Selection rules for RG flows of minimal models

Minimal <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> conformal field theories (CFTs) are usually classified through modular invariant partition functions. There is a finer classification of “noncomplete” models when <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>S</c:mi></c:math> 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 treelike graph. Each tree node corresponds to a local <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>d</e:mi><e:mo>=</e:mo><e:mn>2</e:mn></e:math> 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 (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>c</g:mi><g:mo>&lt;</g:mo><g:mn>1</g:mn></g:math>) <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>d</i:mi><i:mo>=</i:mo><i:mn>2</i:mn></i:math> CFTs. When appropriate, this matches Kawahigashi-Longo’s previous results. We use this finer classification to constrain renormalization group (RG) flows. For a relevant perturbation, the flow can be restricted to the subalgebra associated with it, typically corresponding to a nonmodular 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.