Locality and Conserved Charges in $T\overline{T}$-Deformed Conformal Field Theories: A Perturbative Analysis

Core Concepts

This paper investigates the properties of $T\overline{T}$-deformed conformal field theories (CFTs) within perturbation theory, focusing on the locality of the Hamiltonian operator and conserved charges, and highlighting ambiguities in their definition.

Abstract

• Bibliographic Information: Ruben Monten, Richard M. Myers, Konstantinos Roumpedakis. (2024). Locality and Conserved Charges in $T\overline{T}$-Deformed CFTs. [Journal to be determined]. [Preprint available at arXiv:2411.06261v1 [hep-th]].

• Research Objective: This paper aims to analyze the locality properties of $T\overline{T}$-deformed CFTs using a perturbative approach, focusing on the construction of a local Hamiltonian operator and the behavior of conserved charges under the deformation.

• Methodology: The authors employ a perturbative expansion in the deformation parameter (λ) to study the $T\overline{T}$ deformation of 2D CFTs. They utilize the operator formalism and introduce a smearing regulator to handle UV divergences and ordering ambiguities in the composite $T\overline{T}$ operator. The deformed Hamiltonian is constructed by requiring it to be related to a "fake" Hamiltonian (with the correct spectrum but non-local) through a unitary transformation. The locality of the Hamiltonian and conserved charges is then analyzed order by order in λ.

• Key Findings:

  • The authors successfully construct a local Hamiltonian operator for the $T\overline{T}$-deformed CFT up to third order in perturbation theory, which reproduces the expected energy spectrum and is consistent with the theory's quasi-locality.
  • The Hamiltonian is not uniquely defined, exhibiting a one-parameter family of local solutions at each order, suggesting a non-universality of the deformed Hamiltonian.
  • New terms proportional to the central charge (c) appear in the Hamiltonian, which are crucial for reproducing the correct spectrum and cannot be removed by regularization or field redefinitions.
  • The authors demonstrate that the KdV charges, while not a priori local in the deformed theory, can be made local to first order in perturbation theory.

• Main Conclusions:

  • The study provides a systematic framework for analyzing the locality properties of $T\overline{T}$-deformed CFTs within perturbation theory.
  • The existence of a family of local Hamiltonians suggests an inherent ambiguity in the definition of the $T\overline{T}$ deformation at the quantum level.
  • The new terms proportional to the central charge in the Hamiltonian highlight deviations from the classical Nambu-Goto theory and raise interesting questions about the holographic interpretation of $T\overline{T}$-deformed CFTs with non-zero central charge.

• Significance: This research contributes significantly to the understanding of $T\overline{T}$ deformations in 2D CFTs, particularly concerning the locality properties of the deformed theory. The findings have implications for the study of integrable models, effective string theory, and holography.

• Limitations and Future Research: The analysis is performed perturbatively up to third order in the deformation parameter. It would be interesting to explore the behavior of the Hamiltonian and conserved charges at higher orders and investigate the possibility of non-perturbative effects. Further research could also focus on understanding the implications of the central charge terms in the Hamiltonian for the holographic duals of $T\overline{T}$-deformed CFTs.

Stats

Quotes

Key Insights Distilled From

Locality and Conserved Charges in $T\overline{T}$-Deformed CFTs

by Ruben Monten... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2411.06261.pdf

Deeper Inquiries

How does the non-locality of the unitary transformation used to relate the "fake" and true Hamiltonians manifest itself physically in the $T\overline{T}$-deformed CFT?

The non-locality of the unitary transformation $e^{-\lambda X_\lambda}$ is a direct consequence of the quasi-locality of $T\overline{T}$-deformed CFTs. Let's break down why this is the case:

Fake Hamiltonian: The "fake" Hamiltonian $\tilde{H}\lambda$ is a non-local object. It is constructed directly from the seed CFT Hamiltonian and momentum, which are integrals of local densities. However, $\tilde{H}\lambda$ itself is not the integral of a local density due to its non-linear dependence on these integrated quantities.
True Hamiltonian: The true Hamiltonian $H_\lambda$, on the other hand, is expected to be quasi-local. This means that while it can be written as the integral of a density, this density may contain an increasing number of derivatives of the fundamental fields at higher orders in the deformation parameter $\lambda$.
Unitary Transformation as a Flow: The unitary transformation $e^{-\lambda X_\lambda}$ can be interpreted as generating a flow in the space of states. It maps the energy eigenstates of the seed CFT to the energy eigenstates of the deformed theory. This flow is inherently non-local because it connects a theory with a local Hamiltonian (the seed CFT) to a theory with a quasi-local Hamiltonian (the $T\overline{T}$-deformed CFT).
Physical Manifestations:

Spreading of Operators: The non-locality of the transformation implies that local operators in the seed CFT, when evolved under the $T\overline{T}$ deformation, become non-local operators with support over a finite region. This spreading of operators is a hallmark of the non-local nature of the deformed theory.

Modified Notion of Distance: The quasi-locality of the deformed theory suggests a modification of the usual notion of distance at scales comparable to or smaller than the deformation parameter $\lambda$. This is because the increasing number of derivatives in the Hamiltonian density implies interactions over increasingly larger distances.

Obstacles to a Local UV Completion: The non-locality of the unitary transformation also poses challenges for finding a local UV completion of $T\overline{T}$-deformed CFTs. The non-local nature of the flow suggests that the UV degrees of freedom, if they exist, are likely to be non-locally related to the IR degrees of freedom.

Could the ambiguity in the definition of the deformed Hamiltonian be resolved by imposing additional physical constraints or by considering specific regularization schemes?

The ambiguity in the definition of the deformed Hamiltonian is a subtle issue and it is not yet clear if it can be completely resolved. Here's a balanced perspective:
Arguments for Possible Resolution:

Physical Constraints: Imposing additional physical constraints, beyond the spectrum and conservation of the stress tensor, might reduce the ambiguity. For example:

Factorization of the S-matrix: Demanding that the deformed theory has a factorized S-matrix could impose constraints on the allowed Hamiltonian.
Modular Invariance: For CFTs on the torus, requiring modular invariance of the deformed partition function might restrict the allowed deformations.
Holographic Considerations: If a holographic dual of the deformed CFT is known, demanding consistency with the holographic dictionary could provide further constraints.

Special Regularization Schemes: It's conceivable that certain regularization schemes, perhaps motivated by physical principles, could lead to a preferred definition of the deformed Hamiltonian. For instance, schemes that preserve certain symmetries or have a natural interpretation in a holographic dual might be preferred.
Arguments Against Resolution:

Intrinsic Ambiguity: The ambiguity might be an intrinsic feature of $T\overline{T}$-deformed CFTs, reflecting the non-locality of the theory and the freedom in defining composite operators.

Scheme Dependence: Different regularization schemes, even if physically motivated, might lead to different deformed Hamiltonians. This scheme dependence could be a genuine feature of the deformed theory, suggesting that there is no unique "correct" definition of the Hamiltonian.
Current Status:
The paper you provided finds that the ambiguity persists even after imposing locality and conservation of the stress tensor. This suggests that resolving the ambiguity might require additional physical input or a deeper understanding of the non-local structure of $T\overline{T}$-deformed CFTs.

What are the implications of the findings in this paper for the study of quantum gravity in three dimensions, given the proposed connections between $T\overline{T}$-deformed CFTs and gravity theories?

The findings in this paper have intriguing implications for the study of quantum gravity in three dimensions, particularly in the context of holography:

Non-Locality and Quantum Gravity: The inherent non-locality of $T\overline{T}$-deformed CFTs provides further evidence that quantum gravity is likely to be a non-local theory. This is consistent with other approaches to quantum gravity, such as string theory and loop quantum gravity, which also suggest a departure from locality at the Planck scale.

Beyond AdS/CFT: $T\overline{T}$-deformed CFTs provide a framework for studying holography beyond the familiar AdS/CFT correspondence. The fact that these theories are solvable, despite being non-local, makes them valuable tools for exploring quantum gravity in non-AdS backgrounds.

Understanding the Holographic Dictionary: The ambiguity in the definition of the deformed Hamiltonian raises questions about the precise nature of the holographic dictionary for $T\overline{T}$-deformed CFTs. It suggests that the dictionary might be more involved than in AdS/CFT and could depend on the specific regularization scheme used to define the deformation.

Emergent Geometry: The modification of the notion of distance in $T\overline{T}$-deformed CFTs hints at the possibility of emergent geometry in quantum gravity. The deformation parameter $\lambda$ could be interpreted as a scale at which the classical notion of geometry breaks down, potentially leading to a more fundamental, non-geometric description.
Specific Implications:

Flat Space Holography: The paper's findings are particularly relevant for attempts to formulate a holographic duality for flat spacetime. The $T\overline{T}$ deformation has been proposed as a way to generate asymptotically flat spacetimes from AdS spacetimes. Understanding the non-locality and ambiguity in the deformed theory is crucial for making progress in this direction.

de Sitter Holography: $T\overline{T}$-deformed CFTs have also been studied in the context of de Sitter holography. The deformation has been proposed as a way to regulate the UV divergences that arise in attempts to formulate a holographic dual of de Sitter space. The paper's results could provide insights into the nature of these UV divergences and the challenges in constructing a consistent de Sitter holography.
Overall, the findings in this paper highlight the importance of non-locality and the subtleties of defining composite operators in the context of $T\overline{T}$-deformed CFTs. These insights are likely to be valuable for further exploring the connections between these theories and quantum gravity in three dimensions.