Horacio Casini

The operator product expansion (OPE) of twist operators in the replica trick framework enables a long-distance expansion of the mutual information (MI) in conformal field theories (CFTs). In this expansion, the terms are labeled by primary operators, as contributions from descendant operators can be resummed into conformal blocks. However, for the MI, the expansion involves primaries from the multi-replica theory, which includes far more operators than those in the original theory. In this work, we develop a method to resum this series, yielding an expansion in terms of the primaries of the original theory, specifically restricted to the two-copy sector. This is achieved by expressing the twist operators in a non-local manner across different replicas and using a modular flow representation to obtain the n → 1 limit of the Rényi index. We explicitly compute the resulting “enhanced conformal blocks,” which, surprisingly, provide excellent approximations to the MI of generalized free fields across the full range of cross ratios. Remarkably, this approximation appears to be exact in the limit of large spacetime dimensions.

Minimal d = 2 conformal field theories (CFTs) are usually classified through modular invariant partition functions. There is a finer classification of “noncomplete” 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 treelike 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 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.

Twist operators implement symmetries in bounded regions of the space. Standard twists are a special class of twists constructed using modular tools. The twists corresponding to translations have interesting special properties. They can move continuously an operator from a region to a disjoint one without ever passing through the gap separating the two. In addition, they have generators satisfying the spectrum condition. We compute explicitly these twists for the two-dimensional chiral fermion field. The twist generator gives place to a new type of energy inequality where the smeared energy density is bounded below by an operator. Published by the American Physical Society 2024

Twist operators implement symmetries in bounder regions of the space. Standard twists are a special class of twists constructed using modular tools. The twists corresponding to translations have interesting special properties. They can move continuously an operator from a region to a disjoint one without ever passing through the gap separating the two. In addition, they have generators satisfying the spectrum condition. We compute explicitly these twists for the two dimensional chiral fermion field. The twist generator gives place to a new type of energy inequality where the smeared energy density is bounded below by an operator.