Higher Spin Interacting Quantum Field Theory and Higher Order Conformal Invariant Lagrangians

TL;DR

The work develops a comprehensive framework for higher spin (HS) gauge theory in AdS and flat spacetimes, addressing both propagator structure and interactions. It delivers a detailed analysis of UV behavior and one-loop mass renormalization for HS fields in AdS$_4$, and constructs a hierarchy of conformally invariant Lagrangians via Fefferman–Graham ambient space techniques, including a Weyl-gauged ambient interpretation. The thesis then achieves a complete off-shell covariant construction of trilinear HS interactions with arbitrary spins using Noether’s procedure, providing explicit results for two-, four-, and mixed-spin couplings and a generating-function formulation that unifies free and cubic sectors. The findings illuminate the role of gauge invariance in fixing interactions, reveal a rich structure of HS currents and their derivatives, and establish links to holography and potential string-theoretic origins, while outlining clear routes to quartic order and deeper AdS/CFT connections.

Abstract

This thesis includes several original results. All of them are already published or submitted for publication. I present here the short summary of main results: The ultraviolet singular structure of the bulk-to-bulk propagators for higher spin gauge fields in AdS4 space is analyzed in details. One loop mass renormalization is studied on a simple example. The conformal invariant Lagrangian with the k-th power of Laplacian for the hierarchy of conformally coupled scalars with increasing scaling dimensions connected with the k-th Euler density is rederived using the Fefferman-Graham ambient space approach. The corresponding gauged ambient metric, Fefferman- Graham expansion and extended Penrose-Brown-Henneaux transformations are proposed and analyzed. Linearized gauge invariant interactions of scalar and general higher even spin fields in the AdSD space are obtained. A generalized Weyl transformation is proposed and the corresponding Weyl invariant action for cubic coupling of a scalar to a spin \ell field is constructed. Using Noether's procedure several cubic interactions between different HS gauge fields are derived, including cubic selfinteraction of even spin gauge fields in a flat background. Then the main result - the complete off-shell gauge invariant Lagrangian for the trilinear interactions of Higher Spin Fields with arbitrary spins s1, s2, s3 in a flat background is presented. All possibilities with different numbers of derivatives are discussed. Restrictions on the number of derivatives are obtained. For any possible number of derivatives this interaction is uniquely fixed by gauge invariance up to partial integration and field redefinition. Finally an off-shell generating function for all cubic interactions of Higher Spin gauge fields is presented. It is written in a compact way, and turns out to have a remarkable structure.