry in D = 4

A bstract We realize off-shell, local and gauge invariant $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 8 supergravity in D = 4, to cubic order in fields, as the double copy of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 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 $$ \mathcal{K} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM is compatible with an action of the global supersymmetry algebra. The double copy space is then a subspace of $$ \mathcal{K} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> ⨂ $$ \overset{\sim }{\mathcal{K}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> that inherits an L ∞ algebra on which the two copies of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 action combine into an action of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 8 supersymmetry algebra, with a corresponding enhancement of the R -symmetry group to SU(8).