Nonunitary Lagrangians and unitary non-Lagrangian conformal field theories

TL;DR

The work addresses the challenge of accessing observables in strongly coupled 4D $\mathcal{N}=2$ SCFTs by leveraging a 2D chiral algebra correspondence and turning to non-unitary free field theories. The authors show that the Schur index for the (A$_1$,D$_4$) theory matches the vacuum character of $\widehat{su(3)}_{-3/2}$ and, under modular transformation, maps to the $\widehat{so(8)}_1$ current algebra built from eight Majorana fermions, revealing an affine Kac–Moody-type relation between unitary and non-unitary algebras. They extend the construction to the D$_2$[SU(2N+1)] family, where the chiral algebras are $\widehat{su(2N+1)}_{-(2N+1)/2}$ and the corresponding modular transforms reproduce $\widehat{so(4N(N+1))}_1$ with a 4D Lagrangian consisting of ghost half-hypermultiplets. This framework provides a computable bridge from non-Lagrangian 4D theories to Lagrangian, albeit non-unitary, sectors, and suggests broad generalizations and potential connections to defects and localization in supersymmetric QFTs.

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.