Boundary terms in the AdS/CFT correspondence for spinor fields

TL;DR

This paper shows that the AdS/CFT boundary term for spinor fields is dictated by the variational principle: requiring stationarity of the action for Dirac fields in AdS with prescribed boundary data fixes the necessary surface term and its coefficient. By decomposing spinor solutions into boundary-anchored components and carefully handling the first-order nature of the Dirac equation, the authors derive an improved action $S=S_D+C_ty$ whose on-shell variation vanishes for the admissible histories. The resulting boundary action yields the correct boundary two-point function, matching the CFT predictions, and clarifies the role of boundary data $ psi_0$ and $ chi_0$ in holographic correlators. An alternate Hamiltonian viewpoint is noted, reinforcing the stationarity requirement as the core justification for the boundary term.

Abstract

The requirement that the action be stationary for solutions of the Dirac equations in anti-de Sitter space with a definite asymptotic behaviour is shown to fix the boundary term (with its coefficient) that must be added to the standard Dirac action in the AdS/CFT correspondence for spinor fields.