Excitation basis for (3+1)d topological phases

TL;DR

This work develops a (3+1)d generalization of the Kitaev quantum double by introducing the quantum triple T(G), a trialgebra that extends Ocneanu's tube algebra and the Drinfel'd double. Excitations are labeled by two fluxes and a charge, with a fusion basis constructed from the representation theory of T(G) and organized via manifolds of the form Σ × S^1 to mirror dimensional reduction from 3+1d to 2+1d orders. The construction proceeds from a BF-based lattice gauge theory with finite group G, via the three-cylinder algebra and its gluing rules, to a full excitation basis on three-dimensional manifolds bounded by two tori. The results provide a concrete algebraic framework for (3+1)d topological phases and suggest paths toward twisted (Dijkgraaf–Witten) generalizations and operator constructions that diagonalize a higher-dimensional analogue of ribbon operators.

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 $Σ\times \mathbb{S}_1$, with $Σ$ 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.