# Hopf 2-Algebras: Homotopy Higher Symmetries in Physics

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## Date

2024-06-26

## Authors

Chen, Hank

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## Publisher

University of Waterloo

## Abstract

The theory of Hopf algebras and quantum groups have led to very rich and interesting developments in both mathematics and physics. In particular, they are known to play crucial roles in the interplay between 3d topological quantum field theories, categorical algebras, and the geometry of embedded links and tangles. Moreover, the semiclassical limits of quantum group Hopf algebras, in particular, are vital for the understanding of integrable systems in statistical mechanics and Poisson-Lie dualities in string theory. The goal of this PhD thesis is to study a higher-dimensional version of these correspondences, based on the very successful categorical ladder proposal: higher-dimensional physics and geometry is described by higher-categorical strutures. This is accomplished with the definition of a {\it higher homotopy Hopf algebra}, which can be understood as a quantization of the homotopy Lie bialgebra symmetries that have recently received attention in various fields of theoretical physics. These higher-homotopy symmetries are part of the study of the recently-popular categorical symmetries, which appear in the condensed matter literature, for instance, in relation to 1-form dipole symmetries in topologically ordered phases. However, here I will provide another physical motivation arising from the gauge theoretic perspective, which is natural in the context of the Green-Schwarz anomaly cancellation mechanism in quantum field theories. In particular, I use this perspective to prove various known structural theorems about Lie 2-bialgebras and their associated 2-graded classical $R$-matrices, as well as to provide a new definition and characterization of the so-called "quadratic 2-Casimir" elements. I will apply these higher homotopy symmetries to study the 4d 2-Chern-Simons topological quantum field theory, and to develop a notion of graded classical integrability for 2+1d bulk-boundary coupled systems. By following the philosophy of deformation quantization and the theory of $A_\infty$-algbera, I then introduce the notion of a "Hopf 2-algebra" explicitly, and prove several of their structural theorems. I will in particular derive a novel definition of a universal quantum 2-$R$-matrix and the higher-Yang-Baxter equations they satisfy. The main result of this thesis is that the 2-representation 2-category of Hopf 2-algebras is cohesively braided monoidal iff it is equipped with a universal 2-$R$-matrix, and that (weak) Hopf 2-algebras admit (weak) Lie 2-bialgebras as semiclassical limits. Finally, an application of this quantization framework will be considered, in which I will explicitly compute the higher representation theory of Drinfel'd double Hopf 2-algebras of finite groups. The corresponding 2-group Dijkgraaf-Witten topological field theories are then constructed directly from these Hopf 2-algebras, and I show that they recover the known 2-categorical characterizations of 4d $\mathbb{Z}_2$ symmetry protected topological phases of matter.

## Description

## Keywords

Topological quantum field theory, Quantum algebra, Category theory, Representation theory, Homotopy algebras, Chern-Simons theories, Topological phases of matter