$T \bar{T}$-deformed 2D Quantum Field Theories (1608.05534v2)

Abstract: It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator $T \bar{T}$, built with the components of the energy-momentum tensor, enjoys very special properties in 2D quantum field theories, and can be regarded as a peculiar kind of integrable perturbation. Novel interesting features of this operator have recently emerged from the study of effective string theory models.In this paper we study further properties of this distinguished perturbation. We discuss how it affects the energy levels and one-point functions of a general 2D QFT in finite volume through a surprising relation with a simple hydrodynamic equation. In the case of the perturbation of CFTs, adapting a result by L\"uscher and Weisz we give a compact expression for the partition function on a finite-length cylinder and make a connection with the exact $g$-function method. We argue that, at the classical level, the deformation naturally maps the action of $N$ massless free bosons into the Nambu-Goto action in static gauge, in $N+2$ target space dimensions, and we briefly discuss a possible interpretation of this result in the context of effective string models.

• The paper demonstrates that the T operator induces an integrable perturbation by modifying energy levels through a hydrodynamic-like equation.
• It applies the Burgers equation to model spectral deformations, linking finite-size effects in 2D QFTs with classical hydrodynamics.
• The research connects T-deformations with effective string theories by mapping massless boson actions to the Nambu-Goto framework and exploring spectral transitions.

Analysis of $T$-deformed 2D Quantum Field Theories

This paper undertakes an in-depth exploration of $T$-deformed two-dimensional Quantum Field Theories (QFTs), a rich subject that touches on the fundamental aspects of integrable models and perturbation theory. These investigations apply to various contexts including the paper of effective string theories. The work connects the Y-variable-based Thermodynamic Bethe Ansatz (TBA) method and Non-Linear Integral Equations (NLIE), offering a distinct perspective on the deformation characteristics of these models.

Key Contributions

1. $T$ Operator in 2D QFTs: The paper highlights the role of the $T$ operator, linked to the energy-momentum tensor's components, which gives rise to an exceptional kind of integrable perturbation within two-dimensional QFTs. This operator introduces a deformation of the energy levels and one-point functions in finite volume scenarios through a hydrodynamic equation analog.
2. Relation with Burgers Equation: The authors focus on a Burgers equation of hydrodynamics to describe the observed spectral deformations. The connection to hydrodynamic equations such as the inviscid Burgers type equation is novel in this framework, depicting energy level changes as a flow over time.
3. String Theories and Deformations: The paper explores the $T$-deformation mapped to classical theories, producing intriguing results when applied to the action of massless free bosons, leading toward the Nambu-Goto action. The significance of this lies in the effective string model interpretations, which find new grounding explanations via QFT insights.
4. Exact Spectrum and Spectral Transition: The paper elaborates on deriving the exact finite-size spectrum for massless bosons, modified by a particular scattering phase, illustrating compliance with the Nambu-Goto model energy levels known from string theory. The work gracefully addresses the so-called Hagedorn transition and its correlations to tachyon singularities.
5. Generality of Deformation: The research extends beyond integrability settings, suggesting that integrable frameworks provide insights into a larger class of non-integrable theories, thus hinting at universal features of $T$-deformations.

Implications

• Theoretical Significance: The paper provides a theoretical platform suggesting that certain deformation types transform standard quantum actions into those mimicking string theory models. This has implications for understanding specific perturbative series within fields like condensed matter systems and low-dimensional tecniques.
• Practical Applications: Integral methodologies like TBA or NLIE enhance understanding of energy levels and operator actions which could potentially cascade into effective numerical methods or analytical techniques within quantum simulation fields.

Future Directions

• Non-integrable Models: Speculative extensions might consider non-integrable models with a dual phase factor to comprehend deeper the connections between integrability and criticality within more complex systems.
• Extended Symmetries: A contemplated extension involves embedding these theories into larger symmetry frameworks, such as studying connections with $W$-algebras, possibly enriching the understanding of deformations leading back to familiar symmetry structures often appearing in physically realistic models.

In summary, the work on $T$-deformed 2D QFTs opens a nuanced passage between integrable perturbative frameworks and effective string models. It sets a comprehensive groundwork for string-inspired QFT development and fosters a bridge to communicating string theory ideas within more condensed quantum manifestations.