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Quantisation of the effective string with TBA

Michele Caselle, Davide Fioravanti, Ferdinando Gliozzi, Roberto Tateo

TL;DR

This work shows that the infrared dynamics of a confining string between static sources can be described by a two‑dimensional CFT of central charge $D-2$ perturbed by the integrable operator $T\bar{T}$. Using the thermodynamic Bethe Ansatz with a massless flow S‑matrix, the authors derive the exact NG‑like spectrum for closed strings and extend the construction to ADE perturbed CFTs, including a general treatment of boundary (open string) effects via a boundary TBA. The leading boundary corrections reproduce the known NG open‑string deviations, while the bulk $T\bar{T}$ perturbation accounts for the subleading universal spectrum in the infrared. Although the framework captures nonperturbative spectral data and connects to Hagedorn growth and tachyonic points, it identifies an ultraviolet incompleteness manifested by an infinite negative‑energy sector, suggesting the need for further irrelevant operators at shorter distances. Overall, the paper provides a powerful, nonperturbative route to NG‑type spectra in confining gauge theories and a bridge to broader perturbed CFTs via TBA.

Abstract

In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction - a quartic polynomial in the field derivatives - is exactly the composite field $T\bar{T}$, built with the chiral components, $T$ and $\bar{T}$, of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA),the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an infinite strip, identify the relevant boundary parameter and verify that it modifies the string spectrum as expected.

Quantisation of the effective string with TBA

TL;DR

This work shows that the infrared dynamics of a confining string between static sources can be described by a two‑dimensional CFT of central charge perturbed by the integrable operator . Using the thermodynamic Bethe Ansatz with a massless flow S‑matrix, the authors derive the exact NG‑like spectrum for closed strings and extend the construction to ADE perturbed CFTs, including a general treatment of boundary (open string) effects via a boundary TBA. The leading boundary corrections reproduce the known NG open‑string deviations, while the bulk perturbation accounts for the subleading universal spectrum in the infrared. Although the framework captures nonperturbative spectral data and connects to Hagedorn growth and tachyonic points, it identifies an ultraviolet incompleteness manifested by an infinite negative‑energy sector, suggesting the need for further irrelevant operators at shorter distances. Overall, the paper provides a powerful, nonperturbative route to NG‑type spectra in confining gauge theories and a bridge to broader perturbed CFTs via TBA.

Abstract

In presence of a static pair of sources, the spectrum of low-lying states of whatever confining gauge theory in D space-time dimensions is described, at large source separations, by an effective string theory. In the far infrared the latter flows, in the static gauge, to a two-dimensional massless free-field theory. It is known that the Lorentz invariance of the gauge theory fixes uniquely the first few subleading corrections of this free-field limit. We point out that the first allowed correction - a quartic polynomial in the field derivatives - is exactly the composite field , built with the chiral components, and , of the energy-momentum tensor. This irrelevant perturbation is quantum integrable and yields, through the thermodynamic Bethe Ansatz (TBA),the energy levels of the string which exactly coincide with the Nambu-Goto spectrum. We obtain this way the results recently found by Dubovsky, Flauger and Gorbenko. This procedure easily generalizes to any two-dimensional CFT. It is known that the leading deviation of the Nambu-Goto spectrum comes from the boundary terms of the string action. We solve the TBA equations on an infinite strip, identify the relevant boundary parameter and verify that it modifies the string spectrum as expected.

Paper Structure

This paper contains 10 sections, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The composite perturbation $T\bar{T}$ and its expectation value
  3. The exact $S$ matrix for massless flows
  4. Spectrum degeneracy
  5. The infinite strip
  6. The boundary TBA
  7. Basic boundary conditions
  8. More general boundary conditions
  9. Conclusions
  10. The ADE general case

Figures (2)

  • Figure 1: Possible excited state integration contours for the $\lambda_+$ sector.
  • Figure 2: ${\cal R}$ matrix constraint.