string theory - Geometry of spacetime and spinor bilinears

In this paper (http://arxiv.org/abs/0704.0247) p.20, the author says in the section titled Geometry of spacetime the following:

In order to obtain the spacetime geometry, we consider the spinor bilinears $$V_{\mu}= D(\epsilon, \Gamma_{\mu}\epsilon) \hspace{2cm} B_{\mu}=D(\epsilon, \Gamma_5\Gamma_{\mu}\epsilon) \hspace{2cm}(1.1)$$ whose nonvanishing components are $$V_+ = \sqrt{2}b\bar{b},\hspace{.5cm} V_− = −\sqrt{2}, \hspace{2cm}B_+ = \sqrt{2}b\bar{b},\hspace{.5cm} B_− =\sqrt{2}.$$ As $V^2 = −4b\bar{b} = −B^2$, $V$ is timelike and $B$ is spacelike. Using eqns. (4.1) - (4.4) (therein), it is straightforward to show that V is Killing and B is closed.

What is the importance of these two vectors?

Answer

Equations on bilinears follow directly from the Killing spinor equations. The latter are all you have, there is no additional equations.

For example, look at Appenedix A (in particular in A.1) here. You can construct a lot of different bilinears, but the only bilinears you need to write down the solution are (A.17) and all the following equations (A.18-A.21) follow directly from the Killing spinor equations (A.16).