Smooth Freund–Rubin backgrounds of eleven-dimensional supergravity of the form AdS$_4 \times X^7$ and preserving at least half of the supersymmetry have been recently classified. Requiring that amount of supersymmetry forces $X$ to be a spherical space form, whence isometric to the quotient of the round 7-sphere by a freely acting finite subgroup of SO(8). The classification is given in terms of ADE subgroups of the quaternions embedded in SO(8) as the graph of an automorphism. In this paper, we extend this classification by dropping the requirement that the background be smooth, so that $X$ is now allowed to be an orbifold of the round 7-sphere. We find that if the background preserves more than half of the supersymmetry, then it is automatically smooth in accordance with the homogeneity conjecture, but that there are many half-BPS orbifolds, most of them new. The classification is now given in terms of pairs of ADE subgroups of quaternions fibred over the same finite group. We classify such subgroups and then describe the resulting orbifolds in terms of iterated quotients. In most cases, the resulting orbifold can be described as a sequence of cyclic quotients.

A new formulation of theories of supergravity as theories satisfying a generalized Principle of General Covariance is given. It is a generalization of the superspace formulation of simple 4D-supergravity of Wess and Zumino and it is designed to obtain geometric descriptions for the supergravities that correspond to the super Poincarè algebras of Alekseevsky and Cortés’ classification.

We study the constant contributions to the free energies obtained through the topological recursion applied to the complex curves mirror to toric Calabi–Yau threefolds. We show that the recursion reproduces precisely the corresponding Gromov–Witten invariants, which can be encoded in powers of the MacMahon function. As a result, we extend the scope of the “remodeling conjecture” to the full free energies, including the constant contributions. In the process, we study how the pair of pants decomposition of the mirror curves plays an important role in the topological recursion. We also show that the free energies are not, strictly speaking, symplectic invariants, and that the recursive construction of the free energies does not commute with certain limits of mirror curves.

Recent work applying higher gauge theory to the superstring has indicated the presence of “higher symmetry”. Infinitesimally, this is realized by a “Lie 2-superalgebra” extending the Poincaré superalgebra in precisely the dimensions where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a “Lie 2-supergroup” extending the Poincaré supergroup in the same dimensions.

Briefly, a “Lie 2-superalgebra” is a two-term chain complex with a bracket like a Lie superalgebra, but satisfying the Jacobi identity only up to chain homotopy. Simple examples of Lie 2-superalgebras arise from 3-cocycles on Lie superalgebras, and it is in this way that we constructed the Lie 2-superalgebra above. Because this 3-cocycle is supported on a nilpotent subalgebra, our geometric technique applies, and we obtain a Lie 2-supergroup integrating the Lie 2-superalgebra in the guise of a smooth 3-cocycle on the Poincaré supergroup.