Excitation basis for (3+1)d topological phases

TL;DR

This paper introduces an exactly solvable (3+1)d lattice model based on finite groups, generalizing Kitaev's quantum double, with a new algebraic structure for excitations and a basis for gauge models related to dimensional reduction.

Contribution

It develops a higher-dimensional extension of the tube algebra and constructs a basis for (3+1)d gauge models using fusion of defect excitations, linking to (2+1)d topological orders.

Findings

Defines a higher-dimensional tube algebra extending Drinfel'd double.

Characterizes torus-excitations via irreducible representations labeled by fluxes and charges.

Provides a basis for (3+1)d gauge models related to dimensional reduction from 3+1d to 2+1d.

Abstract

We consider an exactly solvable model in 3+1 dimensions, based on a finite group, which is a natural generalization of Kitaev's quantum double model. The corresponding lattice Hamiltonian yields excitations located at torus-boundaries. By cutting open the three-torus, we obtain a manifold bounded by two tori which supports states satisfying a higher-dimensional version of Ocneanu's tube algebra. This defines an algebraic structure extending the Drinfel'd double. Its irreducible representations, labeled by two fluxes and one charge, characterize the torus-excitations. The tensor product of such representations is introduced in order to construct a basis for (3+1)d gauge models which relies upon the fusion of the defect excitations. This basis is defined on manifolds of the form $\Sigma \times \mathbb{S}_1$, with $\Sigma$ a two-dimensional Riemann surface. As such, our construction is closely related to dimensional reduction from (3+1)d to (2+1)d topological orders.