$C_T$ for Non-unitary CFTs in Higher Dimensions

TL;DR

This work computes the conformal central charge $C_T$ for non-unitary CFTs in higher dimensions, focusing on free higher-derivative scalars (with 4 and 6 derivatives) and on $(n-1)$-form gauge theories in $d=2n{+}2$, including non-gauge-invariant extensions away from integer dimensions. Using Weyl-invariant curved-space actions (involving Paneitz/Branson-type operators) and explicit energy-momentum tensors, the authors derive closed-form expressions for $C_T$ in these theories and verify consistency with large-$N$ results in related models. They also develop a constructive approach to conformal primaries by subtracting descendants, applying it to higher-derivative scalars to recover primaries and the corresponding energy-momentum tensors, and discuss the implications of non-unitarity (e.g., negative $C_T$) for these CFTs. The results illuminate how conformal data interpolate beyond unitary theories and lay groundwork for future checks of three-point functions and heat-kernel corrections in curved backgrounds.

Abstract

The coefficient $C_T$ of the conformal energy-momentum tensor two-point function is determined for the non-unitary scalar CFTs with four- and six-derivative kinetic terms. The results match those expected from large-$N$ calculations for the CFTs arising from the $O(N)$ non-linear sigma and Gross-Neveu models in specific even dimensions. $C_T$ is also calculated for the CFT arising from $(n-1)$-form gauge fields with derivatives in $2n+2$ dimensions. Results for $(n-1)$-form theory extended to general dimensions as a non-gauge-invariant CFT are also obtained; the resulting $C_T$ differs from that for the gauge-invariant theory. The construction of conformal primaries by subtracting descendants of lower-dimension primaries is also discussed. For free theories this also leads to an alternative construction of the energy-momentum tensor, which can be quite involved for higher-derivative theories.