Double field theory algebroid and curved $L_{\mathrm{\infty }}$-algebras

A DFT algebroid is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding the symmetries of double field theory. In particular, a DFT algebroid is a structure defined on a vector bundle over doubled spacetime equipped with the C-bracket of double field theory. In this paper we give the definition of a DFT algebroid as a curved $L_{\mathrm{\infty }}$-algebra and show how implementation of the strong constraint of double field theory can be formulated as an $L_{\mathrm{\infty }}$-algebra morphism. Our results provide a useful step towards coordinate invariant descriptions of double field theory and the construction of the corresponding sigma-model.