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State/Operator Correspondence in Higher-Spin dS/CFT

Gim Seng Ng, Andrew Strominger

TL;DR

This work provides an explicit state/operator dictionary for the $dS_4/CFT_3$ higher-spin holography by mapping boundary CFT$_3$ states on $S^2$ to bulk states on southern $R^3_S$ slices ending on that $S^2$ at ${\rm I}^+$. It identifies Neumann/Dirichlet vacua with free and critical $Sp(N)$ models and constructs bulk wavefunctions associated to boundary insertions, relating them to (anti)quasinormal modes. It shows the bulk Klein–Gordon inner product reproduces the boundary Zamolodchikov two-point function and introduces a southern Hilbert space with a modified adjoint that yields the $C$-norm of the $Sp(N)$ CFT$_3$, thereby linking pseudo-unitarity to the holographic dictionary. It also shows the Euclidean vacuum corresponds to a mixed boundary state on the southern slice and proposes a nonperturbative dS exclusion principle limiting occupation per mode to $N/2$ quanta. Overall, the work strengthens the $dS/CFT$ dictionary in the higher-spin regime and highlights subtle notions of unitarity, holographic norms, and bulk/boundary state correspondences.

Abstract

A recently conjectured microscopic realization of the dS$_4$/CFT$_3$ correspondence relating Vasiliev's higher-spin gravity on dS$_4$ to a Euclidean $Sp(N)$ CFT$_3$ is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary. In particular it is argued that states of the boundary CFT$_3$ on $S^2$ are holographically dual to bulk states on geodesically complete, spacelike $R^3$ slices which terminate on an $S^2$ at future infinity. The dictionary is described in detail for the case of free scalar excitations. The ground states of the free or critical $Sp(N)$ model are dual to dS-invariant plane-wave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT$_3$. CFT$_3$ states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk. A norm is defined on the $R^3$ bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the $Sp(N)$ CFT$_3$.

State/Operator Correspondence in Higher-Spin dS/CFT

TL;DR

This work provides an explicit state/operator dictionary for the higher-spin holography by mapping boundary CFT states on to bulk states on southern slices ending on that at . It identifies Neumann/Dirichlet vacua with free and critical models and constructs bulk wavefunctions associated to boundary insertions, relating them to (anti)quasinormal modes. It shows the bulk Klein–Gordon inner product reproduces the boundary Zamolodchikov two-point function and introduces a southern Hilbert space with a modified adjoint that yields the -norm of the CFT, thereby linking pseudo-unitarity to the holographic dictionary. It also shows the Euclidean vacuum corresponds to a mixed boundary state on the southern slice and proposes a nonperturbative dS exclusion principle limiting occupation per mode to quanta. Overall, the work strengthens the dictionary in the higher-spin regime and highlights subtle notions of unitarity, holographic norms, and bulk/boundary state correspondences.

Abstract

A recently conjectured microscopic realization of the dS/CFT correspondence relating Vasiliev's higher-spin gravity on dS to a Euclidean CFT is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary. In particular it is argued that states of the boundary CFT on are holographically dual to bulk states on geodesically complete, spacelike slices which terminate on an at future infinity. The dictionary is described in detail for the case of free scalar excitations. The ground states of the free or critical model are dual to dS-invariant plane-wave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT. CFT states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk. A norm is defined on the bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the CFT.

Paper Structure

This paper contains 16 sections, 105 equations.

Table of Contents

  1. Introduction
  2. Global dS vacua at $m^2\ell^2=2$
  3. Modes
  4. Neumann and Dirichlet modes
  5. Euclidean modes
  6. Vacua
  7. Boundary operators and quasinormal modes
  8. Highest and lowest weight wavefunctions
  9. General multi-operator insertions
  10. Klein-Gordon inner product
  11. The southern Hilbert space
  12. States
  13. Norm
  14. Boundary dual of the bulk Euclidean vacuum
  15. Pseudounitarity and the C-norm in the $Sp(N)$ CFT$_3$
  16. ...and 1 more sections