[Paper Review] Aspects of higher spin Hamiltonian dynamics: Conformal geometry, duality and charges

Amaury Leonard|arXiv (Cornell University)|Sep 3, 2017

Superconducting Materials and Applications3 citations

TL;DR

This thesis develops a Hamiltonian framework for free higher spin gauge fields in flat and AdS spacetimes, establishing connections between conformal geometry, twisted self-duality, and surface charges. It introduces prepotentials that encode conformal invariance and electric-magnetic duality, enabling a complete Hamiltonian formulation for arbitrary spin bosonic and fermionic fields, with explicit construction of surface charges in AdS and extension to mixed-symmetry tensors.

ABSTRACT

We have studied free higher spin gauge fields through an investigation of their Hamiltonian dynamics. Over a flat space-time, their Hamiltonian constraints were identified and solved through the introduction of prepotentials, enjoying both linearized generalized diffeomorphism and linearized generalized Weyl rescaling gauge invariance, motivating our study of conformal invariants for higher spins. We built these with the Cotton tensor, whose properties (tracelessness, symmetry, divergencelessness; completeness, invariance) we established. With these geometric tools, a first order action was written down in terms of the prepotentials. It is manifestly invariant under electric-magnetic duality which, with the gauge freedom of the prepotentials, completely fixes the action. This action is associated to twisted self-duality conditions. With an interest in supersymmetric extensions, we began to extend this study to fermions, similarly analyzing the spin $5/2$ massless free field, whose prepotential also enjoys conformal gauge invariance. The spin $2$-spin $5/2$ supermultiplet was considered, and a rigid symmetry of its action (a chirality-duality rotation) was built to commute with supersymmetry. We also investigated the properties of a mixed symmetry field on a flat six-dimensional space-time, the so-called chiral $(2,2)$-form: Hamiltonian analysis, prepotentials, and a first order action associated to self-duality conditions. Finally, we studied both fermionic and bosonic higher spin surface charges over a constantly curved background space-time. The Hamitonian constraints are the generators of gauge transformations. Plugging into them appropriate values of the gauge parameters (imposing a physical variation of the fields), their finite and non-vanishing on-shell values were computed and recognized as conserved charges of the theory. Their algebra was checked to be abelian.

Motivation & Objective

• To develop a systematic Hamiltonian formulation for free higher spin gauge fields in flat and AdS spacetimes, extending beyond the standard Fronsdal action.
• To identify and solve constraints using prepotentials that manifest conformal gauge invariance and electric-magnetic duality.
• To generalize twisted self-duality conditions to arbitrary spin fields, reformulating equations of motion in first-order form.
• To compute surface charges for higher spin fields in AdS, linking them to asymptotic symmetries and conformal Killing tensors.
• To extend the formalism to fermionic fields and mixed-symmetry tensors, including the (2,2)-form in six dimensions as a key example of the (4,0) supergravity theory.

Proposed method

• Derives the Hamiltonian and constraints from Fronsdal's action using Dirac's constrained systems formalism, with canonical momenta and primary/secondary constraints identified.
• Introduces prepotentials via a change of variables that solve the momentum and Hamiltonian constraints, preserving gauge invariance.
• Constructs a first-order action using prepotentials that manifestly respects $SO(2)$ electric-magnetic duality and conformal invariance.
• Applies twisted self-duality conditions to rewrite the equations of motion as a duality between electric and magnetic fields, eliminating Lagrange multipliers.
• Uses conformal Killing tensors and spinor-tensors to define boundary conditions and asymptotic symmetries in AdS spacetime.
• Applies the Hamiltonian formalism to hypergravity (spin-2 and spin-5/2) and the (2,2)-form in six dimensions, deriving prepotential actions with chiral and non-chiral formulations.

Experimental results

Research questions

• RQ1 How can the Hamiltonian dynamics of arbitrary spin bosonic fields be consistently formulated in flat spacetime using prepotentials that preserve conformal invariance and duality?
• RQ2 What is the role of the Cotton tensor as a complete set of conformal invariants for higher spin fields in three dimensions?
• RQ3 How do twisted self-duality conditions in higher spin theories relate to the first-order Hamiltonian action and electric-magnetic duality?
• RQ4 Can the prepotential formalism be extended to fermionic higher spin fields, such as the spin-5/2 field in hypergravity, and how does it encode supersymmetry?
• RQ5 How are surface charges for higher spin fields in AdS spacetime computed, and what is their relation to conformal Killing tensors and asymptotic symmetries?

Key findings

• The Cotton tensor is shown to be a complete set of conformal invariants for three-dimensional higher spin fields: it is symmetric, traceless, divergenceless, and fully characterizes conformal flatness.
• For any spin $s$, the momentum and Hamiltonian constraints are solved via prepotentials that transform under conformal gauge symmetries, leading to a unique Hamiltonian action invariant under $SO(2)$ electric-magnetic duality.
• Twisted self-duality conditions in higher spin theories are reformulated as a first-order system equating electric and magnetic fields (up to sign), eliminating auxiliary fields and yielding a consistent variational principle.
• The surface charges for higher spin fields in AdS are explicitly computed in terms of boundary values of the fields and conformal Killing tensors, providing a framework for studying black hole-like solutions in Vasiliev's theory.
• The prepotential formalism successfully extends to fermionic fields, such as the spin-5/2 field in hypergravity, where it encodes both supersymmetry and duality symmetries in a unified way.
• The (2,2)-form in six dimensions is shown to admit a prepotential action derived from twisted self-duality, with chiral and non-chiral formulations, suggesting a path to quantizing the full (4,0) theory.