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Quantum Gravity with Matter via Group Field Theory

Kirill Krasnov

TL;DR

The paper develops a Drinfeld double-based group field theory as a natural framework for 3D quantum gravity coupled to matter. It shows that the Boulatov-type GFT on $D(SU(2))$ reproduces a Ponzano-Regge–like amplitude that, in its simplest form, does not describe point particles, but chain-mail constructions recover particle-like amplitudes via general edge propagators $P$ on the dual complex. By analyzing kernels, dual graphs, and tetrahedral constraints, the work connects GFT Feynman diagrams to triangulated 3-manifolds and to dual particle Feynman diagrams, highlighting a duality between gravity and matter interpretations. It also demonstrates that certain chain-mail models yield genuine 3-manifold invariants, while more general models describe gravity with backreacting point particles, thus establishing a versatile framework for 3D quantum gravity with matter and outlining directions for future generalizations to other algebras and quantum groups.

Abstract

A generalization of the matrix model idea to quantum gravity in three and higher dimensions is known as group field theory (GFT). In this paper we study generalized GFT models that can be used to describe 3D quantum gravity coupled to point particles. The generalization considered is that of replacing the group leading to pure quantum gravity by the twisted product of the group with its dual --the so-called Drinfeld double of the group. The Drinfeld double is a quantum group in that it is an algebra that is both non-commutative and non-cocommutative, and special care is needed to define group field theory for it. We show how this is done, and study the resulting GFT models. Of special interest is a new topological model that is the ``Ponzano-Regge'' model for the Drinfeld double. However, as we show, this model does not describe point particles. Motivated by the GFT considerations, we consider a more general class of models that are defined using not GFT, but the so-called chain mail techniques. A general model of this class does not produce 3-manifold invariants, but has an interpretation in terms of point particle Feynman diagrams.

Quantum Gravity with Matter via Group Field Theory

TL;DR

The paper develops a Drinfeld double-based group field theory as a natural framework for 3D quantum gravity coupled to matter. It shows that the Boulatov-type GFT on reproduces a Ponzano-Regge–like amplitude that, in its simplest form, does not describe point particles, but chain-mail constructions recover particle-like amplitudes via general edge propagators on the dual complex. By analyzing kernels, dual graphs, and tetrahedral constraints, the work connects GFT Feynman diagrams to triangulated 3-manifolds and to dual particle Feynman diagrams, highlighting a duality between gravity and matter interpretations. It also demonstrates that certain chain-mail models yield genuine 3-manifold invariants, while more general models describe gravity with backreacting point particles, thus establishing a versatile framework for 3D quantum gravity with matter and outlining directions for future generalizations to other algebras and quantum groups.

Abstract

A generalization of the matrix model idea to quantum gravity in three and higher dimensions is known as group field theory (GFT). In this paper we study generalized GFT models that can be used to describe 3D quantum gravity coupled to point particles. The generalization considered is that of replacing the group leading to pure quantum gravity by the twisted product of the group with its dual --the so-called Drinfeld double of the group. The Drinfeld double is a quantum group in that it is an algebra that is both non-commutative and non-cocommutative, and special care is needed to define group field theory for it. We show how this is done, and study the resulting GFT models. Of special interest is a new topological model that is the ``Ponzano-Regge'' model for the Drinfeld double. However, as we show, this model does not describe point particles. Motivated by the GFT considerations, we consider a more general class of models that are defined using not GFT, but the so-called chain mail techniques. A general model of this class does not produce 3-manifold invariants, but has an interpretation in terms of point particle Feynman diagrams.

Paper Structure

This paper contains 38 sections, 14 theorems, 145 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Point particles
  3. Ponzano-Regge model in presence of conical singularities
  4. Momentum representation and duality
  5. Point particle Feynman diagrams
  6. Tetrahedron constraints
  7. Pure gravity from point particle Feynman diagrams
  8. Matrix models
  9. Field theory over a group manifold
  10. Outline and organization of the paper
  11. Boulatov field theory over the group
  12. Original Boulatov formulation
  13. Algebra structure
  14. Projectors
  15. Algebraic formulation of Boulatov's theory
  16. ...and 23 more sections

Key Result

Lemma 4.1

The quantity $\eta^2\theta$, where $\eta^{-2}=\delta_e(e)$ is a projector:

Figures (2)

  • Figure 1: A truncated tetrahedron
  • Figure 2: A compound of a tetrahedron $t$ with a truncated tetrahedron $\bar{t}$.

Theorems & Definitions (26)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • proof
  • Lemma 4.6
  • proof
  • Lemma 4.7
  • ...and 16 more