Geometrical representation of the Killing spinors preserving N=4 supersymmetry (I)

In the low energy limit the mysterious M-theory boils down to a much tractable d=11 Supergravity theory (SUGRA). Therefore it is essential to understand the supersymmetric constraints of the theory which have crucial applications in the field of holography.

Supersymmetry is essentially a (very awesome if you ask me!) symmetry which keeps the theory invariant under the bosonic and fermionic variations given by

\delta_\epsilon\Theta =\epsilon\\\delta_\epsilon X^M=i\bar{\epsilon}\Gamma^M\Theta

Here \epsilon is a Killing spinor which satisfies the Killing equation

\nabla_X\epsilon=\lambda X.\epsilon

It becomes covariantly constant for \lambda =0. In the curved solutions of the SUGRA, supersymmetries are broken due to the non trivial covariant derivative. In order to preserve SUSY, the solutions of the Killing equation play essential role. We focus on those spinors which are invariant under the spin lift of the holonomy group of the appropriate manifold.  For d=11 SUGRA, the Killing equation takes the following algebraic form

\nabla_M\epsilon+\frac{1}{288}\left(\Gamma_M^{NPQR}-8\delta^N_M\Gamma^{PQR}\right)G_{NPQR}\epsilon=0

Now the notion of the G-structures essentially classifies the special differential forms which arise in the supersymmetric flux compactifications. As can be deduced from the Killing equation, the solutions characterize the Spin Bundle of the supersymmetries with the metric of the manifold with a spin structure in a very intimate way.

Definition: A spin structure on a manifold (\mathcal{M},g) with signature (s,t) is a principle Spin(s,t)-bundle with Spin(\mathcal{M})\to \mathcal{M} together with a bundle morphism \phi : Spin(\mathcal{M})\to SO(\mathcal{M}).

To define the G-structure, we associate the differential forms with the Killing spinors as follows

\Omega^{ij}_{\mu_1\mu_2\ldots\mu_k}=\bar{\epsilon}^i\Gamma_{\mu_1\mu_2\ldots\mu_k}\epsilon^j

The aim is to show that these differential forms obey the set of the first order differential equations as a natural consequence of the Killing equations. Now it can be shown that for Clabi-Yau manifolds, or manifolds with the G_2 holonomy, one usually finds the Killing spinor bundles trivially defined by an algebraic projection which are some differential forms applied to the complete spin bundle.

So this seems like a good point to start and make an ansatz for the projection operator for the spin bundle structure in the curved spacetime. These projections are essentially the differential forms defined above which give rise to the notion of the G_2 structures.

Here (for the reasons beyond me right now), three projection operators are defined \Pi_j for j=0,1,2 which break the 32 supersymmetries to four. Another factual data is that if there is a holographic dual to the theory with a Coulomb branch, then there is a non-trivial space of the moduli for brane probes. This moduli space will be realized as conformally Kahler section of the metric (for four supersymmetries). And it is on this section of the metric, supersymmetries will satisfy projection conditions \Pi_j\epsilon=0 with the form \Pi_j=\frac{1}{2}(1+\Gamma^{\xi_j}), where \Gamma^{\xi_j} represents the product of gamma matrices parallel to the moduli space of the branes.

Now we can find the equations of motion of the theory by demanding that the fermionic variations vanish, implying the Killing equation! The solution we are considering here essentially has the topology of AdS_4\times S^7. Using the orthonormal frames, https://arxiv.org/pdf/hep-th/0403006.pdf shows the presence of a Kahler structure on the brane-probe moduli space as a conformal multiple of

J_{\text{moduli}}=e^6\wedge e^9+e^7\wedge e^8-e^5\wedge e^{10}

I will continue from here in the next blog-post!

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