Non-local charges from perturbed defects via SymTFT in 2d CFT

TL;DR

This work connects non-local conserved charges in perturbed 2d CFTs to a 3d SymTFT framework, deriving a bulk–defect commutation condition that yields a one-parameter family of mutually commuting, rigidly translation-invariant defects. Applying the construction to Virasoro minimal models, it produces explicit non-local charge families for $(1,2)$, $(1,3)$, and $(1,5)$ bulk perturbations, and finds solutions for certain $(1,7)$ cases, signaling integrability beyond local charges. Through detailed analyses of elementary defects and their fusion graphs, the paper demonstrates how perturbed defects can realize integrable deformations in several minimal models, including a thorough treatment of $M(3,10)$. It also critically assesses the presence of local conserved charges, showing that many perturbations lack them, and discusses outlooks, regularisation issues, and potential links to IR topological behavior and TCSA studies. Overall, the work provides a unified, categorical approach to identifying and organizing both local and non-local conserved structures in perturbed 2d CFTs with an eye toward integrability and RG flows.

Abstract

We investigate non-local conserved charges in perturbed two-dimensional conformal field theories from the point of view of the 3d SymTFT of the unperturbed theory. In the SymTFT we state a simple commutation condition which results in a pair of compatible bulk and defect perturbations, such that the perturbed line defects are conserved in the perturbed CFT. In other words, the perturbed defects are rigidly translation invariant, and such defects form a monoidal category which extends the topological symmetries. As examples we study the A-type Virasoro minimal models $M(p,q)$. Our formalism provides one-parameter families of commuting non-local conserved charges for perturbations by a primary bulk field with Kac label $(1,2)$, $(1,3)$, or $(1,5)$, which are the standard integrable perturbations of minimal models. We find solutions to the commutation condition also for other bulk perturbations, such as $(1,7)$, and we contrast this with the existence of local conserved charges. There has been recent interest in the possibility that in certain cases perturbations by fields such as $(1,7)$ can be integrable, and our construction provides a new way in which integrability can be found without the need for local conserved charges.

Figures (11)

• Figure 1: Bulk commutation condition in the SymTFT picture
• Figure 2: a) A patch of the surface $\Sigma$ on which the 2d CFT is defined. b) The SymTFT presentation on $\Sigma \times [0,1]$ with topological boundary condition $\mathcal{B}$. c) The chiral TFT presentation on $\Sigma \times [-1,1]$ with a topological surface defect with defect condition $\mathcal{B}$ at $\Sigma \times \{0\}$.
• Figure 3: Local identities for line defects in the chiral TFT for the modular fusion category $\mathcal{F}$. Here, $i,j,k,\dots \in \mathrm{Irr}(\mathcal{F})$ denote simple objects, and the small boxes denote the basis element $\lambda_{ij}^k : i\otimes j \to k$ and its dual $k \to i\otimes j$.
• Figure 4: a) CFT: A patch of the surface $\Sigma$ with an insertion of the bulk field $\varphi(z)$ of the 2d CFT. b) SymTFT presentation. c) Chiral TFT presentation. Note that we are now assuming the surface defect in Figure \ref{['fig:geometry']} c) to be trivial.
• Figure 5: a) CFT: A patch of the surface $\Sigma$ with an insertion of a topological line defect $D$, a holomorphic defect field $\psi$ and an antiholomorphic defect field $\, \overline \psi$. b) SymTFT presentation. Here we assume that the endpoints of $x$ and $\overline x$ on the conformal boundary depend holomorphically (resp. antiholomorphically) on the insertion point. c) Chiral TFT presentation.