Weakly Constrained Double Field Theory as the Double Copy of Yang-Mills Theory

Weakly constrained double field theory, in the sense of Hull and Zwiebach, captures the subsector of string theory on toroidal backgrounds that includes gravity, $B$-field and dilaton together with all of their massive Kaluza-Klein and winding modes, which are encoded in doubled coordinates subject to the `weak constraint'. Due to the complications of the weak constraint, this theory was only known to cubic order. Here we construct the quartic interactions for the case that all dimensions are toroidal and doubled. Starting from the kinematic $C_{\infty}$ algebra ${\cal K}$ of pure Yang-Mills theory and its hidden Lie-type algebra, we construct the $L_{\infty}$ algebra of weakly constrained double field theory on a subspace of the `double copied' tensor product space ${\cal K}\otimes\bar{\cal K}$, by doing homotopy transfer to the weakly constrained subspace and performing a non-local shift that is well-defined on the torus. We test the resulting three-brackets, and establish their uniqueness up to cohomologically trivial terms, by verifying the Jacobi identities up to homotopy for the gauge sector.