Regularisation of non-regular translation-invariant defect perturbations compatible with the bulk commutation condition in two-dimensional perturbed CFTs

In two-dimensional quantum field theory, symmetries are encoded by topological line defects, and a conformal field theory perturbed by a relevant bulk field retains, as symmetries, those topological defects that commute with the perturbing field. Ambrosino, Runkel and Watts recently showed that topological defects embed into a larger class of translation invariant defects, which commute with the Hamiltonian of the perturbed theory without being topological, and which are constructed perturbatively from solutions $(D,\eta,\tilde\eta)$ of a commutation condition in a chiral topological field theory. The construction is well-defined only when a short-distance regularity bound $2h_{\phi_X}-h_{\min}<1$ holds. Several algebraic solutions violate this bound, notably the Ising $\sigma$-defect under the thermal bulk perturbation (where $2h_{\phi_X}-h_{\min}=1$ exactly) and the sporadic $(1,7)$-solutions in the Virasoro minimal models $M(3,10)$ and $M(5,18)$. The open problem is to construct an explicit regularisation of the divergent iterated defect-field integrals for these non-regular solutions that is compatible with the bulk commutation condition, i.e. that preserves translation invariance of the perturbed defect operator, and to determine whether such regularisations exist.

Background

In a two-dimensional (2d) quantum field theory, symmetries are organised by topological line defects: line operators that can be freely deformed without changing correlators and whose fusion defines a monoidal (tensor) category containing the global symmetries as the invertible objects. When a 2d conformal field theory (CFT) is perturbed by a relevant bulk field, a renormalisation-group (RG) flow is initiated, and one criterion for which symmetries survive (Chang, Lin, Shao, Wang, Yin 2019) is that a topological defect is preserved if and only if it commutes with the perturbing field. In the Ising CFT, for example, only the identity defect and the $\mathbb{Z}_2$-symmetry $\epsilon$-defect commute with the thermal perturbation; the Kramers-Wannier duality defect $\sigma$ (the topological line implementing Ising self-duality) does not.

Ambrosino, Runkel and Watts (2025) observed that the surviving structure is in fact larger. A translation invariant defect commutes with the full Hamiltonian of the perturbed theory — so moving its position costs no energy — without being topological; such defects still admit non-singular fusion and form a monoidal category containing the topological defects as a full subcategory. Their perturbative construction proceeds in the chiral topological field theory (chiral TFT), a three-dimensional topological description canonically attached to a 2d CFT (related by folding to the symmetry TFT used for categorical symmetries). The bulk field $\varphi$ is represented by chiral data $(X,Y,\omega)$: representations $X$ and $Y$ of the holomorphic and anti-holomorphic chiral algebra, vectors $\phi_X\in X$, $\phi_Y\in Y$, and a topological junction $\omega$. A topological defect $D$ is then perturbed by a holomorphic defect field $\psi$ of weight $h_{\phi_X}$ and an anti-holomorphic defect field $\bar\psi$, encoded by topological junctions $\eta$ and $\tilde\eta$ joining $X$ (respectively $Y$) to $D$. The triple $(D,\eta,\tilde\eta)$ must satisfy the commutation condition, a string-diagram identity in the chiral TFT generalising the commutation condition for integrable perturbations introduced by Runkel (2010) and Bücher-Runkel (2014) via Yetter-Drinfeld modules (a Hopf-algebraic structure packaging the compatibility between the actions of an algebra and its dual). If, in addition, the regularity condition $2h_{\phi_X}-h_{\min}<1$ holds — where $h_{\min}$ is the minimal holomorphic conformal weight generated by the operator product expansions (OPEs) of $\psi$ with itself, which can be negative in non-unitary theories — then every iterated integral in the expansion of the perturbed defect $D(\lambda\psi+\tilde\lambda\bar\psi)$ converges, and the resulting defect operator commutes with the Hamiltonian $H(\mu)$ of the CFT perturbed by $2i\mu\varphi$, at the coupled value $\mu=\lambda\tilde\lambda$, to all orders in $\lambda,\tilde\lambda$.

The framework is illustrated on the Ising CFT (where the $\sigma$-perturbation preserves no non-trivial topological symmetry, yet the non-elementary defect $D=\mathbf{1}\oplus\sigma$ supports a translation invariant defect) and on the Lee-Yang model $M(2,5)$ (the smallest non-unitary minimal model). A search in diagonal Virasoro minimal models $M(p,p')$ (rational CFTs whose fields are labelled by Kac indices; the Ising and Lee-Yang models are the smallest unitary and non-unitary members) found generic solutions for perturbing fields with Kac labels $(1,2)$, $(1,3)$, $(1,5)$ — the well-known integrable deformations — together with sporadic solutions such as the $(1,7)$-perturbations of $M(3,10)$ and $M(5,18)$, flows which have recently been investigated by Katsevich-Klebanov-Sun and Ambrosino-Negro.

The gap this problem addresses: many algebraic solutions of the commutation condition fail the regularity bound — including the $\sigma$-defect under the thermal ($\epsilon$) bulk perturbation of the Ising CFT, where $2h_{\phi_X}-h_{\min}=1$ exactly, and the sporadic minimal-model solutions — so the defining integrals diverge and require regularisation. That regularising such borderline perturbations is possible but subtle is known in neighbouring settings: for the corresponding boundary perturbation of the critical Ising model, Chatterjee (1995) obtained the exact regularised partition function; for the $\sigma$-defect perturbed by the chiral weight-$1/2$ field with the bulk unperturbed, Gaiotto-Lee-Wu (2021) computed the exactly regularised defect via a free-fermion construction based on the Arf invariant (a $\mathbb{Z}_2$-valued invariant of two-dimensional fermion systems), finding that the naive functional relation between defect operators acquires a non-perturbative correction $e^{-2\pi\lambda^2}$, from which the RG endpoint of the defect flow can be read off; Kormos-Runkel-Watts (2009) analysed such defect flows by TCSA (truncated conformal space approach, a Hamiltonian truncation method) and by the thermodynamic Bethe ansatz (TBA, an integral-equation method for finite-volume spectra). However, no regularisation is known that is compatible with the bulk perturbation switched on, i.e. that preserves translation invariance in the combined bulk-plus-defect problem, and the minimal-model sporadic cases admit no free-fermion realisation at all.

Problem Statement

For solutions $(D,\eta,\tilde\eta)$ of the chiral-TFT commutation condition for a bulk field $\varphi$ with chiral data $(X,Y,\omega)$ that violate the short-distance regularity bound $2h_{\phi_X}-h_{\min}<1$ (and analogously for the anti-holomorphic defect field, with $h_{\min}$ the minimal holomorphic weight generated by the OPEs of $\psi$ with itself), the iterated defect-field integrals defining the perturbed defect $D(\lambda\psi+\tilde\lambda\bar\psi)$ diverge, and the all-orders commutation with the perturbed Hamiltonian is not defined. The problem is to construct an explicit regularisation of these integrals that is compatible with the bulk commutation condition, in the precise sense that the regularised defect operator continues to commute with the Hamiltonian $H(\mu)$ of the CFT perturbed by $2i\mu\varphi$ at the coupled value $\mu=\lambda\tilde\lambda$, i.e. translation invariance of the defect is preserved by the regularisation. Concretely: (a) construct such a regularisation and demonstrate it on at least one of the named cases — the Ising $\sigma$-defect under the thermal bulk $\epsilon$-perturbation (where $2h_{\phi_X}-h_{\min}=1$), or the sporadic $(1,7)$-perturbation solutions of the minimal models $M(3,10)$ and $M(5,18)$; and (b) determine, more broadly, whether regularisations preserving translation invariance exist in general for such solutions. Resolving one named case in either direction completes the task: an explicit regularisation for that case, or a rigorous demonstration that no regularisation compatible with translation invariance exists for it, both suffice, while a fully general answer to (b) is desirable but not required for completion.

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