The free $σ$CFTs

TL;DR

The work introduces free, non-unitary σCFTs that describe the shadows of composites from massless scalars and fermions in even dimensions, and shows they can be consistently defined as free CFTs for $d\ge 4$ even. Using conformal OPE and conformal partial waves, the authors analyze the spectrum, identify leading-twist higher-spin towers, and compute the energy-momentum tensor normalization constants $c_T$ and $\tilde{c}_T$ in several even dimensions both via OPE and through explicit construction of higher-derivative stress tensors. In the $d\to\infty$ limit these σCFTs reproduce the spectra of canonical free bosonic CFTs in $d=6$ (σ2) and $d=4$ (σ1), while in finite $d$ they exhibit ghost and null states that cancel poles to maintain consistency. The results provide analytic control over a new class of free, non-unitary CFTs with potential links to higher-spin holography and invite further exploration via bootstrap, $1/d$ expansions, and holographic duals.

Abstract

We introduce the conformal field theories that describe the shadows of the lowest dimension composites made out of massless free scalars and fermions in $d$ dimensions. We argue that these theories can be consistently defined as free CFTs for even $d\geq 4$. We use OPE techniques to study their spectrum and show that for $d\rightarrow\infty$ it matches that of free bosonic CFTs in $d=6$ and $d=4$ dimensions. For these $σ$CFTs we calculate $c_T$ in $d=6,8,10$ and $12$ dimensions using the OPE and also a direct construction of their higher-derivative energy momentum tensors. Our results agree with the general proposal of arXiv:1601.07198.