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Fiber bundle structure in Ashtekar-Barbero-Immirzi formulation of General Relativity

Matteo Bruno

TL;DR

This work develops a rigorous geometric framework placing the Ashtekar-Barbero-Immirzi formulation of General Relativity inside the language of fiber bundles, emphasizing a spin structure and an SU(2) gauge connection as central objects. It shows that the Ashtekar data (A, E) and the ADM variables (q, D, H) are equivalent through a reconstruction based on a solder form and the Levi-Civita connection, with the Gauss, Vector, and Scalar constraints realized as exterior covariant derivatives and a top-form constraint, respectively. The phase space is treated as a symplectic Fréchet manifold, with gauge and diffeomorphism actions described via momentum maps, and symplectic reduction recovers the ADM phase space; the full automorphism group of the spin bundle implements all physical symmetries. The framework provides a background-independent, coordinate-free platform that clarifies the geometric meaning of the constraints and paves the way for mathematically rigorous approaches to quantization and quantum gravity constructions. Overall, the paper bridges Ashtekar variables with fiber-bundle geometry, offering precise tools for interpreting GR in a gauge-theoretic setting and for exploring quantum gravitational dynamics within a rigorous geometric context.

Abstract

We aim to provide a rigorous geometric framework for the Ashtekar-Barbero-Immirzi formulation of General Relativity. As the starting point of this formulation consists in recasting General Relativity as an SU(2) gauge theory, it naturally lends itself to interpretation within the theory of principal bundles. The foundation of our framework is the spin structure, which connects the principal SU(2)-bundle construction with the Riemannian framework. The existence of the spin structure enlightens the geometric properties of the Ashtekar-Barbero-Immirzi-Sen connection and the topological characteristics of the manifold. Within this framework, we are able to express the constraints of the physical theory in a coordinate-free way, using vector-valued forms that acquire a clear geometric interpretation. Using these geometric concepts, we analyze the phase space of the theory and discuss the implementation of symmetries through the automorphism group of the principal SU(2)-bundle. In particular, we demonstrate that the description of the kinematical constraints as vector-valued forms provides a natural implementation as momentum maps for the automorphism group action.

Fiber bundle structure in Ashtekar-Barbero-Immirzi formulation of General Relativity

TL;DR

This work develops a rigorous geometric framework placing the Ashtekar-Barbero-Immirzi formulation of General Relativity inside the language of fiber bundles, emphasizing a spin structure and an SU(2) gauge connection as central objects. It shows that the Ashtekar data (A, E) and the ADM variables (q, D, H) are equivalent through a reconstruction based on a solder form and the Levi-Civita connection, with the Gauss, Vector, and Scalar constraints realized as exterior covariant derivatives and a top-form constraint, respectively. The phase space is treated as a symplectic Fréchet manifold, with gauge and diffeomorphism actions described via momentum maps, and symplectic reduction recovers the ADM phase space; the full automorphism group of the spin bundle implements all physical symmetries. The framework provides a background-independent, coordinate-free platform that clarifies the geometric meaning of the constraints and paves the way for mathematically rigorous approaches to quantization and quantum gravity constructions. Overall, the paper bridges Ashtekar variables with fiber-bundle geometry, offering precise tools for interpreting GR in a gauge-theoretic setting and for exploring quantum gravitational dynamics within a rigorous geometric context.

Abstract

We aim to provide a rigorous geometric framework for the Ashtekar-Barbero-Immirzi formulation of General Relativity. As the starting point of this formulation consists in recasting General Relativity as an SU(2) gauge theory, it naturally lends itself to interpretation within the theory of principal bundles. The foundation of our framework is the spin structure, which connects the principal SU(2)-bundle construction with the Riemannian framework. The existence of the spin structure enlightens the geometric properties of the Ashtekar-Barbero-Immirzi-Sen connection and the topological characteristics of the manifold. Within this framework, we are able to express the constraints of the physical theory in a coordinate-free way, using vector-valued forms that acquire a clear geometric interpretation. Using these geometric concepts, we analyze the phase space of the theory and discuss the implementation of symmetries through the automorphism group of the principal SU(2)-bundle. In particular, we demonstrate that the description of the kinematical constraints as vector-valued forms provides a natural implementation as momentum maps for the automorphism group action.
Paper Structure (20 sections, 79 equations, 1 table)

This paper contains 20 sections, 79 equations, 1 table.

Table of Contents

  1. Introduction
  2. Brief review of the physical notation
  3. Gauge theory structure
  4. Connection and gauge field
  5. Electric field and solder form
  6. Equivalence of the data
  7. Constraints
  8. Gauss constraint
  9. Vector constraint
  10. Scalar constraint
  11. Elements from the physical theory
  12. Independence from the metric
  13. Reconstructing the metric
  14. Symmetries
  15. A note on phase space
  16. ...and 5 more sections

Theorems & Definitions (1)

  • Definition 3.1