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Encoding simplicial quantum geometry in group field theories

Daniele Oriti, Tamer Tlas

TL;DR

The paper develops an extended GFT framework that couples group and Lie algebra variables to encode simplicial geometry directly in the action. In 3d, a modified field symmetry ties the $B$-variables to the discrete connection, producing Feynman amplitudes that form a simplicial path integral with a Regge-like action and a transparent link between triads and connections; a Lorentzian version further yields causal BF-type amplitudes. In 4d, the construction generalizes but does not reproduce the full BF action, highlighting the need for simplicity constraints or a different kinetic term, and suggesting non-commutative reformulations as future avenues. Overall, the work clarifies how geometry can be embedded in GFT actions and points to concrete modifications to align GFT amplitudes with simplicial gravity in higher dimensions. The results have implications for unifying LQG/spin foams with Regge-like and causal triangulations, guiding subsequent refinements of GFT dynamics.

Abstract

We show that a new symmetry requirement on the GFT field, in the context of an extended GFT formalism, involving both Lie algebra and group elements, leads, in 3d, to Feynman amplitudes with a simplicial path integral form based on the Regge action, to a proper relation between the discrete connection and the triad vectors appearing in it, and to a much more satisfactory and transparent encoding of simplicial geometry already at the level of the GFT action.

Encoding simplicial quantum geometry in group field theories

TL;DR

The paper develops an extended GFT framework that couples group and Lie algebra variables to encode simplicial geometry directly in the action. In 3d, a modified field symmetry ties the -variables to the discrete connection, producing Feynman amplitudes that form a simplicial path integral with a Regge-like action and a transparent link between triads and connections; a Lorentzian version further yields causal BF-type amplitudes. In 4d, the construction generalizes but does not reproduce the full BF action, highlighting the need for simplicity constraints or a different kinetic term, and suggesting non-commutative reformulations as future avenues. Overall, the work clarifies how geometry can be embedded in GFT actions and points to concrete modifications to align GFT amplitudes with simplicial gravity in higher dimensions. The results have implications for unifying LQG/spin foams with Regge-like and causal triangulations, guiding subsequent refinements of GFT dynamics.

Abstract

We show that a new symmetry requirement on the GFT field, in the context of an extended GFT formalism, involving both Lie algebra and group elements, leads, in 3d, to Feynman amplitudes with a simplicial path integral form based on the Regge action, to a proper relation between the discrete connection and the triad vectors appearing in it, and to a much more satisfactory and transparent encoding of simplicial geometry already at the level of the GFT action.

Paper Structure

This paper contains 9 sections, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Encoding simplicial geometry in GFT: the 3d model
  3. A modified symmetry requirement for the GFT field
  4. GFT action and geometric meaning of the vertex function
  5. Feynman amplitudes
  6. Properties of the model
  7. Lorentzian model
  8. 4d model
  9. Conclusions

Figures (2)

  • Figure 1: This is a graphical representation of the GFT field. The group element $g$ is the parallel transport going from the centre of the triangle to the edge (as is indicated by the arrow). The Lie algebra element $B'$ is the $B$ field discretized on the edge as seen in the frame of the edge, while the element $B$ is the Lie algebra element as 'seen' from the centre of the triangle. Finally, $h$ is the group element encoding the freedom of choosing a frame at the centre of the triangle.
  • Figure 2: This is the graphical representation of the new vertex. We have depicted only one wedge of this vertex. The arrow indicate the orientation of the parallel transports. The $B$'s are best thought of as being located at the small black squares. The two relations at the top right corner are those enforced by the new vertex.