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Instantons and Conformal Holography

Sebastian de Haro, Anastasios C. Petkou

TL;DR

This paper builds and analyzes a tractable AdS4/CFT3 subsector in which a conformally coupled bulk φ^4 theory is holographically related to a 3D boundary theory with a φ^6 interaction. Through radial quantization and careful handling of instanton backgrounds, the authors establish an exact bulk–boundary correspondence for the scalar sector, identifying the boundary operator O with dimension 1 as the renormalized composite ⟨φ⟩ ≡ φ^2 and showing Φ_hol = -O in the free case, extended to the interacting case with mode-mixing due to broken dilatations. They further show that bulk one-particle states map to boundary two-particle states in a diagonal sector, effectively yielding a tensor-product structure between bulk and boundary Hilbert spaces. The work suggests a controlled realization of conformal holography, with implications for higher-spin holography, instanton physics, and potential cosmological interpretations via instanton decay and de Sitter-like fluctuations.

Abstract

We study a subsector of the AdS_4/CFT_3 correspondence where a class of solutions in the bulk and on the boundary can be explicitly compared. The bulk gravitational theory contains a conformally coupled scalar field with a Phi^4 potential, and is holographically related to a massless scalar with a Phi^6 interaction in three dimensions. We consider the scalar sector of the bulk theory and match bulk and boundary classical solutions of the equations of motion. Of particular interest is the matching of the bulk and the boundary instanton solutions which underlies the relationship between bulk and boundary vacua with broken conformal invariance. Using a form of radial quantization we show that quantum states in the bulk correspond to multiply-occupied single particle quantum states in the boundary theory. This allows us to explicitly identify the boundary composite operator which is dual to the bulk scalar, at the free theory level as well as in the instanton vacuum. We conclude with a discussion of possible implications of our results.

Instantons and Conformal Holography

TL;DR

This paper builds and analyzes a tractable AdS4/CFT3 subsector in which a conformally coupled bulk φ^4 theory is holographically related to a 3D boundary theory with a φ^6 interaction. Through radial quantization and careful handling of instanton backgrounds, the authors establish an exact bulk–boundary correspondence for the scalar sector, identifying the boundary operator O with dimension 1 as the renormalized composite ⟨φ⟩ ≡ φ^2 and showing Φ_hol = -O in the free case, extended to the interacting case with mode-mixing due to broken dilatations. They further show that bulk one-particle states map to boundary two-particle states in a diagonal sector, effectively yielding a tensor-product structure between bulk and boundary Hilbert spaces. The work suggests a controlled realization of conformal holography, with implications for higher-spin holography, instanton physics, and potential cosmological interpretations via instanton decay and de Sitter-like fluctuations.

Abstract

We study a subsector of the AdS_4/CFT_3 correspondence where a class of solutions in the bulk and on the boundary can be explicitly compared. The bulk gravitational theory contains a conformally coupled scalar field with a Phi^4 potential, and is holographically related to a massless scalar with a Phi^6 interaction in three dimensions. We consider the scalar sector of the bulk theory and match bulk and boundary classical solutions of the equations of motion. Of particular interest is the matching of the bulk and the boundary instanton solutions which underlies the relationship between bulk and boundary vacua with broken conformal invariance. Using a form of radial quantization we show that quantum states in the bulk correspond to multiply-occupied single particle quantum states in the boundary theory. This allows us to explicitly identify the boundary composite operator which is dual to the bulk scalar, at the free theory level as well as in the instanton vacuum. We conclude with a discussion of possible implications of our results.

Paper Structure

This paper contains 23 sections, 154 equations.

Table of Contents

  1. Introduction and Motivation
  2. Classical Holography
  3. The Model
  4. Instantons and the Vacuum Structure of the Boundary CFTs
  5. The Proposal for the Dual boundary CFT
  6. Conformal Holography: Free Case
  7. Geometric Set-up for Radial Quantization
  8. Classical Correspondence
  9. Radial Quantization in the Bulk
  10. Mapping of the Hilbert Spaces
  11. Instantons and Conformal Holography
  12. Fluctuations Around the Bulk Instanton
  13. Fluctuations Around the Boundary Instanton
  14. The Bulk-Boundary Correspondence
  15. Quantization
  16. ...and 8 more sections