On the Algebraic Structure of Higher-Spin Field Equations and New Exact Solutions

TL;DR

This work surveys the algebraic structure and unfolded formulation of Vasiliev’s higher-spin gauge theory, detailing free and nonlinear dynamics across diverse space-time signatures. It develops the complex and real HS algebras, adjoint and twisted-adjoint master fields, and oscillator realizations, and introduces the Z-extension to formulate fully nonlinear Vasiliev equations. The unfolded approach is used to derive the central on-mass-shell relations for arbitrary spin and to map between HS degrees of freedom and Harish-Chandra modules, enabling a route to exact solutions. The author then presents original results: new exact four-dimensional HS solutions in varying signatures, including invariant deformations, non-maximally symmetric configurations, and chiral Type D-like constructions, along with a representation-theoretic map that connects master fields to boundary and bulk states. Overall, the work advances background-independent HS dynamics, connects HS algebras to string-theoretic ideas, and opens avenues for regularization and exact solution techniques in nonlinear HS theories.

Abstract

This Thesis reviews Vasiliev's approach to Higher-Spin Gauge Theory and contains some original results concerning new exact solutions of the Vasiliev equations and the representation theory of the higher-spin algebra. The review part covers the various formulations of the free theory as well as Vasiliev's full nonlinear equations, in particular focusing on their algebraic structure and on their properties in various space-time signatures. Then, the original results are presented. First, the 4D Vasiliev equations are formulated in space-times with signatures (4-p,p) and non-vanishing cosmological constant, and some new exact solutions are found, depending on continuous and discrete parameters: (a) an SO(4-p,p)-invariant family of solutions; (b) non-maximally symmetric solutions with vanishing Weyl tensors and higher-spin gauge fields, that differ from the maximally symmetric background solutions in the auxiliary field sector; and (c) solutions of the chiral models with an infinite tower of Weyl tensors proportional to totally symmetric products of two principal spinors. These are apparently the first exact 4D solutions with non-vanishing massless higher-spin fields. Finally, a generalized harmonic expansion of the Vasiliev's master zero-form is performed as a map from the associative algebra A of operators on the singleton phase space to representations of the background isometry algebra that include one-particle states along with linearized runaway solutions. Such Harish-Chandra modules are unitarizable in a Tr_A-norm rather than in the standard Killing norm. We also take the first steps towards a regularization scheme for handling strongly coupled higher-derivative interactions within this operator formalism.

Figures (5)

• Figure 1: Weight diagram of the massless UIR of spin-1 $\mathfrak{D}(2,1)$ in four dimensions.
• Figure 2: Weight diagram of the massless scalar $\mathfrak{D}(1,0)$ in four dimensions.
• Figure 3: Weight diagrams of the scalar singleton $\mathfrak{D}(1/2,0)$ ($\bullet$) and of the spinor singleton $\mathfrak{D}(1,1/2)$ ($\circ$) in four dimensions.
• Figure 4: Weight diagrams of the scalar 2-lineton in $D=9$.
• Figure 5: The adjoint module $\mathfrak{D}(-(s-1);(s-1))$ is connected via the conjugate massless module $\mathfrak{D}(-(s-2);(s))$ to the massless module $\mathfrak{D}(s+2\epsilon_0;(s))$.