Fermions in Geodesic Witten Diagrams

TL;DR

The paper extends the AdS/CFT operator dictionary to fermionic sectors by formulating an embedding formalism for odd-dimensional AdS spinors and applying it to geodesic Witten diagrams. It proves that fermion-exchange GWDs are equivalent to conformal partial waves for spin-1/2 primaries and provides an explicit split-representation-based decomposition of fermionic Witten diagrams into GWDs, revealing the single-trace and double-trace CPWs that appear in the CPW expansion. The work introduces a consistent spinor embedding (including a nontrivial constraint on the auxiliary spinor) and derives the bulk-boundary and bulk-bulk propagators, along with their split representation, to enable systematic CPW decompositions. These results furnish a practical computational basis for bulk calculations with fermions, laying groundwork for supersymmetric extensions and higher-spin generalizations.

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.