M-theory compactified on $S^4$ I am trying to understand what happens to the 32 supersymmetries of M-theory when it is compactified on $S^4$, with the other seven dimensions being flat, e.g., $\mathbb{R^7}$, $\mathbb{R^6} \times S^1$, or $\mathbb{R^5} \times S^1 \times S^1$. From what I have read, $S^4$ admits Killing spinors, so I believe some number of supersymmetries should be preserved in the 11-dimensional spacetime of M-theory. My question is, how many supersymmetries are actually preserved? 
 A: This is answered in section 8.3 of:
Paul de Medeiros, José Figueroa-O'Farrill, "Half-BPS M2-brane orbifolds", Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408 (arXiv:1007.4761)

In this case, we are interested in supersymmetric backgrounds
  $\mathrm{AdS}_7 \times X^4$, with $X$ possibly an orbifold. Bär's
  construction, together with the non-existence of irreducible
  five-dimensional holonomy representations, imply that the only
  (complete) four-dimensional manifold admitting real Killing spinors is
  the round sphere $S^4$, hence any other supersymmetric background must
  be an orbifold of $S^4$ by a finite subgroup of $\mathrm{SO}(5)$,
  lifting isometrically to a subgroup $\Gamma < \mathrm{Sp}(2)$.  The
  space of Killing spinors is again identified with the
  $\Gamma$-invariant parallel spinors on $\mathrm{R}^5$ which is the
  irreducible spinor representation $\Delta$ of $\mathrm{Sp}(2)$.  This
  representation is quaternionic, whence the space of $\Gamma$-invariant
  spinors is a quaternionic subspace: if a spinor is invariant, so is
  its quaternion line, by the quaternion-linearity of the action of
  $\mathrm{Sp}(2)$.  Since $\dim_\mathbb{H}\Delta =2$, necessarily
  $0\leq \dim_\mathbb{H}\Delta^\Gamma \leq 2$.  Hence if we demand some
  supersymmetry, either $\Gamma = \{1\}$ and we have $X = S^4$, or else
  the orbifold is half-BPS.  In this case $\Gamma$ is contained in an
  $\mathrm{Sp}(1)$ subgroup of $\mathrm{Sp}(2)$, leaving a nonzero
  vector invariant in the fundamental representation of
  $\mathrm{Sp}(2)$.  Up to automorphisms, we see that $\Gamma$ is one of
  the ADE subgroups in Table 1, but this time embedded in
  $\mathrm{Sp}(2)$ in such a way that if $u \in \Gamma < \mathrm{Sp}(1)$
  and $(x,y)\in\mathbb{H}^2$, then $u \cdot (x,y) = (ux,y)$.
The action of that $\mathrm{Sp}(1)$ subgroup of $\mathrm{SO}(5)$ on
  $S^4$ is given by restricting the action on $\mathrm{RR}^5$.  This is
  given as follows.  First of all, there is a vector which is fixed,
  call it $\nu$.  If we identify the four-dimensional subspace
  perpendicular to $\nu$ with $\mathbb{H}$, then the action of
  $\mathrm{Sp}(1)$ is by left quaternion multiplication.  Finally, using
  the arguments described in Section 7.1 and in particular Table 11, it
  is a simple exercise to decompose such orbifolds $S^4/\Gamma$, except
  for $\Gamma = \mathbb{E}_8$, into a sequence of cyclic quotients. This
  should become useful if and when we understand the six-dimensional
  superconformal field theory dual to $\mathrm{AdS}_7 \times S^4$.

