Modified crossing in exact S-matrices of 1+1d QFTs with non-invertible symmetries: dynamical verification and completion of the categorical bootstrap

Certain two-dimensional integrable quantum field theories whose vacua realize a spontaneously broken non-invertible (fusion-category) symmetry cannot have kink S-matrices satisfying unitarity, the Yang-Baxter equation and ordinary crossing symmetry simultaneously with the categorical Ward identities. Copetti, Córdova and Komatsu (PRL 133 (2024) 181601) proposed corrected amplitudes obeying modified crossing, $S^{ab}_{dc}(\theta)=\sqrt{d_a d_c/(d_b d_d)}\,S^{bc}_{ad}(i\pi-\theta)$ in terms of quantum dimensions, for the $\phi_{1,3}$ minimal-model flows, the $\phi_{2,1}$ tricritical Ising flow and the anomalous compact-boson deformation SG(β,2), and their follow-up (JHEP 03 (2025) 204) derived the general rule for arbitrary fusion categories and vacuum representations via the symmetry topological field theory, initiating a modified-crossing S-matrix bootstrap. The corrected amplitudes remain unverified by any dynamical, discriminating observable: they are degenerate with the previously conjectured restricted-solid-on-solid (RSOS) amplitudes in the thermodynamic Bethe ansatz, and the finite-volume spectra with non-invertible defect insertions that would distinguish them have not been computed. The exact S-matrices of the remaining anomalous family SG(β,k) for k>2 are unknown. The derivations of modified crossing rest on an LSZ-type reduction formula for non-local kink-creation operators that has not been proven. The categorical bootstrap program - Haagerup-category amplitudes, modified crossing for form factors, and bounds on ultraviolet data such as central charge - remains to be executed. The problem is to develop this rewrite of non-diagonal scattering theory along the directions that remain open: verifying the corrected amplitudes against discriminating dynamical observables, determining the amplitudes of the remaining anomalous family SG(β,k) for $k>2$, and carrying the categorical bootstrap program forward.

Background

In a massive two-dimensional quantum field theory, exact scattering is encoded in the two-particle S-matrix $S^{ab}_{dc}(\theta)$, where $a,b$ label incoming particle species, $c,d$ outgoing ones, and $\theta$ is the rapidity difference. The exact S-matrix bootstrap has conjectured exact amplitudes since the 1970s by imposing three functional equations: unitarity, crossing symmetry (relating a scattering process to the one with a particle crossed to an antiparticle, $\theta\mapsto i\pi-\theta$), and the Yang-Baxter equation (the consistency condition for factorized, particle-production-free scattering). A prominent application is the set of restricted-solid-on-solid (RSOS) kink S-matrices, where the particles are kinks $K_{ab}$ interpolating between degenerate vacua $a,b$ of a spontaneously broken discrete symmetry, as arise in the $\phi_{1,3}$-deformations of the unitary Virasoro minimal models $\mathcal{M}_n$ (Zamolodchikov 1991; Fendley, Saleur and Zamolodchikov 1993).

A non-invertible symmetry is a generalization of an ordinary global symmetry whose symmetry operators form a fusion category rather than a group: topological defect lines $\mathcal{L}$ can be fused with non-invertible fusion rules $\mathcal{L}\times\mathcal{L}'=\sum_{\mathcal{L}''}\mathcal{L}''$. Each simple line carries a quantum dimension $d_{\mathcal{L}}$, the eigenvalue by which a closed loop of the line rescales a state it encloses. The minimal models preserve such an $\mathcal{A}$-type symmetry under the $\phi_{1,3}$ flow, with the $n-1$ gapped vacua transforming as the regular representation and quantum dimensions $d_a=\sin((2a+1)\pi/n)/\sin(\pi/n)$. A 't Hooft anomaly is an obstruction to gauging a symmetry that constrains the infrared dynamics even when the symmetry is spontaneously broken.

Copetti, Córdova and Komatsu showed that, in the flows they analyze, the classic RSOS kink amplitudes fail the Ward identity of the preserved non-invertible symmetry, so that unitarity, the Yang-Baxter equation, standard crossing and the categorical Ward identity are mutually incompatible. Dropping only crossing, they proposed corrected amplitudes satisfying a modified crossing relation

$$S^{ab}_{dc}(\theta)=\sqrt{\frac{d_a d_c}{d_b d_d}}\,S^{bc}_{ad}(i\pi-\theta),$$

tracing the modification to the infrared topological quantum field theory (TQFT) describing the symmetry-breaking sector, which renormalizes in- and out-states. For the $\phi_{2,1}$ deformation of the tricritical Ising model (preserving the Fibonacci category, whose nontrivial object obeys $W\times W=1+W$) their result endorses the old proposals of Smirnov and of Colomo, Koubek and Mussardo, which had never been established precisely because they violate standard crossing. For the $J\bar J$-deformed $SU(2)_1$ WZW model - equivalently a compact boson deformed by $\cos(2X/R)$, called SG(β,2) in the Klassen-Melzer classification of $\cos(kX/R)$ deformations - the $\mathbb{Z}_2$ anomaly produces a sign-type crossing modification, and the general $\mathbb{Z}_k$-anomalous family SG(β,k) is expected to behave analogously. Relatedly, Cordova, Garcia-Sepulveda and Holfester showed that the broken non-invertible symmetry enforces exact degeneracies of particle-soliton spectra, giving independent physical content to the symmetry action on massive spectra. The authors' follow-up derived the general rule for arbitrary fusion categories and spontaneous-breaking patterns (module categories, i.e. the data specifying how the symmetry category acts on the set of vacua), $S^{ab}_{dc}(s)=\sqrt{g_a g_c/(g_b g_d)}\,S^{bc}_{ad}(4m^2-s)$ with $g_a$ the relative Euler term (the disk partition function of the symmetry TQFT on the vacuum boundary condition $a$, which reduces to the quantum dimension $d_a$ for vacua in the regular representation), using the symmetry topological field theory (SymTFT, an auxiliary topological theory in one dimension higher encoding the categorical symmetry data), and implemented a modified-crossing S-matrix bootstrap in which the corrected amplitudes appear at cusps of the allowed region.

Two caveats frame the open problem. First, the derivation extracts S-matrix elements from correlation functions of kink-creation operators, which are non-local because they are attached to topological lines; the LSZ-type reduction formula (a Cauchy-slice, Fourier-transform relation between correlation functions and scattering amplitudes) has not been derived for such non-local operators and is assumed without proof in both papers. Second, the corrected and literature amplitudes differ by a factor $(d_a d_c/d_b d_d)^{i\theta/2\pi}$ that drops out of the thermodynamic Bethe ansatz (TBA, the standard method deriving finite-size ground-state energies from the exact S-matrix), so the two proposals are degenerate for all TBA observables computed so far; the natural discriminating observable is the finite-volume spectrum with an inserted non-invertible topological defect (the defect Hilbert space), which is sensitive to the state normalization.

Problem Statement

The overarching objective is to develop the modified-crossing framework for the exact scattering theory of massive 1+1-dimensional quantum field theories with spontaneously broken non-invertible (fusion-category) symmetries and 't Hooft anomalies along the directions that remain open in the sources: verification of the corrected amplitudes against dynamics, coverage of the remaining symmetry and anomaly classes, and extension of the categorical bootstrap. The concrete open tasks are:

1. Dynamical verification. Compute observables that distinguish the corrected amplitudes from the previously conjectured crossing-symmetric RSOS amplitudes - the prime candidate being the finite-volume spectrum with an inserted non-invertible topological defect (the defect Hilbert space) - for the models in which the corrected amplitudes were proposed, namely the $\phi_{1,3}$ minimal-model flows, the $\phi_{2,1}$ tricritical Ising flow and SG(β,2), given that the two amplitude proposals are exactly degenerate in the thermodynamic Bethe ansatz.

2. Completion of the example set. Determine the exact kink S-matrices of the remaining anomalous family SG(β,k) for $k>2$ (compact bosons deformed by $\cos(kX/R)$ with anomalous $\mathbb{Z}_k$ symmetry and $k$ vacua), satisfying unitarity, the Yang-Baxter equation and the modified crossing appropriate to the $\mathbb{Z}_k$ anomaly. Because the precise relationship between 't Hooft anomalies and the modified crossing is itself listed as open in [1], establishing the $\mathbb{Z}_k$ crossing rule - whether from the general SymTFT formula of [2] or by another route - is part of this task rather than a given input.

3. Completion of the categorical bootstrap. Extend the S-matrix bootstrap with non-invertible Ward identities beyond the $\mathcal A$-type and Fibonacci categories already analyzed, the source-named candidate being the Haagerup fusion category (an exceptional fusion category constructed neither from a finite group nor a quantum group, whose would-be realizing quantum field theories are themselves not yet understood), including the construction of integrable Yang-Baxter-compatible amplitudes; derive the modified crossing rules for form factors (matrix elements of local operators between multi-particle states) and perform the corresponding form-factor bootstrap; and explore bounds on ultraviolet data such as the central charge, which [1] raises as a valuable application of the form-factor bootstrap.

The three directions above are independent: a complete solution of any one of them resolves the task, and the others remain open.

References

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