Holographic Construction of Excited CFT States

TL;DR

The paper develops a systematic holographic construction of bulk solutions dual to excited CFT states, using the real-time Schwinger-Keldysh dictionary and perturbation theory in bulk fields. At leading order, the universal part is captured by a single bulk scalar with $m^2=\Delta(\Delta-2)$ in AdS$_3$, with the state $|\Delta\rangle = \mathcal{O}_\Delta |0\rangle$; higher orders require detailed CFT data from the OPE and backreaction. The authors implement the construction in global AdS$_3$ and in Poincaré AdS$_3$, solving the linearised equations on four connected manifolds (two Euclidean caps and two Lorentzian pieces) and matching them to reproduce exact CFT 1-point functions for $2d$ CFTs on $\mathbb{R}\times S^1$ and $\mathbb{R}^{1,1}$. They demonstrate a precise bulk/boundary correspondence at the level of the 1-point function, establish the universal nature of the leading solution, and outline how this framework can be extended to include backreaction and higher-dimensional cases, connecting to smearing-function pictures and fuzzball constructions.

Abstract

We present a systematic construction of bulk solutions that are dual to CFT excited states. The bulk solution is constructed perturbatively in bulk fields. The linearised solution is universal and depends only on the conformal dimension of the primary operator that is associated with the state via the operator-state correspondence, while higher order terms depend on detailed properties of the operator, such as its OPE with itself and generally involve many bulk fields. We illustrate the discussion with the holographic construction of the universal part of the solution for states of two dimensional CFTs, either on $R \times S^1$ or on $R^{1,1}$. We compute the 1-point function both in the CFT and in the bulk, finding exact agreement. We comment on the relation with other reconstruction approaches.

Figures (4)

• Figure 1: In--in time contour (left) and corresponding AdS manifold (right). The manifolds labeled by $L$ are empty Lorentzian AdS and those labeled by $E$ are empty, Euclidean AdS.
• Figure 2: In--in complex time contour with operator insertions at $t = 0 \pm i \epsilon$.
• Figure 3: In--in time contour (left) and corresponding AdS manifold (right).
• Figure 4: Tracing individual modes through the four segments of the manifold.