On space of integrable quantum field theories

Abstract: We study deformations of 2D Integrable Quantum Field Theories (IQFT) which preserve integrability (the existence of infinitely many local integrals of motion). The IQFT are understood as "effective field theories", with finite ultraviolet cutoff. We show that for any such IQFT there are infinitely many integrable deformations generated by scalar local fields $X_s$, which are in one-to-one correspondence with the local integrals of motion; moreover, the scalars $X_s$ are built from the components of the associated conserved currents in a universal way. The first of these scalars, $X_1$, coincides with the composite field $(T{\bar T})$ built from the components of the energy-momentum tensor. The deformations of quantum field theories generated by $X_1$ are "solvable" in a certain sense, even if the original theory is not integrable. In a massive IQFT the deformations $X_s$ are identified with the deformations of the corresponding factorizable S-matrix via the CDD factor. The situation is illustrated by explicit construction of the form factors of the operators $X_s$ in sine-Gordon theory. We also make some remarks on the problem of UV completeness of such integrable deformations.

• The paper demonstrates that deformations generated by scalars Xₛ preserve integrability in 2D IQFTs through local integrals of motion.
• It establishes a connection between integrable deformations and S-matrix modifications via CDD factors, with the sine-Gordon model as a concrete example.
• The study highlights the role of special form factors and the form-factor bootstrap in constructing integrable S-matrices, enhancing non-perturbative QFT insights.

Overview of "On Space of Integrable Quantum Field Theories"

The paper "On Space of Integrable Quantum Field Theories" by F.A. Smirnov and A.B. Zamolodchikov explores the study of deformations in two-dimensional Integrable Quantum Field Theories (IQFTs). This research presents a comprehensive analysis of the tangent space to the space of IQFTs and makes significant strides in understanding how integrability is preserved under such deformations.

Summary of Findings

The authors explore the structure of the space of IQFTs, denoted as $\Sigma_{\text{Int}}$, within the broader set of two-dimensional Quantum Field Theories (QFTs). For any given IQFT, the paper identifies an infinite set of scalars $X_s$, which generate integrable deformations. These scalars are uniquely associated with the local integrals of motion of the IQFT and are constructed from the components of the conserved currents specific to the theory.

One of the most significant findings is that, in massive IQFTs, the deformations related to the scalars $X_s$ correlate with the modification of the factorizable S-matrix via the CDD factor. The sine-Gordon model serves as an explicit case study to illustrate these properties. Furthermore, the paper rigorously examines the solvability of deformations generated by a specific scalar $X_1$, equivalent to the (TT) operator formed from the energy-momentum tensor components.

Practical and Theoretical Implications

1. Preservation of Integrability: The paper highlights that every field $X_s$ in the tangent space generates an integrable deformation of the IQFT. This finding suggests that the space of integrable deformations is more extensive than previously recognized, potentially impacting the way researchers approach the classification and construction of IQFTs.
2. S-Matrix Deformations: The alignment of the scalars $X_s$ with the deformations of the S-matrix through CDD factors suggests a structured method to explore the parametric space of S-matrix solutions. This insight could facilitate deeper investigations into the symmetries and analytical properties of scattering matrices in massive IQFTs.
3. Form-Factor Bootstrap: The research underscores the role of 'special form factors' which vanish off the energy-momentum shell, except for two-particle states. This property asserts the existence of a linkage between form-factor bootstrap equations and the construction of integrable S-matrices, furthering our understanding of non-perturbative QFT.

Future Directions

As this paper primarily addresses deformations within the context of two-dimensional theories, future research could explore analogous questions in higher dimensions, albeit with the understanding that higher-dimensional integrability invariably presents unique challenges. Additionally, the linkages between integrable deformations and S-matrix theory invite further inquiry into how these deformations might manifest in quantum gravity or through holographic correspondences. Furthermore, the implications of ultraviolet (UV) completeness, particularly concerning finite UV cutoff theories, could be examined in more detail to assess the overall robustness of these integrable structures.

Conclusion

This work by Smirnov and Zamolodchikov represents a significant step in understanding the integrable deformations of two-dimensional IQFTs. By providing a clear connection between the local integrals of motion and the structure of tangent spaces, the paper opens avenues for creating a deeper theoretical framework linking deformations, S-matrices, and fundamental field theoretic properties, all while inviting contemplation of the inherent challenges in UV completeness. Such investigations promise to enrich the fundamental understanding of QFT and its applications to both mathematical physics and potentially to real-world systems modeled by integrable structures.