Quasi hope algebras, group cohomology and orbifold models

Dijkgraaf, Robbert; Pasquier, Vincent; Roche, Ph.

doi:10.1016/0920-5632(91)90123-v

Cited by 317 publications

(444 citation statements)

References 6 publications

Supporting

Mentioning

429

Contrasting

Unclassified

Order By: Relevance

“…The underlying algebraic structure of these theories strongly resembles that of conformal field theories, in particular holomorphic orbifold models [5]. In a previous paper [6], we showed that this underlying structure is the quasi-triangular Hopf algebra D(H) [7]. This insight allowed us to give a complete description of the invariant couplings (i.e.…”

Section: Introductionmentioning

confidence: 98%

See 1 more Smart Citation

Anyons in discrete gauge theories with Chern-Simons terms

Bais¹,

Driel²,

Propitius³

1993

Nuclear Physics B

View full text Add to dashboard Cite

We study the effect of a Chern-Simons term in a theory with discrete gauge group H, which in (2+1)-dimensional space time describes (non-abelian) anyons. As in a previous paper [6], we emphasize the underlying algebraic structure, namely the Hopf algebra D(H). We argue on physical grounds that the addition of a ChernSimons term in the action leads to a non-trivial 3-cocycle on D(H). Accordingly, the physically inequivalent models are labelled by the elements of the cohomology group H 3 (H, U (1)). It depends periodically on the coefficient of the Chern-Simons term which model is realized. This establishes a relation with the discrete topological field theories of Dijkgraaf and Witten. Some representative examples are worked out explicitly.

show abstract

Section: Introductionmentioning

confidence: 98%

“…Consistency of the use of R and ϕ in arbitrary tensor products implies [7] θ g (x, y) = ω(g, x, y) ω(x, y, (xy) −1 gxy) ω(x, x −1 gx, y) (23)…”

Section: The Fact That D ω (H) Is Quasitriangular Implies Something mentioning

confidence: 99%

Anyons in discrete gauge theories with Chern-Simons terms

Bais¹,

Driel²,

Propitius³

1993

Nuclear Physics B

View full text Add to dashboard Cite

show abstract

“…94 All the topological information in these models can be described using the representation theory of the quantum doubles of finite groups. 95 Probably the simplest example which allows for non-Abelian braiding is the model based on the quantum double D͑D 3 ͒ of the smallest non-Abelian group D 3 , the symmetry group of the regular triangle, or equivalently, the permutation group of three objects. This model has been shown to allow for universal quantum computation, if some measurements are allowed as operations in addition to braiding.…”

Section: Discrete Gauge Theory and Orbifoldsmentioning

confidence: 99%

Condensate-induced transitions between topologically ordered phases

2009

View full text Add to dashboard Cite

We investigate transitions between topologically ordered phases in two spatial dimensions induced by the condensation of a bosonic quasiparticle. To this end, we formulate an extension of the theory of symmetrybreaking phase transitions which applies to phases with topological excitations described by quantum groups or modular tensor categories. This enables us to deal with phases whose quasiparticles have noninteger quantum dimensions and obey braid statistics. Many examples of such phases can be constructed from two-dimensional rational conformal field theories, and we find that there is a beautiful connection between quantum group symmetry breaking and certain well-known constructions in conformal field theory, notably the coset construction, the construction of orbifold models, and more general conformal extensions. Besides the general framework, many representative examples are worked out in detail.

show abstract

“…Although these results do not provide an explicit formula for the braiding statistics in Dijkgraaf-Witten models, they do the next best thing: they give a well defined mathematical procedure for how to compute these statistics in terms of the 3-cocycle ω that defines the model. This procedure involves a mathematical structure known as the twisted quantum double algebra 47 . The twisted quantum double formalism is quite general and can be applied to any finite group G, including nonAbelian groups.…”

Section: Review Of Braiding Statistics In 2d Dijkgraaf-witten Modelsmentioning

confidence: 99%

Topological invariants for gauge theories and symmetry-protected topological phases

2015

View full text Add to dashboard Cite

We study the braiding statistics of particle-like and loop-like excitations in 2D and 3D gauge theories with finite, Abelian gauge group. The gauge theories that we consider are obtained by gauging the symmetry of gapped, short-range entangled, lattice boson models. We define a set of quantities -called topological invariants -that summarize some of the most important parts of the braiding statistics data for these systems. Conveniently, these invariants are always Abelian phases, even if the gauge theory supports excitations with non-Abelian statistics. We compute these invariants for gauge theories obtained from the exactly soluble group cohomology models of Chen, Gu, Liu and Wen, and we derive two results. First, we find that the invariants take different values for every group cohomology model with finite, Abelian symmetry group. Second, we find that these models exhaust all possible values for the invariants in the 2D case, and we give some evidence for this in the 3D case. The first result implies that every one of these models belongs to a distinct SPT phase, while the second result suggests that these models may realize all SPT phases. These results support the group cohomology classification conjecture for SPT phases in the case where the symmetry group is finite, Abelian, and unitary.

show abstract

Quasi hope algebras, group cohomology and orbifold models

Cited by 317 publications

References 6 publications

Anyons in discrete gauge theories with Chern-Simons terms

Anyons in discrete gauge theories with Chern-Simons terms

Condensate-induced transitions between topologically ordered phases

Topological invariants for gauge theories and symmetry-protected topological phases

Contact Info

Product

Resources

About