Higher Gauge Structures in Double and Exceptional Field Theory

We review the higher gauge symmetries in double and exceptional field theory from the viewpoint of an embedding tensor construction. This is based on a (typically infinite-dimensional) Lie algebra $\mathfrak{g}$ and a choice of representation $R$. The embedding tensor is a map from the representation space $R$ into $\mathfrak{g}$ satisfying a compatibility condition (`quadratic constraint’). The Lie algebra structure on $\mathfrak{g}$ is transported to a Leibniz–Loday algebra on $R$, which in turn gives rise to an $L_{\mathrm{\infty }}$-structure. We review how the gauge structures of double and exceptional field theory fit into this framework.