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 Letter, 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 N=2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary 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 N=2 superconformal algebra. As a by-product, we find relations between characters in unitary and nonunitary affine Kac-Moody algebras. We conclude by commenting on possible generalizations of our construction.
Highlights
Introduction.—Free fields in two spacetime dimensions are versatile: operators, correlation functions, and partition functions of interacting conformal field theories (CFTs) can often be constructed algebraically from free bosons via the Coulomb gas formalism, and the simplest unitary minimal model—the Ising model—has a free Majorana fermion underlying it
In the case of 4D superconformal field theories (SCFTs) with at least N 1⁄4 2 supersymmetry (SUSY), one such relation was given in [3]: the sector of so-called “Schur” operators of the 4D SCFT is isomorphic to a 2D chiral algebra living on a plane, P ⊂ R4
Where the trace is over the Hilbert space of states associated with the chiral algebra, c2d is the chiral algebra central charge, M⊥ 1⁄4 j1 − j2 is the spin transverse to P [j1;2 are Cartans of SOð4Þ], q ∈ C is a fugacity, and L0 gives the holomorphic scaling dimension, h
Summary
CRST and School of Physics and Astronomy Queen Mary University of London, London E1 4NS, United Kingdom (Received 7 December 2017; published 23 February 2018). In this Letter, 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 N 1⁄4 2 superconformal field theories (SCFTs) by performing simple calculations involving sets of nonunitary 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. The Schur index for the (A1, D4) theory was computed in [9,10,11,12] and was shown to equal the vacuum character of
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