Abstract
We develop the embedding formalism for odd dimensional Dirac spinors in AdS and apply it to the (geodesic) Witten diagrams including fermionic degrees of freedom. We first show that the geodesic Witten diagram (GWD) with fermion exchange is equivalent to the conformal partial waves associated with the spin one-half primary field. Then, we explicitly demonstrate the GWD decomposition of the Witten diagram including the fermion exchange with the aid of the split representation. The geodesic representation of CPW indeed gives the useful basis for computing the Witten diagrams.
Highlights
We develop the embedding formalism for odd dimensional Dirac spinors in AdS and apply it to the Witten diagrams including fermionic degrees of freedom
We first show that the geodesic Witten diagram (GWD) with fermion exchange is equivalent to the conformal partial waves associated with the spin one-half primary field
It is worth noting that the embedding formalism and the split representation are useful to compute usual Witten diagrams with or without loop effects
Summary
We develop the embedding formalism for odd dimensional AdS spinors. By using the embedding formalism, we introduce auxiliary fields, the covariant derivative, the AdS propagators and the quadratic Casimir equation for the spinor fields. We will restrict ourself to Dirac fermions in the odd dimensional AdS space (namely, d is even). The constraint was first briefly discussed in [17], whereas we will introduce a different condition for the AdS fermions. Notice that we can not take the conventional “transverse condition”, XAΓASb = 0 for the AdS spinors. Such a naive condition gives rise to Sb = 0 due to the non-vanishing determinant of XAΓA. S is an auxiliary field in the boundary CFT This S∂ is conventional one in the embedding formalism for CFT fermions [36,37,38]
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