A Bispinor Formalism for Spinning Witten Diagrams

Abstract: We develop a new embedding-space formalism for AdS$_4$ and CFT$_3$ that is useful for evaluating Witten diagrams for operators with spin. The basic variables are Killing spinors for the bulk AdS$_4$ and conformal Killing spinors for the boundary CFT$_3$. The more conventional embedding space coordinates $X^I$ for the bulk and $P^I$ for the boundary are bilinears in these new variables. We write a simple compact form for the general bulk-boundary propagator, and, for boundary operators of spin $\ell \geq 1$, we determine its conservation properties at the unitarity bound. In our CFT$_3$ formalism, we identify an $\mathfrak{so}(5,5)$ Lie algebra of differential operators that includes the basic weight-shifting operators. These operators, together with a set of differential operators in AdS$_4$, can be used to relate Witten diagrams with spinning external legs to Witten diagrams with only scalar external legs. We provide several applications that include Compton scattering and the evaluation of an $R^4$ contact interaction in AdS$_4$. Finally, we derive bispinor formulas for the bulk-to-bulk propagators of massive spinor and vector gauge fields and evaluate a diagram with spinor exchange.