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Topological Graph Polynomials in Colored Group Field Theory

Razvan Gurau

TL;DR

The paper addresses extending topological graph polynomials to open, higher-dimensional colored group field theory graphs by introducing a boundary graph $\mathcal{G}_{\partial}$ and its cellular/homology structure. It develops a generalized polynomial $P_{\mathcal{G}}(\{\beta_l\},\{x_p\},\{y_p\})$ based on active/passive lines that obey a contraction-deletion rule and recover the classical Tutte and Bollobás–Riordan polynomials in suitable limits, linking CGFT combinatorics to Euler characteristics $\chi(\mathcal{G})$ and $\chi(\mathcal{G}_{\partial})$. The approach encodes boundary topology through $\mathcal{G}_{\partial}$ and its boundary homology $H^{\partial}_q$, providing a framework for topological invariants in quantum gravity models. This work enables a principled, dimension-agnostic interface between CGFT graphs and their topological properties, suggesting avenues for further generalizations and applications.

Abstract

In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [arXiv:0907.2582]. We define the boundary graph $\cG_{\partial}$ of an open graph $\cG$ and prove it is a cellular complex. Using this structure we generalize the topological (Bollobas-Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.

Topological Graph Polynomials in Colored Group Field Theory

TL;DR

The paper addresses extending topological graph polynomials to open, higher-dimensional colored group field theory graphs by introducing a boundary graph and its cellular/homology structure. It develops a generalized polynomial based on active/passive lines that obey a contraction-deletion rule and recover the classical Tutte and Bollobás–Riordan polynomials in suitable limits, linking CGFT combinatorics to Euler characteristics and . The approach encodes boundary topology through and its boundary homology , providing a framework for topological invariants in quantum gravity models. This work enables a principled, dimension-agnostic interface between CGFT graphs and their topological properties, suggesting avenues for further generalizations and applications.

Abstract

In this paper we analyze the open Feynman graphs of the Colored Group Field Theory introduced in [arXiv:0907.2582]. We define the boundary graph of an open graph and prove it is a cellular complex. Using this structure we generalize the topological (Bollobas-Riordan) Tutte polynomials associated to (ribbon) graphs to topological polynomials adapted to Colored Group Field Theory graphs in arbitrary dimension.

Paper Structure

This paper contains 7 sections, 4 theorems, 18 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Tutte and Bollobás-Riordan polynomials
  3. Colored Group Field Theory Graphs
  4. Bubbles and cellular structure
  5. External Half Lines and the Boundary Complex
  6. Topological polynomials of GFT graphs
  7. Conclusion

Key Result

Lemma 1

For any line $l_{v_1v_2}\in {\cal L}({\cal G})$, For a graphs with no lines but with $v$ vertices $Z_{{\cal G}}(q,\emptyset)= q^v$.

Figures (7)

  • Figure 1: Ribbon vertices, ribbon lines and strands
  • Figure 2: GFT vertex and a GFT line in four dimensions.
  • Figure 3: Two vertices and a line in a colored graph.
  • Figure 4: A closed colored graph in 3 dimensions.
  • Figure 5: Tetrahedron dual to a CGFT vertex.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Definition 1: Sum over subgraphs
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 2
  • Lemma 3
  • ...and 4 more