Tensors, State-Sum Topological Field Theories and the Evaluation of Extruded Graphs

Explore a comprehensive lecture on the intersection of tensor networks, topological field theories, and graph evaluations. Delve into the geometric realization of two-dimensional tensor network models through three-dimensional topological field theories with boundaries. Examine how tensor entries in PEPS and MPO arise from evaluations of tetrahedral graphs on spheres, representing mixed 6j-symbols. Investigate the extension of graph evaluations on spheres to include surface defects and bulk Wilson lines, leading to a three-dimensional calculus with tensors computable through traces. This talk, part of the Thematic Programme on "Tensor Networks: Mathematical Structures and Novel Algorithms" at the Erwin Schrödinger International Institute, offers deep insights into the mathematical structures underlying tensor networks and their applications in physics.