Pin(d,d) covariance of pure spinor equations for supersymmetric vacua and Non-Abelian T-duality

Abstract

In a supersymmetric compactification of Type II supergravity, preservation of ${\cal{N}} = 1$ supersymmetry in four dimensions requires that the structure group of the generalized tangent bundle $TM \oplus T^*M$ of the six dimensional internal manifold $M$ is reduced from $SO(6,6)$ to $SU(3) \times SU(3)$. This topological condition on the internal manifold implies existence of two globally defined compatibe pure spinors $\Phi_1$ and $\Phi_2$ of non-vanishing norm. Furthermore, these pure spinors should satisfy certain first order differential equations. In this paper, we show that Non-Abelian T-duality (NATD) is a solution generating transformation for these pure spinor equations. We first show that the pure spinor equations are covariant under $Pin(d,d)$ transformations. Then, we use the fact NATD is generated by a coordinate dependent $Pin(d,d)$ transformation. The key point is that the flux produced by this transformation is the same as the geometric flux associated with the isometry group, with respect to which one implements NATD. We demonstrate our method by studying NATD of certain solutions of Type IIB supergravity with $SU(2)$ isometry and $SU(3)$ structure.