Conformal nets I: coordinate-free nets

Bartels, Arthur; Douglas, Christopher L.; Henriques, André

doi:10.48550/arxiv.1302.2604

Cited by 9 publications

(53 citation statements)

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“…Conformal nets are a mathematical formalization of the notion of conformal field theory [5,6,7,9,10,11,13]. This paper is the fourth in a series investigating coordinate-free conformal nets and their defects [1,2,3]. The notion of coordinatefree conformal nets is a modification of the usual notion of conformal nets, in which one does not demand the positive-energy condition and does not require the existence of a vacuum vector in the vacuum sector.…”

Section: Background and Resultsmentioning

confidence: 99%

“…The notion of coordinatefree conformal nets is a modification of the usual notion of conformal nets, in which one does not demand the positive-energy condition and does not require the existence of a vacuum vector in the vacuum sector. 1 Here we will use the term 'conformal net' to refer to coordinate-free conformal nets.…”

Section: Background and Resultsmentioning

confidence: 99%

“…Theorem 1 The collections of all finite-index conformal nets, defects, sectors, and intertwiners form the objects, 1-morphisms, 2-morphisms, and 3-morphisms of a tricategory.…”

Section: Background and Resultsmentioning

confidence: 99%

“…In this case, given a subinterval I ⊆ K , we will often not distinguish between A(I) and its image in A(K). A conformal net A is an isotonic net subject to a number of axioms [1,Def 1.1]. Conformal nets form a symmetric monodial category, whose morphisms are natural transformations and whose tensor product is the tensor product of VN applied objectwise.…”

Section: Overviewmentioning

confidence: 99%

“…The two marked points are the points where the color changes. In particular, every subinterval of S 1 6 that contains at most one of the marked points (and none on the boundary) inherits a bicoloring.…”

Section: [0-2] Sectorsmentioning

confidence: 99%

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Conformal nets IV: The 3-category

Bartels,

Douglas,

Henriques

2016

Preprint

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Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. We previously introduced a notion of composition, called fusion, between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal 3-category. This 3-category encodes the algebraic structure of the possible interactions among conformal field theories. 81T05, 18D05

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Section: Background and Resultsmentioning

confidence: 99%

Section: Background and Resultsmentioning

confidence: 99%

“…Theorem 1 The collections of all finite-index conformal nets, defects, sectors, and intertwiners form the objects, 1-morphisms, 2-morphisms, and 3-morphisms of a tricategory.…”

Section: Background and Resultsmentioning

confidence: 99%

Section: Overviewmentioning

confidence: 99%

Section: [0-2] Sectorsmentioning

confidence: 99%

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Conformal nets IV: The 3-category

Bartels,

Douglas,

Henriques

2016

Preprint

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Conformal Nets II: Conformal Blocks

Bartels

Douglas²,

Henriques

2017

Commun. Math. Phys.

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Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the 'bundle of conformal blocks', a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.

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A Skeletal Model for $$2\hbox {d}$$ Conformal AQFTs

Benini

Giorgetti

Schenkel

2022

Commun. Math. Phys.

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A simple model for the localization of the category $$\mathbf {CLoc}_2$$ CLoc 2 of oriented and time-oriented globally hyperbolic conformal Lorentzian 2-manifolds at all Cauchy morphisms is constructed. This provides an equivalent description of 2-dimensional conformal algebraic quantum field theories (AQFTs) satisfying the time-slice axiom in terms of only two algebras, one for the 2-dimensional Minkowski spacetime and one for the flat cylinder, together with a suitable action of two copies of the orientation preserving embeddings of oriented 1-manifolds. The latter result is used to construct adjunctions between the categories of 2-dimensional and chiral conformal AQFTs whose right adjoints formalize and generalize Rehren’s chiral observables.

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Conformal nets I: coordinate-free nets

Cited by 9 publications

References 0 publications

Conformal nets IV: The 3-category

Conformal nets IV: The 3-category

Conformal Nets II: Conformal Blocks

A Skeletal Model for $$2\hbox {d}$$ Conformal AQFTs

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