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.
Highlights
Properties such as soft and collinear limits and factorization (e.g. [8]), their origin — and in particular the role of the underlying Riemann sphere — seems mysterious from the perspective of space-time field theory
These ambitwistor strings do more than just reproduce tree-level amplitude formulae, though: they have led to novel representations for higher-loop field theory integrands in terms of localized expressions both on higher genus Riemann surfaces [14, 15] and on degenerate Riemann spheres [16,17,18]
We find that each model encodes information about a specific space-time conformal field theories (CFTs): the bosonic model leads to d = 6 biadjoint cubic scalar theory; the heterotic model leads to d = 4 gauge theory; and the type II model leads to d = 2 gravity
Summary
Consider complexified d-dimensional Euclidean space; this is Cd with the flat, holomorphic Euclidean metric. PN is a d-dimensional space with a natural action of SO(d + 1, 1) To see that this action is equivalent to the action of the conformal group in d dimensions, we can consider a particular coordinate patch of the projective space PN .3. Since PN is a projective space and the conformal primaries are just tensors of fixed homogeneity, conformal integrals over PN are constrained (up to an overall factor) by homogeneity, leading to myriad applications of the projective null cone, and the embedding space more generally, to the study of CFTs in d > 2 (see for instance [43,44,45,46,47,48,49]).
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