Modified boundary crossing for non-invertible symmetries beyond regular module categories: breather spectra, non-regular vacua, and boundary RG flows in 1+1 dimensions

When a non-invertible symmetry (a fusion category of topological defect lines rather than a group) of a 1+1-dimensional quantum field theory is spontaneously broken, the associated topological quantum field theory renormalizes the kink operators and modifies both the bulk and the boundary crossing relations of the exact S-matrix. Shimamori and Yamaguchi derived the modified boundary crossing relation for weakly-symmetric boundaries and constructed the resulting boundary S-matrix for the $\Phi_{(1,3)}$-deformed unitary minimal models, under three working assumptions: the degenerate vacua lie in the regular module category of the fusion category, the spectrum contains only kinks, and the boundary condition is fixed. This problem asks for the general framework beyond these assumptions: derive the modified boundary crossing relation when vacua belong to non-regular module categories (the bulk analogue is known from symmetry-TFT analysis); construct boundary S-matrices for spectra containing breathers together with kinks, as arise from other relevant deformations such as the $\Phi_{(2,1)}$-deformed tricritical Ising model, for general weakly-symmetric boundaries; and determine how boundary renormalization-group flows interact with the TQFT normalization and the modified crossing relation. The first breather-containing instance (one weakly-symmetric boundary of the $\Phi_{(2,1)}$-deformed tricritical Ising model) was solved in 2026 up to an unidentified free parameter, but the non-regular module category derivation and the boundary RG flow analysis remain untouched.

Background

In an integrable relativistic QFT in 1+1 dimensions, two-body scattering is elastic and encoded in a rapidity-dependent S-matrix constrained by unitarity, the Yang-Baxter equation (the consistency condition guaranteeing that multi-particle scattering factorizes into two-body amplitudes independently of the scattering order), and crossing symmetry (which relates a particle's scattering amplitude to that of the corresponding antiparticle at the complementary rapidity $i\pi-\theta$). A non-invertible symmetry is a collection of topological defect lines whose fusion obeys the rules of a fusion category rather than a group, so a line's action cannot be undone by an inverse line. If such a symmetry is preserved along the renormalization-group (RG) flow of a relevant deformation but spontaneously broken in the infrared, the massive theory has degenerate vacua and kink excitations $K_{ab}$ interpolating between neighboring vacua; the kinks are twisted-sector states of the topological lines. Copetti-Cordova-Komatsu showed that the topological QFT (TQFT) associated with the broken symmetry fixes a non-standard normalization of kink operators, so the conventional kink crossing relation $S^{ab}_{dc}(\theta)=S^{bc}_{ad}(i\pi-\theta)$ must be modified by quantum-dimension factors, $S^{ab}_{dc}(\theta)=\sqrt{d_a d_c/(d_b d_d)}\,S^{bc}_{ad}(i\pi-\theta)$, where $d_a$ is the quantum dimension of the topological line $\mathcal{L}_a$ attached to the vacuum label $a$ (the positive Perron-Frobenius eigenvalue of the fusion matrix of $\mathcal{L}_a$, i.e. the factor by which fusing with $\mathcal{L}_a$ rescales state spaces).

On a half-line, an integrable boundary is described by the boundary (reflection) S-matrix $R^{a}_{bc}(\theta)$ introduced by Ghoshal and Zamolodchikov, which reflects a kink $K_{bc}$ in the sector with boundary-adjacent vacuum $a$; it obeys boundary unitarity, the boundary Yang-Baxter equation (BYBE, the boundary analogue of the bulk factorization condition, constraining successive reflections of two excitations off the boundary), and a boundary crossing relation. A boundary is weakly symmetric under the non-invertible symmetry when every topological line can end on it, i.e. the junction space $\mathrm{Hom}(\mathcal{L}_a\otimes\mathcal{B},\mathcal{B})$ is non-trivial. Shimamori and Yamaguchi proved that for such boundaries the boundary crossing relation is modified by the same TQFT effects:

$R^{b}_{ca}\left(\frac{i\pi}{2}-\theta\right)=\sum_{d}\sqrt{\frac{d_d}{d_b}}\,S^{ba}_{cd}(2\theta)\,R^{d}_{ca}\left(\frac{i\pi}{2}+\theta\right).$

A module category over the fusion category $\mathcal{C}$ is a category on which $\mathcal{C}$ acts by fusion; physically it classifies the possible patterns of spontaneous breaking of $\mathcal{C}$, and the vacua of a given gapped phase are objects of one module category. In the regular module category ($\mathcal{C}$ acting on itself) every vacuum is obtained by fusing a line into a reference vacuum, $|a\rangle=\mathcal{L}_a|\tilde 0\rangle$; non-regular module categories give inequivalent patterns with different vacuum content. As the concrete example, Shimamori and Yamaguchi solved the $\Phi_{(1,3)}$-deformation of the unitary minimal models $\mathcal{M}_p$ ($p\ge 4$): the preserved categorical symmetry $\mathcal{A}_p$ (the $su(2)$-type fusion category with simple objects $1,\dots,p-1$ and adjacency $|a-b|=1$) is spontaneously broken to $p-1$ vacua in the regular module category, the spectrum contains only kinks, and the Ward-Takahashi identities (the constraints that a symmetry's currents impose on correlation functions) of the non-invertible symmetry forbid non-diagonal boundary scattering for $p\ge 5$ and fix the minimal boundary S-matrix. Their analysis explicitly assumes (i) vacua in the regular module category, (ii) kink-only spectra from $\Phi_{(1,3)}$ deformations, and (iii) fixed boundary conditions. A breather - a kink-antikink bound state created by a local operator - does appear in other integrable relevant deformations, notably the $\Phi_{(2,1)}$-deformation of the tricritical Ising model (the unitary minimal model immediately above the Ising model in the unitary minimal series), which preserves the Fibonacci categorical symmetry (the fusion category with a single non-trivial line $\varphi$ fusing as $\varphi\times\varphi=1+\varphi$) and whose bulk kink-breather S-matrix is known and consistent with the modified bulk crossing. The bulk modified crossing for arbitrary fusion categories and module categories has been derived from the symmetry TFT (the topological field theory in one higher dimension that encodes the symmetry), but the boundary analogue for non-regular module categories has not.

Problem Statement

Replace each of the three working assumptions of the Shimamori-Yamaguchi analysis by its general counterpart and determine the resulting boundary crossing relations and boundary S-matrices for 1+1-dimensional integrable QFTs with spontaneously broken non-invertible symmetries and weakly-symmetric boundaries. The three components share one objective - the general modified-boundary-crossing framework, of which the solved $\Phi_{(1,3)}$/regular-module/fixed-boundary case is the first instance.

(1) Non-regular module categories. Derive the modification of the boundary crossing relation when the degenerate vacua are objects of a non-regular module category $\mathcal{M}$ over the fusion category $\mathcal{C}$ of the preserved non-invertible symmetry, expressing the result in terms of the module-category data (module quantum dimensions, module $F$-symbols (the associativity data of the module action), junction normalizations between the topological lines and the boundary). The relation must reduce to

$R^{b}_{ca}\left(\frac{i\pi}{2}-\theta\right)=\sum_{d}\sqrt{\frac{d_d}{d_b}}\,S^{ba}_{cd}(2\theta)\,R^{d}_{ca}\left(\frac{i\pi}{2}+\theta\right)$

when $\mathcal{M}$ is the regular module category, and be consistent with the bulk modified crossing derived from the symmetry TFT for general module categories.

(2) Spectra with breathers and other relevant deformations. Construct the boundary S-matrices for gapped theories obtained from relevant deformations whose spectra contain both kinks and breathers - paradigmatically the $\Phi_{(2,1)}$-deformed tricritical Ising model with its Fibonacci categorical symmetry, beyond the one boundary condition already solved - for general weakly-symmetric boundary conditions: determine the Ward-Takahashi constraints on kink-breather and breather-breather reflection amplitudes, and solve the boundary Yang-Baxter equations, boundary unitarity, and the modified boundary crossing.

(3) Boundary RG flows. Determine how boundary renormalization-group flows in which the boundary condition changes non-trivially interact with the TQFT normalization and the modified boundary crossing relation: how the boundary S-matrix evolves along integrable boundary perturbations.

The source also proposes two further downstream applications - incorporating the modified crossing into numerical boundary S-matrix bootstrap bounds, and applying these boundary results to the s-wave (angular-momentum-zero partial-wave) reduction of 4D monopole-fermion scattering - which are separate problems with distinct objectives and are not part of this problem.

References

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