Crossing with the circle in Dijkgraaf–Witten theory and applications to topological phases of matter

  title={Crossing with the circle in Dijkgraaf–Witten theory and applications to topological phases of matter},
  author={Alex Bullivant and Clement Delcamp},
  journal={Journal of Mathematical Physics},
Given a fully extended topological quantum field theory, the “crossing with the circle” conditions establish that the dimension, or categorification thereof, of the quantum invariant assigned to a closed k-manifold Σ is equivalent to that assigned to the ( k + 1)-manifold [Formula: see text]. We compute in this paper these conditions for the 4-3-2-1 Dijkgraaf–Witten theory. In the context of the lattice Hamiltonian realization of the theory, the quantum invariants assigned to the circle and the… 
2 Citations

Dualities in one-dimensional quantum lattice models: topological sectors

It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Building on ref. [arXiv:2112.09091], whereby dualities are defined

Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

  • C. Delcamp
  • Mathematics
    Journal of High Energy Physics
  • 2022
Abstract Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group G, we consider a family of tensor network representations of its ground state subspace. This family



Tube algebras, excitations statistics and compactification in gauge models of topological phases

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

Topological quasiparticles and the holographic bulk-edge relation in (2+1) -dimensional string-net models

String-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wavefunction, which is

State sum invariants of 3 manifolds and quantum 6j symbols

Excitation basis for (3+1)d topological phases

A bstractWe 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

On unitary 2-representations of finite groups and topological quantum field theory

This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

Abstract We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped boundaries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk dyonic

Higher algebraic structures and quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons

String-net condensation: A physical mechanism for topological phases

We show that quantum systems of extended objects naturally give rise to a large class of exotic phases---namely topological phases. These phases occur when extended objects, called ``string-nets,''

Universal topological data for gapped quantum liquids in three dimensions and fusion algebra for non-Abelian string excitations

Recently we conjectured that a certain set of universal topological quantities characterize topological order in any dimension. Those quantities can be extracted from the universal overlap of the