A 2025 construction of Ambrosino, Runkel and Watts produces one-parameter families of mutually commuting non-local conserved charges in a two-dimensional conformal field theory (CFT) perturbed by a relevant primary field: a topological line defect perturbed by a defect field commutes with the Hamiltonian of the perturbed theory whenever a 'bulk commutation condition', stated in the three-dimensional symmetry topological field theory (SymTFT) of the unperturbed CFT, is satisfied. For A-type Virasoro minimal models this reproduces the standard integrable perturbations with Kac labels (1,2), (1,3) and (1,5), and yields new commuting families for (1,7)-perturbations, including the non-unitary models M(3,10) and M(5,18), whose renormalization-group flows are candidate integrable trajectories without local conserved currents. The (1,7) construction is, however, formal: apart from the degenerate (1,4)=(1,7) case of M(2,11), its defect perturbations need short-distance regularisation, no scheme compatible with the bulk commutation condition is known, and the non-perturbative functional relations (T-systems, Y-systems) among the charges — the entry point to non-linear integral equations (NLIEs) and exact finite-size spectra — have been derived neither for the new (1,7) examples nor as a verification for (1,2) and (1,5). The problem asks for (i) a regularisation/renormalisation preserving the bulk commutation relation order by order, (ii) the T-/Y-system functional relations among the non-local charges, benchmarked on (1,2) and (1,5) and determined for the (1,7) families, and (iii) a decision on whether these relations pass through the integrable tool chain to NLIEs and exact spectral data for the (1,7)-flows, to be tested against the conjectural NLIEs of Ambrosino-Negro and Hamiltonian-truncation spectra. A positive resolution would indicate that commuting non-local charges furnish an integrability criterion independent of local conserved currents. No work resolving any of these components is known in the literature.

Background

Two-dimensional conformal field theories (CFTs) can be deformed by adding a relevant primary field (one of scaling dimension below two, so that its coupling grows at long distance) to the action; the loss of scale invariance defines a renormalization-group (RG) flow away from the conformal fixed point. A distinguished class of deformations leads to integrable quantum field theories, in which infinitely many conservation laws constrain the theory so strongly that spectra and scattering are exactly solvable. In two dimensions the conventional diagnostic of integrability is an infinite family of local conserved currents of higher spin (chiral currents whose Lorentz weight, or spin, exceeds two); the standard integrable perturbations of the Virasoro minimal models carry Kac labels (1,2)(1,2), (1,3)(1,3) and (1,5)(1,5) (and their p↔qp\leftrightarrow q partners).

The Virasoro minimal models M(p,q)M(p,q) — the rational CFTs whose only chiral symmetry is the Virasoro algebra, labelled by coprime integers p,qp,q — are non-unitary unless ∣p−q∣=1|p-q|=1. 'Kac label' (r,s)(r,s) denotes the position of a representation in the Kac table; primary bulk fields and, in the diagonal (A-type) theories considered here, also the elementary topological line defects carry these labels. A topological line defect is a line operator that can be freely deformed without changing correlators; such defects implement the (generally non-invertible) topological symmetries of a 2d QFT. The modern organisational framework for these symmetries is the SymTFT (symmetry topological field theory): the 2d CFT is represented as a 3d topological field theory on an interval, with a topological boundary condition carrying the symmetry data and a 'conformal' boundary condition carrying the conformal data.

Ambrosino-Runkel-Watts (2025) use this picture to ask which symmetries survive a relevant bulk perturbation μφ\mu\varphi. They state a graphical bulk commutation condition in the SymTFT: a bulk perturbation and a perturbation of a topological defect D\mathcal{D} by holomorphic and antiholomorphic defect fields, λψ+λ~ ψˉ\lambda\psi+\tilde\lambda\,\bar\psi, are compatible when the condition holds together with the coupling constraint λλ~=μ\lambda\tilde\lambda=\mu. If so, the perturbed defect D(λψ+μ/λ ψˉ)\mathcal{D}(\lambda\psi+\mu/\lambda\,\bar\psi) commutes with the Hamiltonian of the perturbed theory — it is 'rigidly translation invariant' — and constitutes a non-local conserved charge ('non-local' because it is built from fields that live only on the defect, not from local bulk currents). In the examples analysed, the charges for different λ\lambda mutually commute, giving a one-parameter commuting family, and the rigidly translation invariant defects form a monoidal category (a category equipped with a compatible tensor product, realised by defect fusion) that extends the category of topological defects.

For A-type minimal models the authors find solutions of the bulk commutation condition for the standard integrable perturbations (1,2)(1,2), (1,3)(1,3), (1,5)(1,5) and for new cases, most prominently (1,7)(1,7)-perturbations including M(3,10)M(3,10) and M(5,18)M(5,18). The (1,7)(1,7) cases matter because a recent body of work has proposed RG flows between non-unitary minimal models driven by (1,7)(1,7) (and more generally (1,2k+1)(1,2k+1)) fields: Ginzburg-Landau descriptions of M(3,8)M(3,8) and related models with emergent supersymmetry, an infinite family of flows protected by non-invertible symmetries, Hamiltonian-truncation evidence (from diagonalising the perturbed Hamiltonian on a truncated basis of conformal states) for the flow M(3,10)+ϕ1,7→M(3,8)M(3,10)+\phi_{1,7}\to M(3,8), and a proposed family of non-linear integral equations (NLIEs) encoding the exact finite-size ground-state energies of these flows. Whether these flows are integrable is unresolved; remarkably, the (1,7)(1,7)-perturbation of M(5,18)M(5,18) has commuting non-local charges but no local conserved currents (none found up to spin 16), so if it is integrable it is integrable without local charges.

Two gaps remain in the construction. First, all listed (1,7)(1,7) solutions except the degenerate (1,4)=(1,7)(1,4)=(1,7) perturbation of M(2,11)M(2,11) require short-distance regularisation of the defect-perturbation integrals, so those non-local charges are only formal. Second, the non-perturbative functional relations among the charges are unknown. In the integrable toolkit, commuting transfer matrices satisfy T-systems — bilinear difference equations for functions Ta(θ)T_a(\theta), analogous to the Hirota equation, with the index aa labelling fusion channels — and Y-systems, functional equations for ratio functions YaY_a built from the T-functions; these convert into non-linear integral equations (NLIEs), solvable with the same techniques as the thermodynamic Bethe ansatz (TBA, a method that replaces the Hilbert space by integral equations for pseudo-energy densities), and yield exact finite-size spectra. Such relations are known for chiral minimal models via quantum inverse scattering, for W3W_3/Boussinesq theory, and in general from solvable lattice models. On the defect side, Runkel (2008, 2010) derived the T-system for the Lee-Yang model (the smallest non-unitary minimal model, M(2,5)M(2,5)) and for the (1,3)(1,3)-perturbation, and Bücher-Runkel (2014) recast the algebra of perturbed defects as Yetter-Drinfeld modules (modules over a Hopf-algebra-like structure adapted to defect fusion), in which a T-system corresponds to exact sequences of tensor products of such modules. Extending this to (1,2)(1,2) and (1,5)(1,5) — a verification against known expectations — and to the new (1,7)(1,7) families is the open problem, together with the regularisation it depends on and the question of whether the resulting relations deliver exact spectra for the (1,7)(1,7)-flows.

Problem Statement

The problem, as posed in the outlook of Ambrosino-Runkel-Watts (2025), is to complete the construction of non-local conserved charges from perturbed defects non-perturbatively for the Virasoro minimal models, in three linked steps:

(1) Regularisation. Construct a short-distance regularisation/renormalisation scheme for the defect perturbations — required by all listed (1,7)(1,7)-solutions of the bulk commutation condition except the degenerate (1,4)=(1,7)(1,4)=(1,7) perturbation of M(2,11)M(2,11), and in particular for the (1,7)(1,7)-perturbations of M(3,10)M(3,10) and M(5,18)M(5,18) — that preserves the bulk commutation relation (the equality of the relevant commutators in the unperturbed CFT) order by order in the perturbation, so that the operators D(λψ+μ/λ ψˉ)\mathcal{D}(\lambda\psi+\mu/\lambda\,\bar\psi) are genuine conserved (rigidly translation invariant) operators in the fully regularised perturbed theory; or prove that no such scheme exists.

(2) Functional relations. Derive the non-perturbative functional relations among the commuting non-local charges: verify that the (1,2)(1,2)- and (1,5)(1,5)-perturbations recover the T-system functional relations expected from the chiral quantum-inverse-scattering results, extending the (1,3)(1,3) analysis of Runkel (2010); and determine the T-system — and the Y-system following from it — for the new (1,7)(1,7) families, notably those of M(3,10)M(3,10) and M(5,18)M(5,18).

(3) Spectral consequences. Decide whether these functional relations can be passed through the integrable tool chain — T-system to Y-system to non-linear integral equations — to obtain exact finite-size spectral data for the (1,7)(1,7)-flows, and confront the result with the NLIE family proposed by Ambrosino-Negro (2025) and with Hamiltonian-truncation / truncated-conformal-space spectra.

The overarching objective is a verdict on the non-perturbative status of the defect-based non-local charges for the (1,7)(1,7)-perturbations. A positive outcome of (1)-(3) would indicate that commuting non-local charges furnish an integrability criterion that does not rest on local conserved currents (which are absent for the (1,7)(1,7)-perturbation of M(5,18)M(5,18)); a negative outcome of step (1) or step (2) — no compatible regularisation, or no finite functional relation of T-/Y-system type — would block this particular route from the charges to exact spectral data and is by itself a resolution of the objective, without by itself excluding other routes, since mutually commuting families of conserved charges are not in themselves sufficient to prove integrability. The (1,2)(1,2) and (1,5)(1,5) verifications in step (2) are consistency benchmarks required for any claimed step-(2) derivation, and the (1,7)(1,7) determination is asked for the (1,7)(1,7) family as a whole, of which M(3,10)M(3,10) and M(5,18)M(5,18) are the named instances a complete answer must cover.

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