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Matrix model formulation of four dimensional gravity

R. De Pietri

TL;DR

The paper investigates extending the successful $2$-D matrix-model description of quantum gravity to $4$-D by using $n$-tensor models whose Feynman diagrams encode $n$-simplicial complexes. It develops a formalism where fat graphs correspond to the dual $2$-skeletons of triangulations, and derives manifold conditions (Cycl, Surf) needed for the resulting space to be a $4$-manifold, while highlighting that not all fat graphs satisfy these conditions. Recognizing this limitation, the authors advocate moving to group field theories on homogeneous spaces to weight diagrams by topological invariants such as the Ooguri–Crane–Yetter ($OCY$) and Barrett–Crane invariants, thereby discriminating admissible space-times and connecting to spin-foam approaches. The work thus links dynamical triangulations, tensor models, and group-field theory in an effort to construct a viable $4D$ quantum gravity framework with manifest combinatorial and topological control.

Abstract

The attempt of extending to higher dimensions the matrix model formulation of two-dimensional quantum gravity leads to the consideration of higher rank tensor models. We discuss how these models relate to four dimensional quantum gravity and the precise conditions allowing to associate a four-dimensional simplicial manifold to Feynman diagrams of a rank-four tensor model.

Matrix model formulation of four dimensional gravity

TL;DR

The paper investigates extending the successful -D matrix-model description of quantum gravity to -D by using -tensor models whose Feynman diagrams encode -simplicial complexes. It develops a formalism where fat graphs correspond to the dual -skeletons of triangulations, and derives manifold conditions (Cycl, Surf) needed for the resulting space to be a -manifold, while highlighting that not all fat graphs satisfy these conditions. Recognizing this limitation, the authors advocate moving to group field theories on homogeneous spaces to weight diagrams by topological invariants such as the Ooguri–Crane–Yetter () and Barrett–Crane invariants, thereby discriminating admissible space-times and connecting to spin-foam approaches. The work thus links dynamical triangulations, tensor models, and group-field theory in an effort to construct a viable quantum gravity framework with manifest combinatorial and topological control.

Abstract

The attempt of extending to higher dimensions the matrix model formulation of two-dimensional quantum gravity leads to the consideration of higher rank tensor models. We discuss how these models relate to four dimensional quantum gravity and the precise conditions allowing to associate a four-dimensional simplicial manifold to Feynman diagrams of a rank-four tensor model.

Paper Structure

This paper contains 5 sections, 10 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. GENERALIZED MATRIX MODELS
  3. ASSOCIATED COMPLEXES
  4. MANIFOLD CONDITIONS
  5. GENERALIZED MODEL IN DIMENSION FOUR

Figures (3)

  • Figure 1: Feynman rules of a four-tensor generalized matrix model and the analogy of the vertex diagram with the four simplex.
  • Figure 2: The basic building block in three-dimension. The shaded areas, after gluing, will become the boundary components of $X^\partial$
  • Figure 3: Links of the barycentres of a triangle (a) and an edge (b) in dimension four. The link of the midpoint of an edge is the double cone on the link in a cross-section (c). The triangle involved in the Surf condition is shaded.