vacua, which should be identified as one and the same.

The appearance of a non-compact global symmetry group $G$ is a characteristic feature of the supergravity theories that represent the low-energy effective action for the massless modes of a superstring compactification. Typically, the group $G$ is realized nonlinearly by scalar fields that parametrize the homogeneous space $G/H$, where $H$ is the maximal compact subgroup of $G$. The first example of this phenomenon, with $G = SL(2,\RR)$ and $H =
U(1)$, was uncovered in 1976 in a version of $N = 4$ 4d supergravity by Cremmer, Ferrara, and Scherk.[@cremmer78a] Curiously, a discrete subgroup of the symmetry of this particular example corresponds precisely to the example of $S$ duality that was first recognized in string theory – that of the toroidally compactified heterotic string. An analogous non-compact $E_7$ symmetry was found in $N = 8$ 4d supergravity by Cremmer and Julia in 1978,[@cremmer78b] and many other examples were worked out thereafter.[@salam89] The Cremmer–Julia example corresponds to the toroidally compactified type II string, and combines $S$, $T$, and $U$ dualities in a single discrete group. (As mentioned above, this usage of the term ‘$U$ duality’ differs a bit from the one proposed by 