On Killing spinors in general dimensions

The following property is true in four spacetime dimensions [1] [2]

If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations \begin{equation*} \nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0 \end{equation*} and there is a spinor $\psi$ such that \begin{equation}\label{eq:20161115b} (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi = 0 \end{equation} and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well: \begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}

This property (property A from now on) can for example be used to obtain the metric and electromagnetic field of the Israel-Wilson-Perjés (IWP) black holes. Because I wanted to generalize the IWP black holes to higher dimensions, I wanted to find the generalization of property A in higher dimensions. In $D=4$, the super covariant derivative of a spinor is \begin{equation} \hat\nabla_{\mu}\psi = (\nabla_{\mu} + i \sqrt{4 \pi} \not F \gamma_{\mu} ) \psi \end{equation} Because $\not F \gamma_{\mu}= \dfrac{1}{4} F^{ij}\gamma_{ij\mu} - \dfrac{1}{2} F_{\mu}^{\ \ j}\gamma_j$, I decided to define in general dimension $D$ \begin{equation}\label{eq:20161205a} \hat\nabla_{\mu}\psi = (\nabla_{\mu} + \dfrac{1}{4} F^{ij}\gamma_{ij\mu} + \theta F_{\mu}^{\ \ j}\gamma_j ) \psi \end{equation} with $\theta$ a number that I will fix later. After rescaling the gauge field $A$, this is consistent with the definition in $D=4$ if $\theta = -\frac{1}{2}$
Property A is proved by using the equation \begin{equation} \gamma^{\nu}[\hat\nabla_{\mu}, \hat\nabla_{\nu}]\psi = 0 \end{equation} I therefore calculated this commutator for the ansatz \eqref{eq:20161205a}. The result is [3] \begin{align} \gamma^{\nu}[\hat\nabla_{\mu}, \hat\nabla_{\nu}]\psi = \Big(& -\frac{1}{2} R_{i \mu} \gamma^{i} + \frac{1}{8} (- D + 5 - 4 \theta) V_{\mu} +(\frac{1}{2} - 2 \theta^2 - \frac{\theta}{2} D + 2 \theta) III_{\mu} + ( 2 \theta^2 + \frac{D-3}{2} ) I_{\mu} \nonumber \\ & + \frac{D-3}{4} F^2 \gamma_{\mu} -\frac{1}{2} \nabla_i F^{ij} \gamma_{j \mu} + \theta \nabla_j F^j_{\ \ \mu} \label{eq:20161205b}\\ & -\frac{1}{4} \nabla^{\nu} F^{ij} \gamma_{\nu ij \mu} - (\frac{3}{4} - \frac{D}{4} -\theta ) \nabla_{\mu}F^{kj} \gamma_{kj} - \theta \nabla_k F_{\mu j} \gamma^{kj}\Big) \psi \nonumber \end{align} with \begin{align*} I_{\mu} &= (F^2)_{\mu}^{\ \ i}\gamma_i\\ III_{\mu} &= F_{\mu}^{\ \ i} F^{jk} \gamma_{ijk}\\ V_{\mu} &= F^{ij}F^{kl} \gamma_{\mu ijkl} \end{align*} If I now assume that $\nabla_{[ i } F_{jk ]} =0$ then the last three terms in \eqref{eq:20161205b} combine to \begin{equation*} ( \frac{3}{4} - \frac{D}{4} - \theta + \frac{\theta}{2} )\nabla_{\mu} F^{kj} \gamma_{kj} \end{equation*} This is zero if I set $\theta = \dfrac{3 - D}{2}$. This agrees with the special case $D=4$ [4]. For this value of $\theta$, I can simplify expression \eqref{eq:20161205a}. The result is \begin{align} \gamma^{\nu}[\hat\nabla_{\mu}, \hat\nabla_{\nu}]\psi = \Big(& -\frac{1}{2} R_{i \mu} \gamma^{i} + \frac{D-1}{8} V_{\mu} -\frac{1}{4} (D-1) (D-4) III_{\mu} -\frac{1}{2} (D-2) (D-3) ( T_{\mu i} - \frac{T} {D-2} \eta_{\mu i} ) \gamma^i\nonumber \\ & -\frac{1}{2} \nabla_i F^{ij} \gamma_{j \mu} -\frac{D-3}{2} \nabla_j F^j_{\ \ \mu}\Big) \psi \nonumber \end{align} with $T_{\mu\nu}$ the electromagnetic stress tensor \begin{align*} T_{\mu\nu} &= - ( F_{\mu \alpha} F^{\alpha}_{\ \ \nu} + \frac{1}{4} \eta_{\mu\nu} F^2)\\ T &= T_{\mu}^{\ \mu} \end{align*} For general $D$, the terms $V_{\mu}$ and $III_{\mu}$ survive and the commutator $\gamma^{\nu}[\hat\nabla_{\mu}, \hat\nabla_{\nu}]\psi$ contains terms that are not zero if the fields satisfy the equations of motion. I conclude that for general dimension $D$, I cannot use a generalization of property $A$ to obtain a generalization of the IWP black holes. For $D=4$, $V_{\mu}= 0 $ and the coefficient of $III_{\mu}$ is zero, hence property $A$ follows. Something can also be salvaged in $D=5$, this is the subject of another post. Of course, I can also force the terms involving $V_{\mu}$ and $III_{\mu}$ to zero, then I get the following property (after rescaling of the electromagnetic field)

If the electromagnetic field $F_{ab}$ satisfies Maxwell's equations \begin{equation*} \nabla_{[a}F_{bc]}= 0 \quad\text{and}\quad \nabla^a F_{ab} =0 \end{equation*} and $F_{\mu}^{\ \ [i} F^{jk]} =0$ and there is a spinor $\psi$ such that \begin{equation} \left(\nabla_{\mu} + i \sqrt{ \frac{8 \pi}{(D-2)(D-3)}} \left( F^{ij}\gamma_{ij\mu} + \frac{D-3}{2} F_{\mu}^{\ \ j}\gamma_j \right)\right) \psi =0 \end{equation} and $i \bar \psi \gamma^{\mu} \psi$ is time-like

then the Einstein equations are satisfied as well: \begin{equation*} R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 8 \pi T_{\mu\nu} \end{equation*}

All in all what I learned from this calculation is that naive generalizations of general relativity in $D=4$ to higher dimensions often do not work. The higher dimensional generalizations that seem to work are bosonic sectors of various supergravities.

References and comments

[1] Black hole solutions in string theory, Maeda and Nozawa, 2011, hep-th/1104.1849

[2] See the previous blog post for the notation that I use.

[3] In $D=4$, the result can be found in [1], where it is said that is "a simple exercise". However, I found the calculation very tedious and it took me a long time to calculate the commutator for general dimension $D$. There are many manipulations one can do with gamma matrices, but I think that the calculation proceeds in the most efficient way if I use the commutators \begin{align*} [\gamma_i , \gamma_j] &= 2 \gamma_{ij}\\ [\gamma_i , \gamma_{jkl}] &= 2 \gamma_{ijkl}\\ [\gamma^{ijk} , \gamma_{abc}] &= 2 \gamma^{ijk}_{\quad \ abc} - 36 \delta^{[i}_{[a}\delta^{j}_{b} \gamma^{k]}_{\ \ \ c]} \end{align*}

[4] My result is similar to, but different from, equation A10 in hep-th/1104.1849