Open classification of UV completions of integrable $T\bar{T}$-deformed CFTs: admissible CDD factors, remaining spectra, and UV field content

The $T\bar{T}$ deformation (the irrelevant deformation of a 2D quantum field theory generated by the composite operator built from the energy-momentum tensor components) preserves integrability, and the deformation of a massive or massless factorized S-matrix by CDD factors — multiplicative scalar factors that can be inserted without violating unitarity or crossing — can convert an infrared (IR) scattering theory into an ultraviolet (UV)-complete quantum field theory flowing to a new conformal fixed point. Ahn and LeClair (2022) proposed classifying all such UV completions but could only do so under concrete restrictions (diagonal scalar S-matrices, finite products of specific left-right CDD factors, rational UV central charge), left exhaustiveness open for high-multiplicity spectra such as the $E_8$ Ising magnetic spectrum, and left the UV field content unidentified in most enumerated cases. Later work by the same collaboration has resolved the $E_8$ Ising case (exactly four completions), the $A_n$ coset family, the RSOS minimal models, and one non-unitary family. The open problem is to complete the classification — covering the remaining simply-laced spectra ($D_n$, $E_6$, $E_7$) and non-diagonal scattering theories satisfying the Yang–Baxter equation — to decide whether the imposed restrictions are necessary or can be relaxed, to identify the UV conformal field theories of the completions known only through thermodynamic Bethe ansatz data, and to develop the suggested generalized Goldstone theorem for fractional supersymmetry.

Background

Two-dimensional integrable quantum field theories (QFTs) are special in that their exact S-matrices can be determined by a bootstrap program: one fixes a factorized, elastic two-particle S-matrix using unitarity, crossing symmetry and analyticity, together with constraints from the spectrum of stable particles. A long-standing ambiguity in this program is the CDD ambiguity (named after work of Castillejo, Dalitz and Dyson in old S-matrix theory): any factorized S-matrix can be multiplied by additional scalar factors without spoiling unitarity or crossing, so the S-matrix alone does not determine the theory. The $T\bar{T}$ deformation — deforming the action by the composite operator $\int T\bar{T}$ built from the holomorphic and antiholomorphic components of the energy-momentum tensor, where $\bar{T}$ denotes the antiholomorphic component — is a distinguished irrelevant, solvable, integrability-preserving deformation of a 2D QFT, and its deformation of the S-matrix is exactly a CDD-type factor. Smirnov and Zamolodchikov showed that a massive integrable QFT remains exactly solvable under this deformation — its finite-size energy levels satisfy the inviscid Burgers equation — but the deformed theory generically develops a short-distance singularity in the ultraviolet rather than a new fixed point; resolving this singularity raised the systematic question: for which CDD factors does the deformed S-matrix correspond to a genuine local QFT with a well-behaved ultraviolet limit, and what conformal field theory (CFT) does that UV limit flow to?

For massless scattering theories the question is richer. Massless integrable QFTs describe renormalization-group (RG) flows between CFTs: the particles split into left-moving (L) and right-moving (R) species, the S-matrix has LL, RR (massless, often trivial) and LR blocks, and the IR fixed point of the flow is encoded in the massless S-matrix itself. LeClair (2021) initiated the program by analyzing the $T\bar{T}$ deformation of the Ising model, finding two UV completions with central charges $c=7/10$ (the tricritical Ising model) and $c=3/2$, both supersymmetric (the central charge $c$ is the number characterizing a 2D CFT; the Ising model has $c=1/2$). Ahn and LeClair (2022) then posed the general classification problem: starting from a given IR massless S-matrix (they treated the free-fermion/Ising spectrum and the high-multiplicity spectra appearing in coset constructions, notably the $su(3)$ and $su(4)$ spectra and the $E_8$ spectrum of the scaling Ising model in a magnetic field, whose eight stable particles interact through the exceptional $E_8$ symmetry), determine which deformations by products of the left-right CDD factors $T_\beta$ and paired $F_\gamma$ (their specific building blocks for the LR block, defined in Section III of arXiv:2205.10905, with real parameters $\beta$, $\gamma>0$) produce UV-complete theories; for the $E_8$ spectrum they found three such completions, with $c_{\mathrm{UV}}=21/22$, $15/2$ and $31/2$, and argued that more cases are likely.

The tool connecting the S-matrix to the UV fixed point is the thermodynamic Bethe ansatz (TBA): the exact finite-size ground-state energy is expressed through a system of integral equations whose UV limit yields plateaux (constant solutions) whose value gives the effective central charge $c_{\mathrm{UV}}$ of the UV fixed point. Ahn and LeClair classified candidate completions by solving plateau equations for the CDD-deformed kernels $\widehat{K}$, obtaining tables of candidate UV central charges. The classification was carried out under three restrictions: (i) diagonal scalar S-matrices; (ii) admissible CDD factors restricted to finite products of $T_\beta$ and paired $F_\gamma$ with $\gamma>0$; (iii) rational $c_{\mathrm{UV}}$. Their conclusions section explicitly listed what remains open: whether these restrictions can be relaxed or should instead be strengthened (additional physical constraints might exclude some listed candidates); exhaustiveness for high-multiplicity spectra such as $E_8$ (they exhibited several completions without claiming completeness); the field-theoretic identification of the UV operator content — the central charge from TBA does not identify the UV CFT, its field content, or the relevant operator with UV conformal dimension $\Gamma_{\mathrm{UV}}$ driving the flow; the extension to non-diagonal scattering theories (matrix-valued S-matrices constrained by the Yang–Baxter equation, the consistency condition guaranteeing factorization of many-body scattering); and a suggested generalized Goldstone theorem for fractional supersymmetry — a proposed extension of ordinary supersymmetry whose bosonic subgroup has fractional spin, invoked because some flows produce massless degrees of freedom resembling Goldstone modes of a broken fractional supersymmetry.

Subsequent work has substantially advanced the program: the massless S-matrix bootstrap (imposing the bootstrap functional equations directly on the LR block, beyond the plateau equations) resolved the $A_n$ coset family and the Ising $E_8$ case; massless RSOS scattering (relevant to the minimal models $M_p$, the unitary CFTs $M(p,p+1)$) and a non-unitary family were classified; and a formula for the vacuum energy density of these UV completions was derived. What remains is the completion of the classification for the remaining spectra, the non-diagonal case, the field-theoretic identification of the UV endpoints, and the conceptual symmetry questions.

Problem Statement

Complete the classification of ultraviolet completions of integrable $T\bar{T}$-deformed conformal field theories posed by Ahn and LeClair (2022). Starting from an integrable massless (or massive) factorized S-matrix describing an infrared fixed point or RG flow, determine all deformations by CDD factors that yield a UV-complete local quantum field theory, together with the UV conformal field theory that each such deformation flows to. Concretely, the still-open parts of this classification problem are: (1) for the remaining simply-laced coset spectra — $D_n$, $E_6$, $E_7$ — and other high-multiplicity particle spectra, enumerate all solutions of the massless S-matrix bootstrap (all admissible LR blocks $S^{RL}$ and their CDD-factor decompositions) that generate RG flows to UV-complete theories, and decide exhaustiveness, extending what was done for the free-fermion, $A_n$, RSOS $M_p$, Ising $E_8$, and one non-unitary family; (2) extend the classification to non-diagonal scattering theories, where the S-matrix is matrix-valued and must satisfy the Yang–Baxter equation together with unitarity and crossing — no example is currently worked out; (3) decide whether the restrictions under which the existing classification was obtained — diagonal scalar S-matrices, admissible CDD factors built as finite products of $T_\beta$ and paired $F_\gamma$ with $\gamma>0$, and rational $c_{\mathrm{UV}}$ — are necessary, can be relaxed (including, but not limited to, more general CDD factors, other parameter regions of $\beta$ and $\gamma$, irrational UV central charges, non-unitary theories), or should be strengthened by additional physical constraints that exclude some presently listed candidates; (4) identify the UV field content of the completions so far known only through thermodynamic Bethe ansatz data — the cases left as '$?$' in the 2022 classification, including the exceptional $su(4)$ candidates with $c_{\mathrm{UV}}=9/7$, $7/5$, $13/7$, $11/5$, $12/5$, $13/5$ and the $su(3)$ candidate with $c_{\mathrm{UV}}=8/5$ — namely the specific conformal field theory, its operator content, and the relevant operator of UV conformal dimension $\Gamma_{\mathrm{UV}}$ driving the flow; and (5) develop the suggested generalized Goldstone theorem for fractional supersymmetry, clarifying when the massless degrees of freedom of these flows are its Goldstone modes. These five parts are the open fronts of the classification: each may be pursued independently, a definite result on any one of them is a self-contained partial advance, and the classification is complete only when all five are resolved.

References

1. Changrim Ahn and André LeClair, 'On the classification of UV completions of integrable $T\bar{T}$ deformations of CFT', JHEP 08 (2022) 179. DOI 10.1007/JHEP08(2022)179, arXiv:2205.10905, https://arxiv.org/abs/2205.10905
2. André LeClair, '$T\bar{T}$ deformation of the Ising model and its ultraviolet completion', J. Stat. Mech. (2021) 113104. DOI 10.1088/1742-5468/ac2a99, arXiv:2107.02230, https://arxiv.org/abs/2107.02230
3. Changrim Ahn, 'Massless S-matrix bootstraps and RG flows', JHEP 12 (2025) 103. DOI 10.1007/JHEP12(2025)103, arXiv:2509.20740, https://arxiv.org/abs/2509.20740
4. Changrim Ahn and Minkyoo Kim, 'UV completion of 2D Ising CFT: a golden $E_8$ massless S-matrix', arXiv:2606.28783 (preprint, June 2026), https://arxiv.org/abs/2606.28783
5. Changrim Ahn and Zoltan Bajnok, 'New integrable RG flows with parafermions', JHEP 11 (2024) 078. DOI 10.1007/JHEP11(2024)078, arXiv:2407.06582, https://arxiv.org/abs/2407.06582
6. Changrim Ahn, Zoltan Bajnok and Soma Elek, 'New RG flows between non-Unitary CFTs from exact massless scattering theories', arXiv:2608.24385 (preprint, August 2026), https://arxiv.org/abs/2608.24385
7. André LeClair, 'Mingling of the infrared and ultraviolet and the "cosmological constant" for interacting QFT in 2d', JHEP 05 (2023) 222. DOI 10.1007/JHEP05(2023)222, arXiv:2301.09019, https://arxiv.org/abs/2301.09019
8. F. A. Smirnov and A. B. Zamolodchikov, 'On space of integrable quantum field theories', Nucl. Phys. B 915 (2017) 363–383. DOI 10.1016/j.nuclphysb.2016.12.014, arXiv:1608.05499, https://arxiv.org/abs/1608.05499