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From gauge to higher gauge models of topological phases

Clement Delcamp, Apoorv Tiwari

TL;DR

This work develops a unified framework for topological phases in (3+1)d by linking exactly solvable lattice models to higher gauge theory based on 2-groups. By interpreting Pachner move coherence through the pentagonator of a monoidal 2-category and then weakening axioms to pass to a weak 2-category, the authors construct 2-group–based state-sum models and their lattice realizations. They show how bosonic SPTs protected by higher-form and 2-group symmetries arise, and how gauging these symmetries yields higher-form topological gauge theories and 2-group DW-type theories; they also analyze how ’t Hooft anomalies of such higher symmetries are captured by bulk SPT inflow. The results provide a cohesive categorical and gauge-theoretic perspective that subsumes DW theories, higher-form theories, and higher-symmetry SPTs, with significant implications for classifying and realizing gapped bosonic phases in four dimensions. Overall, the paper advances a systematic program to harness higher categories and higher gauge structures for constructing and diagnosing topological phases and their boundary anomalies in (3+1)d systems.

Abstract

We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and 't Hooft anomalies in the context of these higher group symmetries.

From gauge to higher gauge models of topological phases

TL;DR

This work develops a unified framework for topological phases in (3+1)d by linking exactly solvable lattice models to higher gauge theory based on 2-groups. By interpreting Pachner move coherence through the pentagonator of a monoidal 2-category and then weakening axioms to pass to a weak 2-category, the authors construct 2-group–based state-sum models and their lattice realizations. They show how bosonic SPTs protected by higher-form and 2-group symmetries arise, and how gauging these symmetries yields higher-form topological gauge theories and 2-group DW-type theories; they also analyze how ’t Hooft anomalies of such higher symmetries are captured by bulk SPT inflow. The results provide a cohesive categorical and gauge-theoretic perspective that subsumes DW theories, higher-form theories, and higher-symmetry SPTs, with significant implications for classifying and realizing gapped bosonic phases in four dimensions. Overall, the paper advances a systematic program to harness higher categories and higher gauge structures for constructing and diagnosing topological phases and their boundary anomalies in (3+1)d systems.

Abstract

We consider exactly solvable models in (3+1)d whose ground states are described by topological lattice gauge theories. Using simplicial arguments, we emphasize how the consistency condition of the unitary map performing a local change of triangulation is equivalent to the coherence relation of the pentagonator 2-morphism of a monoidal 2-category. By weakening some axioms of such 2-category, we obtain a cohomological model whose underlying 1-category is a 2-group. Topological models from 2-groups together with their lattice realization are then studied from a higher gauge theory point of view. Symmetry protected topological phases protected by higher symmetry structures are explicitly constructed, and the gauging procedure which yields the corresponding topological gauge theories is discussed in detail. We finally study the correspondence between symmetry protected topological phases and 't Hooft anomalies in the context of these higher group symmetries.

Paper Structure

This paper contains 35 sections, 197 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of (2+1)d string net models
  3. Fixed point wave functions
  4. Pentagon relation
  5. Triangle relations
  6. Categorical aspects
  7. State-sum invariant and lattice Hamiltonian
  8. (3+1)d string net models
  9. Fixed point wave functions
  10. 2-categorical aspects
  11. Generalizations
  12. Higher gauge theory models
  13. 2-connections and strict 2-groups
  14. Weak 2-groups
  15. Topological gauge theories from groups and group-like generalizations
  16. ...and 20 more sections

Figures (1)

  • Figure 1: Graphical depiction of the fifth Stasheff polytope. Each vertex of this three-dimensional polytope is associated with a triangulation of a $6$-gon while every edge represents a 2--2 Pachner move. The combinatorics of this object is exactly the same as the one of the coherence relation of the pentagonator which is one of the structural 2-morphisms entering the definition of the monoidal 2-category we are interested in.