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Pachner moves in a 4d Riemannian holomorphic Spin Foam model

Andrzej Banburski, Lin-Qing Chen, Laurent Freidel, Jeff Hnybida

TL;DR

This work develops a four-dimensional, Riemannian holomorphic Spin Foam model with holomorphic simplicity constraints imposed on BF propagators. Using the homogeneity map and loop identities, it provides analytic expressions for Pachner moves and reveals that the constrained model is not invariant under the $3$-$3$, $4$-$2$, or $5$-$1$ moves due to nonlocal strand mixing, interpreted as insertions of nonlocal operators. A natural truncation scheme keeps only the unmixed (J' = 0) sector, restoring invariance for $4$-$2$ and $5$-$1$ up to a boundary factor and enabling a controlled coarse-graining program. The paper also analyzes divergences, showing parameter regions where certain moves remain finite, which offers a potential route to recovering diffeomorphism invariance in the continuum limit through renormalization flow toward fixed points.

Abstract

In this work we study a Spin Foam model for 4d Riemannian gravity, and propose a new way of imposing the simplicity constraints that uses the recently developed holomorphic representation. Using the power of the holomorphic integration techniques, and with the introduction of two new tools: the homogeneity map and the loop identity, for the first time we give the analytic expressions for the behaviour of the Spin Foam amplitudes under 4-dimensional Pachner moves. It turns out that this behaviour is controlled by an insertion of nonlocal mixing operators. In the case of the 5-1 move, the expression governing the change of the amplitude can be interpreted as a vertex renormalisation equation. We find a natural truncation scheme that allows us to get an invariance up to an overall factor for the 4-2 and 5-1 moves, but not for the 3-3 move. The study of the divergences shows that there is a range of parameter space for which the 4-2 move is finite while the 5-1 move diverges. This opens up the possibility to recover diffeomorphism invariance in the continuum limit of Spin Foam models for 4D Quantum Gravity.

Pachner moves in a 4d Riemannian holomorphic Spin Foam model

TL;DR

This work develops a four-dimensional, Riemannian holomorphic Spin Foam model with holomorphic simplicity constraints imposed on BF propagators. Using the homogeneity map and loop identities, it provides analytic expressions for Pachner moves and reveals that the constrained model is not invariant under the -, -, or - moves due to nonlocal strand mixing, interpreted as insertions of nonlocal operators. A natural truncation scheme keeps only the unmixed (J' = 0) sector, restoring invariance for - and - up to a boundary factor and enabling a controlled coarse-graining program. The paper also analyzes divergences, showing parameter regions where certain moves remain finite, which offers a potential route to recovering diffeomorphism invariance in the continuum limit through renormalization flow toward fixed points.

Abstract

In this work we study a Spin Foam model for 4d Riemannian gravity, and propose a new way of imposing the simplicity constraints that uses the recently developed holomorphic representation. Using the power of the holomorphic integration techniques, and with the introduction of two new tools: the homogeneity map and the loop identity, for the first time we give the analytic expressions for the behaviour of the Spin Foam amplitudes under 4-dimensional Pachner moves. It turns out that this behaviour is controlled by an insertion of nonlocal mixing operators. In the case of the 5-1 move, the expression governing the change of the amplitude can be interpreted as a vertex renormalisation equation. We find a natural truncation scheme that allows us to get an invariance up to an overall factor for the 4-2 and 5-1 moves, but not for the 3-3 move. The study of the divergences shows that there is a range of parameter space for which the 4-2 move is finite while the 5-1 move diverges. This opens up the possibility to recover diffeomorphism invariance in the continuum limit of Spin Foam models for 4D Quantum Gravity.

Paper Structure

This paper contains 35 sections, 4 theorems, 123 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Holomorphic Spin Foam Models
  3. BF theory and Cable Diagrams
  4. The Holomorphic Representation and Diagramatics
  5. Holomorphic Simplicity Constraints
  6. Imposing constraints
  7. DL prescription
  8. Constrained projector
  9. The Homogeneity Map
  10. The Homogeneity Map
  11. Pachner moves in 3d quantum gravity
  12. Definition of Pachner Moves
  13. Fixing the gauge
  14. The Loop Identity
  15. Alternative method
  16. ...and 20 more sections

Key Result

Theorem 3.1

Any simplicial piecewise linear manifold $\mathcal{M}$ can be transformed into any other simplicial piecewise linear manifold $\mathcal{M}'$ homeomorphic to $\mathcal{M}$ by a finite sequence of Pachner moves. For proof, see Pachner.

Figures (30)

  • Figure 1: A tetrahedron and its dual 2-complex
  • Figure 2: A tetrahedron and its cable diagram
  • Figure 3: Graph for the 4-simplex amplitude in the DL model. The contractions inside correspond to two copies of BF 20j symbols, constrained on the boundary.
  • Figure 4: Graph for the amplitude of contraction of two 4-simplices. Propagators $P_\rho^1$ and $P_\rho^2$ belong to the same edge but two different 4-simplices. The spinors belonging on the same strand but belonging to different propagators are contracted according to the strand orientation. For example, spinors $w^1_i = \check{z}^2_i$.
  • Figure 5: Two dimensional Pachner moves: a) 3--1 move, in which three triangles are merged into one by removing a vertex inside; b) 2--2 move, in which two triangles exchange the edge, along which they are glued.
  • ...and 25 more figures

Theorems & Definitions (7)

  • Theorem 3.1
  • Lemma 3.2
  • Lemma B.1
  • proof
  • Theorem C.1
  • proof
  • proof