Two-dimensional algebra in lattice gauge theory

Abstract

We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor networks, to compute the parallel transport along a surface using approximations on a lattice. Although this work is mainly intended as expository, we prove a convergence theorem for the surface transport in the continuum limit. Locality is used to define infinitesimal parallel transport and two- dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data. The correct surface ordering is dictated by two-dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice. The surface ordering is inherently complicated, but we prove a simplification theorem confirming earlier results of Schreiber and Waldorf. Assuming little background, we present a simple way to understand some abstract concepts of higher category theory. In doing so, we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory.

arXiv comment85 pages, 132 figures, updated version of a part of the author's PhD thesis

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Cited by in corpus (6)

• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• Topological non-linear $\sigma$-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems
• Generalized symmetries in singularity-free nonlinear $\sigma$ models and their disordered phases
• Stinespring's construction as an adjunction
• Noncommutative Differential K-theory