Derivation of the GKP-Witten relation by symmetry without Lagrangian - Sinya Aoki; Janos Balog; Kengo Shimada (Physics Letters B, 2025)
We derive the GKP-Witten relation in terms of correlation functions by symmetry without referring to a Lagrangian or the large N expansion. By constructing bulk operators from boundary operators in conformal field theory (CFT) by the conformal smearing, we first determine bulk-boundary 2-pt functions for an arbitrary spin using both conformal and bulk symmetries, then evaluate their small z behaviors, where z is the (d+1)-th coordinate in the bulk. Next, we explicitly determine small z behaviors of bulk-boundary-boundary 3-pt functions also by the symmetries, while small z behaviors of correlation functions among one bulk and n boundary operators with n≥3 are fixed by the operator product expansion (OPE). Combining all results, we construct the GKP-Witten relation in terms of these correlation functions at all orders in an external source J. We compare our non-Lagrangian approach with the standard approach employing the bulk action. Our results indicate that the GKP-Witten relation holds not only for holographic CFTs but also for generic CFTs as long as certain conditions are satisfied.