GKP-Witten Formula

Definition

The Gubser-Klebanov-Polyakov-Witten (GKP-W) formula is the operational statement of the holographic dictionary: $Z_{\text{CFT}}[J]=Z_{\text{gravity}}[{\varphi }_{\partial }=J]$, equating the CFT generating functional with sources $J$ to the on-shell gravity partition function with boundary conditions ${\varphi }_{\partial }=J$. The bulk field $\varphi$ near the boundary has the leading expansion $\varphi (z,x)\sim z^{d-\Delta }J(x)+z^{\Delta }⟨\mathcal{O}(x)⟩+\cdots$ for an operator $\mathcal{O}$ of dimension $\Delta$.

Role in Research

The GKP-W formula is the foundational machinery of Lecture 15 of the holography course, providing the basis for all subsequent boundary-correlator computations in holography. It operationalizes the holographic-dictionary by giving a precise prescription for extracting CFT data from bulk solutions.

Relations

• holographic-dictionary — the GKP-W formula is the central entry in the dictionary.
• ryu-takayanagi-formula — the entropy entry in the dictionary, complementing the correlator prescription of GKP-W.

Papers

• Gubser–Klebanov–Polyakov 1998, arXiv:hep-th/9802109.
• Witten 1998, arXiv:hep-th/9802150.

Notes

Course coverage: Lecture 15, with the large-N counting of Lecture 14.