Generalization of the Bollobás-Riordan polynomial for tensor graphs

Abstract

Tensor models are used nowadays for implementing a fundamental theory of quantum gravity. We define here a polynomial $\mathcal{T}$ encoding the supplementary topological information. This polynomial is a natural generalization of the Bollobás-Riordan polynomial (used to characterize matrix graphs) and is different of the Gur\uau polynomial, (R. Gur\uau, "Topological Graph Polynomials in Colored Group Field Theory", Annales Henri Poincare {\bf 11}, 565-584 (2010)) defined for a particular class of tensor graphs, the colorable ones. The polynomial $\mathcal{T}$ is defined for both colorable and non-colorable graphs and it is proved to satisfy the contraction/deletion relation. A non-trivial example of a non-colorable graphs is analyzed.

22 pages, 20 figures

References in corpus (16)

• The massless higher-loop two-point function
• Group field theory renormalization - the 3d case: power counting of divergences
• Scaling behaviour of three-dimensional group field theory
• A translation-invariant renormalizable non-commutative scalar model
• EPRL/FK Group Field Theory
• Lost in Translation: Topological Singularities in Group Field Theory
• Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models
• Parametric Representation of Noncommutative Field Theory
• The multivariate signed Bollobas-Riordan polynomial
• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• Parametric representation of "critical" noncommutative QFT models
• Parametric representation of a translation-invariant renormalizable noncommutative model
• Graph polynomials and their applications I: The Tutte polynomial
• Comment on `Lost in Translation: Topological Singularities in Group Field Theory'
• Graph polynomials and their applications II: Interrelations and interpretations
• A recipe theorem for the topological Tutte polynomial of Bollobas and Riordan

Cited by in corpus (9)

• Colored Tensor Models - a Review
• A generalization of the Virasoro algebra to arbitrary dimensions
• The Tensor Track, III
• Multi-orientable Group Field Theory
• Some combinatorial aspects of quantum field theory
• Extending the Tutte and Bollobás-Riordan Polynomials to Rank 3 Weakly-Colored Stranded Graphs
• Classical Group Field Theory
• Polynomial Invariants for Arbitrary Rank $D$ Weakly-Colored Stranded Graphs
• Universality for polynomial invariants on ribbon graphs with half-ribbons