Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories

Abstract

In various dimensions, we can sometimes compute observables of interacting conformal field theories (CFTs) that are connected to free theories via the renormalization group (RG) flow by computing protected quantities in the free theories. On the other hand, in two dimensions, it is often possible to algebraically construct observables of interacting CFTs using free fields without the need to explicitly construct an underlying RG flow. In this note, we begin to extend this idea to higher dimensions by showing that one can compute certain observables of an infinite set of unitary strongly interacting four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs) by performing simple calculations involving sets of non-unitary free four-dimensional hypermultiplets. These free fields are distant cousins of the Majorana fermion underlying the two-dimensional Ising model and are not obviously connected to our interacting theories via an RG flow. Rather surprisingly, this construction gives us Lagrangians for particular observables in certain subsectors of many "non-Lagrangian" SCFTs by sacrificing unitarity while preserving the full $\mathcal{N}=2$ superconformal algebra. As a byproduct, we find relations between characters in unitary and non-unitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.

7 pages; 1 appendix; v2: version to appear in PRL; minor typo corrected; references added; title changed to conform to PRL guidelines (old title: "A Non-Unitary Surprise")

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Cited by in corpus (22)

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• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• Towards a Classification of Two-Character Rational Conformal Field Theories
• Modular Constraints on Superconformal Field Theories
• Infinitely many 4d $\mathcal{N}=2$ SCFTs with $a=c$ and beyond
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Rationalizing CFTs and Anyonic Imprints on Higgs Branches
• Schur sector of Argyres-Douglas theory and $W$-algebra
• On the Correspondence between Surface Operators in Argyres-Douglas Theories and Modules of Chiral Algebra
• Flowing from 16 to 32 Supercharges
• OPE Selection Rules for Schur Multiplets in 4D $\mathcal{N}=2$ Superconformal Field Theories
• The Chiral Algebra of Genus Two Class $\mathcal{S}$ Theory
• Emergent N=4 supersymmetry from N=1
• Spinors in Supersymmetric dS/CFT
• W algebra, Cosets and VOA for 4d N = 2 SCFT from M5 branes
• Peculiar Index Relations, 2D TQFT, and Universality of SUSY Enhancement
• Superconformal index of higher derivative $\mathcal{N}=1$ multiplets in four dimensions
• A Family of Vertex Algebras from Argyres-Douglas Theory
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Quasi-Characters in $\widehat{su(2)}$ Current Algebra at Fractional Levels
• 4d $\mathcal{N}=2$ SCFTs and lisse W-algebras
• Vanishing short multiplets in rank one 4d/5d SCFTs