On the refined counting of graphs on surfaces

Abstract

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with S_d gauge group which gives them a topological membrane interpretation.

57 pages, 12 figures; v2: Typos corrected; references added

References in corpus (6)

• Dimer models and toric diagrams
• Brane Tilings and Their Applications
• Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes
• From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group
• Structure of the MHV-rules Lagrangian
• What is the Simplest Gauge-String Duality?

Cited by in corpus (12)

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• Flavour singlets in gauge theory as Permutations
• Consistency and Derangements in Brane Tilings
• Counting surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group with motivations from string theory and QFT
• Topological String Correlators from Matrix Models
• Permutation invariant Gaussian 2-matrix models
• Permutation combinatorics of worldsheet moduli space
• Holographic Hierarchy in the Gaussian Matrix Model via the Fuzzy Sphere
• Interactions as conformal intertwiners in 4D QFT
• Spectral curves for hypergeometric Hurwitz numbers
• Enumeration of N-rooted maps using quantum field theory