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arXiv · 2411.16269

Derivation of the GKP-Witten relation by symmetry without Lagrangian

Sinya Aoki·Janos Balog·Kengo Shimada
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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.

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Sinya Aoki, Janos Balog, Kengo Shimada. 2024-11-25. Derivation of the GKP-Witten relation by symmetry without Lagrangian. https://arxiv.org/abs/2411.16269

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