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Spinors and Twistors in Loop Gravity and Spin Foams

Maite Dupuis, Simone Speziale, Johannes Tambornino

Abstract

Spinorial tools have recently come back to fashion in loop gravity and spin foams. They provide an elegant tool relating the standard holonomy-flux algebra to the twisted geometry picture of the classical phase space on a fixed graph, and to twistors. In these lectures we provide a brief and technical introduction to the formalism and some of its applications.

Spinors and Twistors in Loop Gravity and Spin Foams

Abstract

Spinorial tools have recently come back to fashion in loop gravity and spin foams. They provide an elegant tool relating the standard holonomy-flux algebra to the twisted geometry picture of the classical phase space on a fixed graph, and to twistors. In these lectures we provide a brief and technical introduction to the formalism and some of its applications.

Paper Structure

This paper contains 12 sections, 4 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Twisted geometries, spinors and null twistors
  3. Twisted geometries
  4. Twistors
  5. Gauge-invariant spinor networks and polyhedral geometries
  6. ${\mathrm U}(N)$ formalism
  7. Spinorial Hilbert space for loop gravity
  8. Spinorial Hilbert space for loop gravity
  9. Twistors and covariant twisted geometries
  10. $GL(N,{\mathbb C})$ formalism
  11. Holomorphic simplicity constraints
  12. Summary and Outlook

Key Result

Theorem 1

nameref-thm1 fith LAB: thm1 Freidel:2010bw The symplectic reduction of the space ${\mathbb C}^2_\ast \times {\mathbb C}^2_\ast$ by the constraint ${\mathcal{M}}$ is isomorphic to the cotangent bundle $T^*{\rm SU}(2) - \{ |X| = 0 \}$ as a symplectic space.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4