Light-Front approach to $4d$ massless Higher-Spin interactions

This thesis studies $4d$ massless higher-spin interactions in the Light-Front approach by analysing the closure of the Poincar\'e algebra at quartic order. We first solve the light-cone quartic holomorphic constraint in flat space and show the existence of infinitely many interacting local higher-spin theories with either a finite or infinite number of fields. We classify all one- and two-derivative theories, corresponding to higher-spin extensions of gauge and gravitational interactions. These are consistent subsectors of higher-spin extensions of self-dual Yang--Mills and gravity, themselves truncations of Chiral Higher-Spin Gravity. We then clarify the relation between the OPE associativity in celestial CFT, the vanishing of tree-level amplitudes for generic kinematics, the Jacobi identity of the associated ''gauge algebra'' (kinematical algebra), and the light-cone holomorphic constraints. Finally, we investigate the non-holomorphic quartic constraint involving both MHV and anti-MHV vertices. We recover the existence of Yang-Mills theory and gravity, and the inconsistency of interacting multi-graviton theories. We then show that once higher-derivative cubic vertices are included, nontrivial solutions to the full quartic constraint exist. We classify all unitary local higher-spin theories, identify new families of local quasi-chiral theories, and determine all local higher-spin four-point amplitudes using the spinor-helicity formalism together with locality in the form of consistent factorisation.