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Driven inhomogeneous CFT as a theory in curved space-time

Johanna Erdmenger, Jani Kastikainen, Tim Schuhmann

TL;DR

This work establishes a Lorentzian, operator-based framework for 2D CFTs driven by evolving curved space-time, showing that unitary evolution is realized as a Virasoro quantum circuit in a chirally split renormalization scheme. It clarifies how the Weyl anomaly is redistributed between diffeomorphism and Weyl sectors, yielding a clean separation of left- and right-moving stress-tensor sectors and enabling a state interpretation of curved-space observables. The authors compute the stress tensor, correlation functions, and entanglement entropy in both diffeomorphism-invariant and chirally split schemes, proving that a state-compatible entanglement entropy arises only in the chirally split scheme. They further derive a holographic dual in AdS$_3$ for arbitrary driving, showing exact matches with field-theory results when the appropriate scheme is used, and identify a boundary term needed to reproduce chirally split results holographically. Overall, the paper provides a comprehensive operational dictionary linking driven inhomogeneous CFTs, curved-space Ward identities, and holography, with broad potential for extending to boundaries, additional background fields, and quantum-information measures.

Abstract

For two-dimensional conformal field theories driven by evolving background space-time metrics in a closed universe, we present an operator formulation as a driven inhomogeneous CFT. The Hamiltonian of this theory is given by a background space-time dependent smearing of the stress tensor over the spatial slice. Emphasis is placed on the treatment of the curved-space Weyl anomaly, which we show is realized by the difference between Schrödinger and Heisenberg picture Hamiltonians once an appropriate renormalization scheme, the chirally split scheme, is chosen. As a result, the unitary evolution generated by the background metric coincides with that of a Virasoro quantum circuit. To showcase our formalism, we consider the stress tensor one-point function and the entanglement entropy of an interval in both operator and curved-space formulations. We find that these curved-space observables admit a state interpretation only in the chirally split scheme. Finally, we derive the holographic dual of the driven CFT in three-dimensional gravity, extending previous works to arbitrary driving. The holographic dictionary reproduces the stress tensor one-point function and the entanglement entropy in a diffeomorphism invariant scheme.

Driven inhomogeneous CFT as a theory in curved space-time

TL;DR

This work establishes a Lorentzian, operator-based framework for 2D CFTs driven by evolving curved space-time, showing that unitary evolution is realized as a Virasoro quantum circuit in a chirally split renormalization scheme. It clarifies how the Weyl anomaly is redistributed between diffeomorphism and Weyl sectors, yielding a clean separation of left- and right-moving stress-tensor sectors and enabling a state interpretation of curved-space observables. The authors compute the stress tensor, correlation functions, and entanglement entropy in both diffeomorphism-invariant and chirally split schemes, proving that a state-compatible entanglement entropy arises only in the chirally split scheme. They further derive a holographic dual in AdS for arbitrary driving, showing exact matches with field-theory results when the appropriate scheme is used, and identify a boundary term needed to reproduce chirally split results holographically. Overall, the paper provides a comprehensive operational dictionary linking driven inhomogeneous CFTs, curved-space Ward identities, and holography, with broad potential for extending to boundaries, additional background fields, and quantum-information measures.

Abstract

For two-dimensional conformal field theories driven by evolving background space-time metrics in a closed universe, we present an operator formulation as a driven inhomogeneous CFT. The Hamiltonian of this theory is given by a background space-time dependent smearing of the stress tensor over the spatial slice. Emphasis is placed on the treatment of the curved-space Weyl anomaly, which we show is realized by the difference between Schrödinger and Heisenberg picture Hamiltonians once an appropriate renormalization scheme, the chirally split scheme, is chosen. As a result, the unitary evolution generated by the background metric coincides with that of a Virasoro quantum circuit. To showcase our formalism, we consider the stress tensor one-point function and the entanglement entropy of an interval in both operator and curved-space formulations. We find that these curved-space observables admit a state interpretation only in the chirally split scheme. Finally, we derive the holographic dual of the driven CFT in three-dimensional gravity, extending previous works to arbitrary driving. The holographic dictionary reproduces the stress tensor one-point function and the entanglement entropy in a diffeomorphism invariant scheme.

Paper Structure

This paper contains 67 sections, 360 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Stress tensor in chirally split renormalization scheme.
  4. Equivalence with inhomogeneous driven CFT.
  5. Evolution of correlation functions.
  6. Evolution of entanglement entropy.
  7. Holographic dual.
  8. The renormalized stress tensor in evolving curved space-time
  9. Parametrization of the background metric
  10. Static space-time.
  11. Renormalized Ward identities in the diffeomorphism invariant scheme
  12. Chirally split renormalization scheme
  13. The connection to the Virasoro algebra.
  14. Example: free massless scalar field.
  15. CFTs driven by background metrics: operators and observables
  16. ...and 52 more sections

Figures (2)

  • Figure 1: Outline of the article. Section \ref{['sec:CFTCurvedSpacetime']} introduces CFT$_{1+1}$ on evolving curved space-time. In Section \ref{['sec:CCrevisited']}, we then provide the equivalent perspective of unitary evolution with a deformed Virasoro Hamiltonian and the holographic perspective is given in Section \ref{['sec:holography']}.
  • Figure 2: The renormalized $\text{Diff}\ltimes \text{Weyl}$ Ward identities in two different renormalization schemes: diffeomorphism invariant and chirally split scheme. The derivation of this form of the Ward identities can be found in Appendix \ref{['app:actions_anomalies']}.