Abstract We consider lattice Hamiltonian realizations of (d+1)-dimensional Dijkgraaf- Witten theory. In (2+1) d, it is well-known that the Hamiltonian yields point-like excita- tions classified by irreducible representations of the twisted quantum double. This can be confirmed using a tube algebra approach. In this paper, we propose a generalisation of this strategy that is valid in any dimensions. We then apply this generalisation to derive the algebraic structure of loop-like excitations in (3+1) d, namely the twisted quantum triple. The irreducible representations of the twisted quantum triple algebra correspond to the simple loop-like excitations of the model. Similarly to its (2+1) d counterpart, the twisted quantum triple comes equipped with a compatible comultiplication map and anR-matrix that encode the fusion and the braiding statistics of the loop-like excitations, respectively. Moreover, we explain using the language of loop-groupoids how a model defined on a man- ifold that isn-times compactified can be expressed in terms of another model inn-lower dimensions. This can in turn be used to recast higher-dimensional tube algebras in terms of lower dimensional analogues.
Abstract We consider an exactly solvable model for topological phases in (3+1) d whose input data is a strict 2-group. This model, which has a higher gauge theory interpretation, provides a lattice Hamiltonian realisation of the Yetter homotopy 2-type topological quantum field theory. The Hamiltonian yields bulk flux and charge composite excitations that are either point-like or loop-like. Applying a generalised tube algebra approach, we reveal the algebraic structure underlying these excitations and derive the irreducible modules of this algebra, which in turn classify the elementary excitations of the model. As a further application of the tube algebra approach, we demonstrate that the ground state subspace of the three-torus is described by the central subalgebra of the tube algebra for torus boundary, demonstrating the ground state degeneracy is given by the number of elementary loop-like excitations.
We consider an exactly solvable model for topological phases in (3+1)d whose input data is a strict 2-group. This model, which has a higher gauge theory interpretation, provides a lattice Hamiltonian realisation of the Yetter homotopy 2-type topological quantum field theory. The Hamiltonian yields bulk flux and charge composite excitations that are either point-like or loop-like. Applying a generalised tube algebra approach, we reveal the algebraic structure underlying these excitations and derive the irreducible modules of this algebra, which in turn classify the elementary excitations of the model. As a further application of the tube algebra approach, we demonstrate that the ground state subspace of the three-torus is described by the central subalgebra of the tube algebra for torus boundary, demonstrating the ground state degeneracy is given by the number of elementary loop-like excitations.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
Given a gapped boundary of a
(3+1)-dimensional [
(3+1)D] topological order (TO), one can stack on it a decoupled
(2+1)D TO to get another boundary theory. Should one view these two boundaries as ``different''? A natural choice would be no. Different classes of gapped boundaries of
(3+1)D TO should be defined modulo the decoupled
(2+1)D TOs. But is this enough? We examine the possibility of coupling the boundary of a
(3+1)D TO to additional
(2+1)D TOs or fractonic systems, which leads to even more possibilities for gapped boundaries. Typically, the bulk pointlike excitations, when touching the boundary, become excitations in the added
(2+1)D phase, while the stringlike excitations in the bulk may end on the boundary but with end points dressed by some other excitations in the
(2+1)D phase. For a good definition of ``class'' for gapped boundaries of
(3+1)D TO, we choose to quotient out the different dressings as well. We characterize a class of gapped boundaries by the stringlike excitations that can end on the boundary, whatever their end points are. A concrete example is the
(3+1)D bosonic toric code. Using group cohomology and category theory, three gapped boundaries have been found previously: rough boundary, smooth boundary, and twisted smooth boundary. We can construct many more gapped boundaries beyond these, which all naturally fall into two classes corresponding to whether the
m-string can or cannot end on the boundary. According to this classification, the previously found three boundaries are grouped as {rough}, {smooth, twisted smooth}. For a
(3+1)D TO characterized by a finite group
G, different classes correspond to different normal subgroups of
G. We illustrate the physical picture from various perspectives including coupled layer construction, Walker-Wang model, and field theory.
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×S1 , 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.