Abstract
We study the general requirement for supersymmetric AdS$_6$ solutions in type IIB supergravity. We employ the Killing spinor technique and study the differential and algebraic relations among various Killing spinor bilinears to find the canonical form of the solutions. Our result agrees precisely with the work of Apruzzi et. al. \cite{Apruzzi:2014qva} which used the pure spinor technique. We also obtained the four-dimensional theory through the dimensional reduction of type IIB supergravity on AdS$_6$. This effective action is essentially a nonlinear sigma model with five scalar fields parametrizing $\textrm{SL}(3,\mathbb{R})/\textrm{SO}(2,1)$, modified by a scalar potential and coupled to Einstein gravity in Euclidean signature. We argue that the scalar potential can be explained by a subgroup CSO(1,1,1) $\subset\textrm{SL}(3,\mathbb{R})$ in a way analogous to gauged supergravity.
Highlights
C (2015) 75:484 conjectured enhancement of global symmetry to EN f +1 was verified from the analysis of the superconformal index in
Supergravity, using the pure spinor approach. They found that the four-dimensional internal space is a fibration of S2 over a two-dimensional space, and they showed that the supersymmetry conditions boil down to two coupled partial differential equations
One can study the integrability conditions of the Killing spinor equations and check whether the supersymmetry conditions satisfy the equations of motion and the Bianchi identities automatically
Summary
We consider the most general supersymmetric AdS6 solutions of type IIB supergravity. We take the D = 10 metric as a warped product of AdS6 with a four-dimensional Riemannian space M4. To respect the symmetry of AdS6, we should set the five-form flux to zero. The complex threeform flux G is non-vanishing only on M4. The warp factor U , the dilation φ and the axion C, are functions on M4 and independent of coordinates in AdS6. There are two differential and four algebraic-type equations: Dm ξ1±. With the assumption that there exists at least one nowherevanishing solution to the equations in the above, we can construct various spinor bilinears. The supersymmetric condition is translated into various algebraic and differential relations between the spinor bilinears. We have recorded them in Appendix C.1 and C.2
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