An Introduction to Hopf Algebra Gauge Theory

          @misc{ pirsa_PIRSA:17080001,
            doi = {10.48660/17080001},
            url = {https://pirsa.org/17080001},
            author = {Wise, Derek},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {An Introduction to Hopf Algebra Gauge Theory},
            publisher = {Perimeter Institute},
            year = {2017},
            month = {aug},
            note = {PIRSA:17080001 see, \url{https://pirsa.org}}
          }
          

Abstract

A variety of models, especially Kitaev models, quantum Chern-Simons theory, and models from 3d quantum gravity, hint at a kind of lattice gauge theory in which the gauge group is generalized to a Hopf algebra. However, until recently, no general notion of Hopf algebra gauge theory was available. In this self-contained introduction, I will cover background on lattice gauge theory and Hopf algebras, and explain our recent construction of Hopf algebra gauge theory on a ribbon graph (arXiv:1512.03966). The resulting theory parallels ordinary lattice gauge theory, generalizing its structure only as necessary to accommodate more general Hopf algebras. All of the key features of gauge theory, including gauge transformations, connections, holonomy and curvature, and observables, have Hopf algebra analogues, but with a richer structure arising from non-cocommuntativity, the key property distinguishing Hopf algebras from groups. Main results include topological invariance of algebras of observables, and a gauge theoretic derivation of algebras previously obtained in the combinatorial quantization of Chern-Simons theory.