4
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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 $\mathrm{AdS}_7 \times S^4$.

$\endgroup$
  • 1
    $\begingroup$ I copied the 8.3 here, but I am not sure it is correct. Could you please delete the not important parts? $\endgroup$ – user259412 Aug 25 '16 at 13:49
  • $\begingroup$ I think it should be added that $S^4$ admits four Killing (Dirac) spinors. $\endgroup$ – Meer Ashwinkumar Sep 13 '16 at 6:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.