# Space-time CFTs from the Riemann sphere

## TL;DR

This paper develops a 2D chiral conformal field theory framework on the Riemann sphere for describing lower-dimensional CFTs, revealing critical dimensions where anomalies vanish and connecting to known space-time theories.

## Contribution

It introduces a novel 2D CFT approach with gauge constraints and BRST analysis that reproduces key features of space-time conformal theories and their amplitudes.

## Key findings

- Critical dimensions match those of classical conformal invariance in space-time theories.
- BRST cohomology contains vertex operators for full conformal multiplets.
- Provides a new prescription for computing three-point functions and relates to scattering equations.

## Abstract

We consider two-dimensional chiral, first-order conformal field theories governing maps from the Riemann sphere to the projective light cone inside Minkowski space -- the natural setting for describing conformal field theories in two fewer dimensions. These theories have a SL(2) algebra of local bosonic constraints which can be supplemented by additional fermionic constraints depending on the matter content of the theory. By computing the BRST charge associated with gauge fixing these constraints, we find anomalies which vanish for specific target space dimensions. These critical dimensions coincide precisely with those for which (biadjoint) cubic scalar theory, gauge theory and gravity are classically conformally invariant. Furthermore, the BRST cohomology of each theory contains vertex operators for the full conformal multiplets of single field insertions in each of these space-time CFTs. We give a prescription for the computation of three-point functions, and compare our formalism with the scattering equations approach to on-shell amplitudes.