Superspin projection operators and off-shell higher-spin supermultiplets on Minkowski and anti-de Sitter superspace

This thesis is dedicated to the study of (super)spin projection operators and their applications on maximally symmetric (super)space backgrounds in three and four dimensions. On such backgrounds, the irreducible representations of the associated isometry (super)algebra may be realised on the space of tensor (super)fields satisfying certain differential constraints. The (super)spin projectors isolate the component of an unconstrained (super)field which furnishes the irreducible representation with maximal (super)spin. The explicit form of these (super)projectors are found in the following backgrounds: three-dimensional ($3d$) $\mathcal{N}$-extended Minkowski superspace $\mathbb{M}^{3|2 \mathcal{N}}$; $3d$ (anti-)de Sitter space (A)dS$_3$; $3d$ $\mathcal{N} = 1$ anti-de Sitter superspace AdS$^{3|2}$; four-dimensional ($4d$) $\mathcal{N} = 1$ anti-de Sitter superspace AdS$^{4|4}$; and $4d$ $\mathcal{N} = 2$ anti-de Sitter superspace AdS$^{4|8}$. An array of novel applications are investigated, with an emphasis placed on the interplay between (super)projectors and (super)conformal higher-spin theory. Another major component of this thesis consists of a detailed study of massless higher-spin gauge models with $\mathcal{N} = 2$ AdS supersymmetry in three dimensions. We find that every known higher-spin theory with $(1, 1)$ AdS supersymmetry decomposes into a sum of two off-shell $(1, 0)$ supermultiplets which belong to three series of inequivalent higher-spin gauge models.