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Feynman Diagrams of Generalized Matrix Models and the Associated Manifolds in Dimension 4

Roberto De Pietri, Carlo Petronio

TL;DR

The paper develops a unified framework to encode and analyze four-dimensional manifolds as Feynman diagrams of generalized tensor theories, extending the well-known 2D matrix-model picture to higher ranks. By mapping fat graphs to simplicial gluings and imposing explicit PL-topological conditions (Cycl, Dir, Surf, Ori), it provides an actionable criterion to determine when a given Feynman diagram defines a manifold and, in that case, a weight equal to the discretized Einstein–Hilbert action on the associated triangulation. This connects tensorial group field theories, dynamical triangulations, and spin-foam models, offering a concrete route to sum over 4D geometries via controlled diagrammatics. The results yield both theoretical insight into manifoldness from combinatorial data and practical algorithmic checks for identifying valid 4-manifolds within the tensor-model Feynman expansion.

Abstract

The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which relying on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space-time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space-times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial 4-manifolds to the Feynman diagrams of certain tensor theories.

Feynman Diagrams of Generalized Matrix Models and the Associated Manifolds in Dimension 4

TL;DR

The paper develops a unified framework to encode and analyze four-dimensional manifolds as Feynman diagrams of generalized tensor theories, extending the well-known 2D matrix-model picture to higher ranks. By mapping fat graphs to simplicial gluings and imposing explicit PL-topological conditions (Cycl, Dir, Surf, Ori), it provides an actionable criterion to determine when a given Feynman diagram defines a manifold and, in that case, a weight equal to the discretized Einstein–Hilbert action on the associated triangulation. This connects tensorial group field theories, dynamical triangulations, and spin-foam models, offering a concrete route to sum over 4D geometries via controlled diagrammatics. The results yield both theoretical insight into manifoldness from combinatorial data and practical algorithmic checks for identifying valid 4-manifolds within the tensor-model Feynman expansion.

Abstract

The problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which relying on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space-time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space-times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial 4-manifolds to the Feynman diagrams of certain tensor theories.

Paper Structure

This paper contains 18 sections, 9 theorems, 32 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Generalized matrix models and fat graphs
  3. Two-dimensional quantum gravity as a matrix model
  4. Higher-rank generalized matrix models
  5. The three-tensor and four-tensor generalized matrix model
  6. Encoding of Feynman diagrams: fat graphs
  7. Complex associated to a fat graph
  8. Equivalence of fat graphs
  9. General vertex function
  10. Simplicial glueings which define manifolds
  11. PL category
  12. Dimension 2
  13. Dimension 3
  14. Orientation in dimension 3
  15. Dimension 4
  16. ...and 3 more sections

Key Result

Proposition 2.1

The free partition function in presence of source of the generalized matrix models (Z:ntensor) whose fundamental field fulfills the symmetry requirements (GMM:real), defined as is given by where the propagator is defined as and $G_{\alpha_1\ldots\alpha_n;\beta_1\ldots\beta_n}^{ (\! \tau \!)} = \delta_{\alpha_{\tau(1)}\beta_1} \ldots \delta_{\alpha_{\tau(n)}\beta_n}$. Moreover, if integration is

Figures (10)

  • Figure 1: The three inequivalent Feynman diagrams of the two-dimensional matrix model (\ref{['eq:Zmatrixmodel']}) at order $\lambda^2$. The association of triangles to vertices is explicitly shown. Interpreting each propagator as a gluing instruction between two edges of triangles, it is easy to see that $D1$ and $D2$ correspond to different triangulation of the sphere, while $D3$ corresponds to a triangulation of the torus.
  • Figure 2: Feynman rules of the 3-tensor generalized matrix model. The analogy of the vertex diagram with the tetrahedron is explicitly shown.
  • Figure 3: Feynman rules of the 4-tensor generalized matrix model. The analogy of the vertex diagram with the 4-simplex is explicitly shown.
  • Figure 4: The move which corresponds to a change of numbering of vertices of a simplex
  • Figure 5: The links of the various points $p$ are shown in bold.
  • ...and 5 more figures

Theorems & Definitions (20)

  • Proposition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 4.5
  • Proposition 4.6
  • ...and 10 more