Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories

TL;DR

This paper demonstrates how to compute certain observables of 4D $ ext{N}=2$ superconformal field theories using non-unitary free fields, revealing new Lagrangian descriptions for non-Lagrangian theories by sacrificing unitarity.

Contribution

It introduces a novel method to obtain Lagrangian descriptions of observables in non-Lagrangian 4D SCFTs through non-unitary free field constructions, extending ideas from 2D CFTs.

Findings

Constructed Lagrangians for observables in non-Lagrangian SCFTs.

Linked characters of unitary and non-unitary affine Kac-Moody algebras.

Extended 2D algebraic methods to higher-dimensional 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.