Derivation of the GKP-Witten relation by symmetry without Lagrangian

TL;DR

This paper derives the GKP-Witten relation using symmetry principles and correlation functions without relying on a Lagrangian or large N expansion, broadening its applicability to general CFTs.

Contribution

The authors present a novel, symmetry-based derivation of the GKP-Witten relation that does not depend on a Lagrangian formulation or large N limit, applicable to a wide class of CFTs.

Findings

GKP-Witten relation derived from correlation functions and symmetry.

Applicable to generic CFTs beyond holographic models.

Valid at all orders in external sources J.

Abstract

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 $n\ge 3$ 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.