Tractors and twistors from conformal Cartan geometry: a gauge theoretic approach II. Twistors

2016/09/30 by Jordan François, Jérémy Attard, Jeremy Attard · 16 citations
Mathematics · Physics and Astronomy · #BRST quantization #Black Holes and Theoretical Physics #Conformal map #Cosmology and Gravitation Theories #Covariant transformation #Differential form #Differential geometry #Gauge anomaly #Gauge covariant derivative #Gauge symmetry #Gauge theory #Geometry #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Supersymmetric gauge theory #Symmetry (geometry) #Theoretical physics #math-ph #math.MP

paper · pdf · doi:10.1088/1361-6382/aa627d

published in Classical and Quantum Gravity 34(8), 085004 (IOP Publishing) · 29 pages, minor typos corrected

openalex created_date 2016/10/07 · arxiv created 2016/11/04 · openalex publication_date 2017/03/21 · arxiv updated 2017/03/29 · openalex updated_date 2026/08/05

Abstract

Abstract Tractor and Twistor bundles provide natural conformally covariant calculi on 4D-Riemannian manifolds. They have different origins but are closely related, and usually constructed bottom–up through prolongation of defining differential equations. We propose alternative top–down gauge theoretic constructions, starting from the conformal Cartan bundle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="script">P</mml:mi> </mml:mrow> </mml:mrow> </mml:mstyle> </mml:math> and its vectorial E and spinorial <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mrow> <mml:mrow> <mml:mtext mathvariant="sans-serif">E</mml:mtext> </mml:mrow> </mml:mrow> </mml:mstyle> </mml:math> associated bundles. Our key ingredient is the dressing field method of gauge symmetry reduction, which allows tractors and twistors and their associated connections to exhibit as gauge fields of a non-standard kind as far as Weyl rescaling transformation is concerned. By non-standard we mean that they implement the gauge principle of physics, but are of a different geometric nature than the well-known differential geometric objects usually underlying gauge theories. We provide the corresponding BRST treatment. In a companion paper we dealt with tractors, in the present one we address the case of twistors.

Citations

• A Remark on the Spontaneous Symmetry Breaking Mechanism in the Standard Model
• Spinors and Space-Time
• The Principles of Quantum Mechanics
• The normal conformal Cartan connection and the Bach tensor
• Twistor theory: An approach to the quantisation of fields and space-time
• Gauge invariant accounts of the Higgs mechanism
• Nucleon spin decomposition and differential geometry
• Local BRST cohomology in gauge theories
• Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions
• Polyakov Soldering and Second-Order Frames: The Role of the Cartan Connection
• Invariant Theoretical Interpretation of Interaction

Cited by

• The dressing field method of gauge symmetry reduction, a review with examples

Related

• Subtleties Concerning Conformal Tractor Bundles
• An Embedding Theorem for tractor bundles, and an application in conformal pseudo-Riemannian geometry
• Current Algebra and Generalised Cartan Geometry
• On Conformal Powers of the Dirac Operator on Spin Manifolds
• Conformal geometry as a gauge theory of gravity: covariant equations of motion & conservation laws