On space of integrable quantum field theories

TL;DR

The authors formulate a geometric view of the space of 2D integrable quantum field theories, showing that the tangent space at any IQFT is spanned by infinitely many local, integrable deformations X_s tied to the theory's local integrals of motion, with X_1 identified as the $T\bar{T}$ operator. They demonstrate that adding X_s preserves integrability, and in massive theories these deformations correspond to CDD-factor deformations of the S-matrix, while the form-factor bootstrap provides a bridge between local fields and S-matrix data. The sine-Gordon model serves as a concrete testbed where X_s acquire explicit fermionic representations and their form factors can be computed, confirming the general construction and revealing additional principal deformations. The work also discusses the implications for UV completeness and the broader landscape of IQFT deformations, including potential non-Lorentz-invariant extensions and lattice connections. Overall, the paper establishes a rich structure for classifying and understanding integrable perturbations in 2D QFTs and highlights the central role of current densities in shaping the space of solvable deformations.

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.