Unoriented HQFT and its underlying algebra

Tagami, Keiji

doi:10.1016/j.topol.2011.11.057

Cited by 5 publications

(12 citation statements)

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“…We will use Proposition 2.3 to identify orientation twisted homotopy field theories with target T × BZ 2 with unoriented homotopy field theories with target T . In the non-extended setting, unoriented homotopy field theories with various targets have been studied by many authors; see [45], [44] and, when T = pt, also [24], [2], [48].…”

Section: 4mentioning

confidence: 99%

“…Definition. An orientation twisted G-equivariant topological field theory is an orientation twisted homotopy field theory Z : Ĝ-Cob π n,n−1,n−2 → C. If we restrict attention to non-extended (and pointed) theories, then we recover the equivariant unoriented topological field theories of [45], [23].…”

Section: 4mentioning

confidence: 99%

“…is an equivalence onto the non-full subcategory of TFT Unoriented Ĝ-Turaev algebras were discovered by Kapustin-Turzillo [23, §4.2], who argued along the lines of Turaev's derivation of G-Turaev algebras. See [45], [44] for earlier work in this direction. The above discussion shows that unoriented Ĝ-Turaev algebras are simply unoriented Frobenius algebras in Vect C (Grpd) whose underlying Frobenius algebra is G-Turaev.…”

Section: Of R Xmentioning

confidence: 99%

“…Our definition of T -Cob Π n,n−1,n−2 is motivated by Atiyah's oriented cobordism categories with coefficients in a double cover [3] and by mathematical approaches to unoriented Wess-Zumino-Witten theory [36] and string theory with orientifolds [9]. In the non-extended setting, various specializations of orientation twisted field theories have been studied under different names; see [45], [44], [23]. A key result of the present paper is Theorem 2.5, which associates to an n-cocycle λ ∈ Z n ( T ; U(1) T ), whose coefficients are twisted by the double cover T → T , an n-dimensional orientation twisted homotopy field theory P λ T .…”

Section: Introductionmentioning

confidence: 99%

“…2,1 (Vect C (Grpd)) spanned by unoriented Ĝ-Turaev algebras.Proof. For special Ĝ, the proposition was proved at the level of objects in[45, Theorem 3.11],[44, Proposition 3.2.7]. For the general case in the pointed setting, see[23, Proposition 1].…”

mentioning

confidence: 99%

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Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Young

2019

Commun. Math. Phys.

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Given a finite Z 2 -graded groupĜ with ungraded subgroup G and a twisted cocycleλ ∈ Z n (BĜ; U(1) π ) which restricts to λ ∈ Z n (BG; U(1)), we construct a lift of λ-twisted G-Dijkgraaf-Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientation twisted G-equivariant topological field theory.

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Section: 4mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Of R Xmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Young

2019

Commun. Math. Phys.

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Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

Shiozaki

Ryu

2017

J. High Energ. Phys.

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Matrix Product States (MPSs) provide a powerful framework to study and classify gapped quantum phases -symmetry-protected topological (SPT) phases in particular -defined in one dimensional lattices. On the other hand, it is natural to expect that gapped quantum phases in the limit of zero correlation length are described by topological quantum field theories (TFTs or TQFTs). In this paper, for (1+1)-dimensional bosonic SPT phases protected by symmetry G, we bridge their descriptions in terms of MPSs, and those in terms of G-equivariant TFTs. In particular, for various topological invariants (SPT invariants) constructed previously using MPSs, we provide derivations from the point of view of (1+1) TFTs. We also discuss the connection between boundary degrees of freedom, which appear when one introduces a physical boundary in SPT phases, and "open" TFTs, which are TFTs defined on spacetimes with boundaries.

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A Khovanov Type Invariant Derived From an Unoriented HQFT for Links in Thickened Surfaces

Tagami

2013

Int. J. Math.

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Abstract. Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented TQFT to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equivalence class. Moreover, our homology yields an invariant of links (under ambient isotopy) in the oriented I-bundle of a compact surface.

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Unoriented HQFT and its underlying algebra

Cited by 5 publications

References 7 publications

Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions

A Khovanov Type Invariant Derived From an Unoriented HQFT for Links in Thickened Surfaces

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