Master thesis: Generation and algebra of strand graphs

Feynman diagrams are an interesting intersection point between combinatorics and graph theory, algebra, as well as quantum field theory. They are graphs which label the perturbative expansion of amplitudes of local quantum field theories at weak coupling. Such expansions are thus generating functions of certain classes of graphs. In realistic quantum field theories it is however necessary to renormalize divergent amplitudes of given graphs taking into account all divergent subgraphs; underlying is the mathematical structure of the Connes-Kreimer Hopf algebra of divergent graphs. These structures generalize to the bigger class of “2-graphs” or “strand graphs” as generated in combinatorially non-local field theories [arXiv:2102.12453].

To improve the understanding of such combinatorial and algebraic structures and to allow for explicit calculations, computer algebra will be useful for generating such graphs and applying the coproduct. In the case of local field theories this has been implemented in arXiv:1402.2613 on the basis of the graph generating package nauty [arXiv:1301.1493]. The main objective of this master project is to generalize these algorithms to strand graphs. Basic coding skills are necessary to this aim. Further focus may be either on the mathematical, physical or computer-science aspects of the project according to the interest of the student.

If interested please contact Johannes Thürigen for further information!