Selection rules for RG flows of minimal models

TL;DR

This work imposes locality (T-invariance) without requiring S-invariance to achieve a finer, constructive classification of unitary minimal models with $c<1$. By starting from the smallest chiral algebra and sequentially adding local primaries, the authors build tree-like submodel structures, quantify Haag duality violation via the Jones index, and identify their intrinsic superselection categories. These ingredients yield unified RG-flow selection rules: perturbations must preserve the Jones indices, the submodel completions, and the DHR fusion category above the perturbed node, thereby constraining UV-to-IR flows across diagonal and non-diagonal minimal models. The framework recovers known Zamolodchikov-type flows and expounds new constraints, including parity-sensitive and $su(2)$-symmetric flows, offering a cohesive, non-perturbative perspective on RG trajectories within the ADE-family of minimal models.

Abstract

Minimal d=2 CFTs are usually classified through modular invariant partition functions. There is a finer classification of ``non complete'' models when S-duality is not imposed. We approach this classification by starting with the local chiral algebra and adding primaries sequentially. At each step, we only impose locality (T-duality) and closure of the operator algebra. For each chiral algebra, this produces a tree-like graph. Each tree node corresponds to a local d=2 CFT, with an intrinsic Jones index measuring the size of Haag duality violation. This index can be computed with the partition function and is related to the total quantum dimension of the category of superselection sectors of the node, and to the relative size between the node and a modular invariant completion. In this way, we find in a very explicit manner a classification of local minimal (c<1) d=2 CFTs. When appropriate, this matches Kawahigashi-Longo's previous results. We use this finer classification to constrain RG flows. For a relevant perturbation, the flow can be restricted to the subalgebra associated with it, typically corresponding to a non-modular invariant node in the tree. The structure of the graph above such node needs to be preserved by the RG flow. In particular, the superselection sector category for the node must be preserved. This gives selection rules that recover in a unified fashion several known facts while unraveling new ones.

Figures (25)

• Figure 1: Allowed representations of the $m=3$ minimal model, classified by their Kac label $(r,s)$ in the Kac table (left), and table with the corresponding conformal dimensions $h$ and quantum dimensions $d$ (right).
• Figure 2: Classification of $d=2$ CFTs for $m=3$. Each node has a global Jones index $\mu$ and we also write the relative Jones index between all immediate inclusions $\lambda$. The blue boxes represent algebras composed purely of spin zero primary fields (excepting the stress tensor) and the green ones include spin $1/2$ fermion fields.
• Figure 3: Allowed representations of the $m=4$ minimal model, classified by their Kac label $(r,s)$ in the Kac table (left), and table with the correspondig conformal dimensions $h$ and quantum dimensions $d$ (right).
• Figure 4: Classification of $d=2$ CFTs for $m=4$. Each node has a global Jones index $\mu$ and we also write the relative Jones index between all immediate inclusions $\lambda$. The blue boxes represent algebras composed purely of spin zero primary fields and the green ones include spin $1/2$ and/or $3/2$ fermion fields.
• Figure 5: Allowed representations of the $m=5$ minimal model, classified by their Kac label $(r,s)$ in the Kac table (left), and table with the corresponding conformal dimensions $h$ and quantum dimensions $d$ (right).