A Spin and a Twist on Conformal Field Theory

Abstract

The modern scattering amplitudes program has revolutionized our understanding of quantum field theory in Minkowski spacetime, leading to insights such as novel recursion relations, uncovering hidden symmetries like dual conformal and Yangian invariance, providing a dual geometric interpretation of amplitudes such as the associahedron, amplituhedron, and much more. Beginning with the remarkably simple Parke-Taylor formula for n−point tree-level MHV gluon amplitudes, this program has led to powerful methods to compute higher multiplicity and loop amplitudes which are extremely difficult to obtain using the usual Lagrangian and Feynman diagram framework. A key ingredient in these pursuits was the use of the right set of kinematic variables such as spinor helicity, twistors and momentum twistors that make the simplicity manifest from the start. Given these remarkable successes, it is natural to ask whether similar structures exist for boundary conformal correlators in Anti-de Sitter and cosmological spacetimes or more generally, for conformal field theory correlators. This forms the main subject of this thesis. We first develop a momentum-space and spinor helicity approach to conformal field theories, particularly focusing on the physically relevant case of three dimensional conformal field theory. We also determine that n−point functions in the simpler, yet illuminating setting of conformal quantum mechanics take the form of Lauricella functions. We apply our formalism to Chern-Simons matter theories which unveils an anyonic form for current correlators. We also use spinor helicity to uncover the holographic dual of chiral higher spin theory in four dimensional Anti-de Sitter spacetime. We then discuss the twistor framework for three dimensional conformal field theories which presents an advantage over the spinor framework as it makes conformal symmetry completely manifest. We derive the Penrose transform for conserved currents and extend the formalism to accommodate arbitrary conformal primaries and for super-currents. Finally, we discuss the construction of twistor space boundary correlators in AdS4 finding novel factorization properties, connections to conformal partial waves and a four point double copy relation between Yang-Mills theory and Einstein gravity. Our results provide a starting point towards a potentially conceptually and technically rich understanding of conformal correlators. More broadly, this perspective points toward a common analytic and geometric language that could potentially bring together holography, conformal bootstrap and the S-matrix program.