Module categories over graded fusion categories

Meir, Ehud; Musicantov, Evgeny

doi:10.1016/j.jpaa.2012.03.014

Cited by 20 publications

(28 citation statements)

References 9 publications

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“…Remark 3.8. Viewing the splitting of the exact sequence (7) as an obstruction bears many similarities to the results of [MM12]. There, the authors investigate module categories over G-graded fusion categories, and they quickly reduce their problem to studying G-equivariant module categories.…”

Section: The Splitting Obstruction and G-equivariancementioning

confidence: 92%

Spontaneous symmetry breaking from anyon condensation

Bischoff

Jones

Lü

et al. 2019

J. High Energ. Phys.

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In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a Gcrossed braided extension C ⊆ C × G , we show that physical considerations require that a connected etale algebra A ∈ C admit a G-equivariant algebra structure for symmetry to be preserved under condensation of A. Given any categorical action G → Aut br ⊗ (C) such that g(A) ∼ = A for all g ∈ G, we show there is a short exact sequence whose splittings correspond to G-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of A. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of C loc A , and gauging this symmetry commutes with anyon condensation.

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Section: The Splitting Obstruction and G-equivariancementioning

confidence: 92%

Spontaneous symmetry breaking from anyon condensation

Bischoff

Jones

Lü

et al. 2019

J. High Energ. Phys.

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“…6 At the same time this paper was completed, Meir and Musicantov posted the paper [14] containing results similar to some of ours. In this paper module categories over C are classified in terms of module categories over C e and the extension data (c, M, α) of C defined in [6].…”

mentioning

confidence: 58%

Clifford theory for graded fusion categories

Galíndo

2012

Isr. J. Math.

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We develop a categorical analogue of Clifford theory for strongly graded rings over graded fusion categories. We describe module categories over a fusion category graded by a group G as induced from module categories over fusion subcategories associated with the subgroups of G. We define invariant Ce-module categories and extensions of Ce-module categories. The construction of module categories over C is reduced to determining invariant module categories for subgroups of G and the indecomposable extensions of this modules categories. We associate a G-crossed product fusion category to each G-invariant Cemodule category and give a criterion for a graded fusion category to be a group-theoretical fusion category. We give necessary and sufficient conditions for an indecomposable module category to be extendable.1991 Mathematics Subject Classification. 16W30. 1 PreliminariesThroughout the paper we work over an algebraically closed field k of characteristic 0. All categories considered in this paper are finite, abelian, semisimple, and k-linear. All functors and bifunctors considered in this paper are additive and k-linear.

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“…Although C " UCoreppG T Y q -T YpG, χ, τ q, these categories have different associativities, so we cannot apply directly the classification of module categories from [7], Section 9, however, we will use similar reasoning. The category C is Z 2 -graded, i.e., C " C 0 ' C 1 , where C 0 -V ec G (both these categories have trivial associativities) and C 1 is generated by a single simple object U m .…”

Section: Classification Of Indecomposable Finite Dimensional G T Y -Cmentioning

confidence: 99%