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Gravity Coupled with Scalar, SU$(n)$, and Spinor Fields on Manifolds with Null-Boundary

Alberto S. Cattaneo, Filippo Fila-Robattino, Valentino Huang, Manuel Tecchiolli

TL;DR

This work develops a boundary-focused formulation of gravity in the Palatini–Cartan coframe framework on manifolds with null boundaries, incorporating scalar, SU$(n)$ gauge, and spinor matter. Using the Kijowski–Tulczijew approach, it constructs a reduced boundary phase space and identifies a degeneracy-friendly constraint structure, where the structural and degeneracy constraints fix boundary representatives and produce a second-class constraint $R_ au$. Across scalar, Yang–Mills, and spinor couplings, the boundary constraint algebra remains non-first-class, reducing the local degrees of freedom to two on null boundaries, a feature with potential relevance for horizon physics and the formulation of gauge theories on spaces with degenerate boundary metrics.

Abstract

In this paper, we present a theory for gravity coupled with scalar, SU$(n)$ and spinor fields on manifolds with null-boundary. We perform the symplectic reduction of the space of boundary fields and give the constraints of the theory in terms of local functionals of boundary vielbein and connection. For the three different couplings, the analysis of the constraint algebra shows that the set of constraints does not form a first class system.

Gravity Coupled with Scalar, SU$(n)$, and Spinor Fields on Manifolds with Null-Boundary

TL;DR

This work develops a boundary-focused formulation of gravity in the Palatini–Cartan coframe framework on manifolds with null boundaries, incorporating scalar, SU gauge, and spinor matter. Using the Kijowski–Tulczijew approach, it constructs a reduced boundary phase space and identifies a degeneracy-friendly constraint structure, where the structural and degeneracy constraints fix boundary representatives and produce a second-class constraint . Across scalar, Yang–Mills, and spinor couplings, the boundary constraint algebra remains non-first-class, reducing the local degrees of freedom to two on null boundaries, a feature with potential relevance for horizon physics and the formulation of gauge theories on spaces with degenerate boundary metrics.

Abstract

In this paper, we present a theory for gravity coupled with scalar, SU and spinor fields on manifolds with null-boundary. We perform the symplectic reduction of the space of boundary fields and give the constraints of the theory in terms of local functionals of boundary vielbein and connection. For the three different couplings, the analysis of the constraint algebra shows that the set of constraints does not form a first class system.
Paper Structure (11 sections, 29 theorems, 155 equations, 1 table)

This paper contains 11 sections, 29 theorems, 155 equations, 1 table.

Table of Contents

  1. Introduction
  2. Geometrical background of gravity
  3. Coframe formalism
  4. Symplectic reduction on the boundary
  5. The structural and degeneracy constraints
  6. Fixing the representative
  7. Coupling terms: degenerate structure
  8. Scalar field
  9. Yang--Mills field
  10. Spinor field
  11. First and second class constraints

Key Result

Proposition 3

The inner product on $V$ allows the identification $\mathfrak{so}(N-1,1) \cong {\bigwedge^2} {V}$.

Theorems & Definitions (97)

  • Definition 1
  • Remark 2
  • Proposition 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Remark 8
  • Remark 9
  • Definition 10
  • ...and 87 more