  Double field theory was originally introduced as the subsector of closed
string field theory on a toroidal background given by the massless fields
together with all their massive Kaluza-Klein and winding modes. These massive
modes are encoded in the dependence of the massless fields on doubled toroidal
coordinates, subject to the so-called 'weak constraint'. This theory was
constructed by Hull and Zwiebach in 2009 to cubic order in fields, but due to
the weak constraint it is a highly non-trivial problem to extend this to
quartic and higher order. In this letter we announce and outline the
construction of weakly constrained double field theory to quartic order, in
which all coordinates are toroidal and doubled. To this end we use the
framework of homotopy algebras and obtain double field theory as a double copy
of the kinematic homotopy algebra of Yang-Mills theory.
