scholarly journals String-net models for nonspherical pivotal fusion categories

2020 ◽  
Vol 29 (06) ◽  
pp. 2050035
Author(s):  
Ingo Runkel
Keyword(s):  
Field Theory ◽  
Marked Point ◽  
Simple Object ◽  
Fusion Category ◽  
Mapping Class ◽  
State Spaces ◽  
Spin Structures ◽  
Conformal Field ◽  
Fusion Categories ◽  
Topological Quantum

A string-net model associates a vector space to a surface in terms of graphs decorated by objects and morphisms of a pivotal fusion category modulo local relations. String-net models are usually considered for spherical fusion categories, and in this case, the vector spaces agree with the state spaces of the corresponding Turaev–Viro topological quantum field theory. In the present work, some effects of dropping the sphericality condition are investigated. In one example of nonspherical pivotal fusion categories, the string-net space counts the number of [Formula: see text]-spin structures on a surface and carries an isomorphic representation of the mapping class group. Another example concerns the string-net space of a sphere with one marked point labeled by a simple object [Formula: see text] of the Drinfeld center. This space is found to be nonzero iff [Formula: see text] is isomorphic to a nonunit simple object determined by the nonspherical pivotal structure. The last example mirrors the effect of deforming the stress tensor of a two-dimensional conformal field theory, such as in the topological twist of a supersymmetric theory.

scholarly journals Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Chao-Ming Jian ◽  
Andreas W. W. Ludwig ◽  
Zhu-Xi Luo ◽  
Hao-Yu Sun ◽  
Zhenghan Wang
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Quantum Gravity ◽  
Central Charge ◽  
Planck Scale ◽  
Mapping Class ◽  
Partition Functions ◽  
Conformal Field ◽  
Topological Quantum ◽  
Genus One

Abstract We study 3D pure Einstein quantum gravity with negative cosmological constant, in the regime where the AdS radius l is of the order of the Planck scale. Specifically, when the Brown-Henneaux central charge c = 3l/2GN (GN is the 3D Newton constant) equals c = 1/2, we establish duality between 3D gravity and 2D Ising conformal field theory by matching gravity and conformal field theory partition functions for AdS spacetimes with general asymptotic boundaries. This duality was suggested by a genus-one calculation of Castro et al. [Phys. Rev. D85 (2012) 024032]. Extension beyond genus-one requires new mathematical results based on 3D Topological Quantum Field Theory; these turn out to uniquely select the c = 1/2 theory among all those with c < 1, extending the previous results of Castro et al. Previous work suggests the reduction of the calculation of the gravity partition function to a problem of summation over the orbits of the mapping class group action on a “vacuum seed”. But whether or not the summation is well-defined for the general case was unknown before this work. Amongst all theories with Brown-Henneaux central charge c < 1, the sum is finite and unique only when c = 1/2, corresponding to a dual Ising conformal field theory on the asymptotic boundary.

scholarly journals The Relative Drinfeld Commutant of a Fusion Category and α-Induction

2018 ◽  
Vol 2019 (20) ◽  
pp. 6304-6316
Author(s):  
Yasuyuki Kawahigashi
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Fusion Category ◽  
Central Projections ◽  
Conformal Field ◽  
Fusion Categories ◽  
Relative Commutant

Abstract We establish a correspondence among simple objects of the relative commutant of a full fusion subcategory in a larger fusion category in the sense of Drinfeld, irreducible half-braidings of objects in the larger fusion category with respect to the fusion subcategory, and minimal central projections in the relative tube algebra. Based on this, we explicitly compute certain relative Drinfeld commutants of fusion categories arising from α-induction for braided subfactors. We present examples arising from chiral conformal field theory.

Skeins and mapping class groups

1994 ◽  
Vol 115 (1) ◽  
pp. 53-77 ◽  
Author(s):  
Justin Roberts
Keyword(s):  
Field Theory ◽  
Mapping Class Groups ◽  
Topological Quantum Field Theory ◽  
Unitary Representations ◽  
Mapping Class ◽  
Heegaard Splittings ◽  
Quantum Field ◽  
Class Groups ◽  
Topological Quantum

AbstractThe protective unitary representations of the mapping class groups of surfaces corresponding to the Jones–Witten topological quantum field theory for SU(2) are expressed as representations in algebras of skeins in the surface. The skein-theoretic construction of the representations uses neither Kirby's surgery theorem nor a presentation of the group. Using these representations and the Reidemeister–Singer classification of Heegaard splittings gives a proof of the existence of the moduli of the Witten invariants of 3-manifolds.

scholarly journals Galois Conjugated Tensor Fusion Categories and Nonunitary Conformal Field Theory

2020 ◽  
Vol 124 (12) ◽  
Author(s):  
Laurens Lootens ◽  
Robijn Vanhove ◽  
Jutho Haegeman ◽  
Frank Verstraete
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Conformal Field ◽  
Fusion Categories

scholarly journals State sum invariants of three manifolds from spherical multi-fusion categories

2017 ◽  
Vol 26 (14) ◽  
pp. 1750104 ◽  
Author(s):  
Shawn X. Cui ◽  
Zhenghan Wang
Keyword(s):  
Field Theories ◽  
Fusion Category ◽  
Usual Sense ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Quantum Invariants ◽  
Fusion Categories ◽  
Topological Quantum ◽  
Weakened Version ◽  
Theory Formula

We define a family of quantum invariants of closed oriented [Formula: see text]-manifolds using spherical multi-fusion categories (SMFCs). The state sum nature of this invariant leads directly to [Formula: see text]-dimensional topological quantum field theories ([Formula: see text]s), which generalize the Turaev–Viro–Barrett–Westbury ([Formula: see text]) [Formula: see text]s from spherical fusion categories. The invariant is given as a state sum over labeled triangulations, which is mostly parallel to, but richer than the [Formula: see text] approach in that here the labels live not only on [Formula: see text]-simplices but also on [Formula: see text]-simplices. It is shown that a multi-fusion category in general cannot be a spherical fusion category in the usual sense. Thus, we introduce the concept of a SMFC by imposing a weakened version of sphericity. Besides containing the [Formula: see text] theory, our construction also includes the recent higher gauge theory [Formula: see text]-[Formula: see text]s given by Kapustin and Thorngren, which was not known to have a categorical origin before.

scholarly journals Extended TQFTs via generators and relations I: The extended toric code

2020 ◽  
pp. 2050054
Author(s):  
Bruce Bartlett ◽  
Gerrit Goosen
Keyword(s):  
Class Group ◽  
Fusion Category ◽  
Mapping Class ◽  
Quantum Field Theories ◽  
Generators And Relations ◽  
3 Dimensional ◽  
Topological Quantum ◽  
Linear Maps ◽  
Phd Thesis ◽  
Elementary String

In his PhD thesis [G. Goosen, Oriented 123-tqfts via string-nets and state-sums, PhD thesis, Stellenbosch University, Stellenbosch (2018)], Goosen combined the string-net and the generators-and-relations formalisms for arbitrary once-extended 3-dimensional topological quantum field theories (TQFTs). In this paper, we work this out in detail for the simplest nontrivial example, where the underlying spherical fusion category is the category of [Formula: see text]-graded vector spaces. This allows us to give an elementary string-net description of the linear maps associated to 3-dimensional bordisms. The string-net formalism also simplifies the description of the mapping class group action in the resulting TQFT. We conclude the paper by performing some example calculations from this viewpoint.

scholarly journals ON OCNEANU'S THEORY OF ASYMPTOTIC INCLUSIONS FOR SUBFACTORS, TOPOLOGICAL QUANTUM FIELD THEORIES AND QUANTUM DOUBLES

1995 ◽  
Vol 06 (02) ◽  
pp. 205-228 ◽  
Author(s):  
DAVID E. EVANS ◽  
YASUYUKI KAWAHIGASHI
Keyword(s):  
Field Theory ◽  
Hilbert Space ◽  
Finite Depth ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Natural Basis ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Topological Quantum

A fully detailed account of Ocneanu's theorem is given that the Hilbert space associated to the two-dimensional torus in a Turaev-Viro type (2+1)-dimensional topological quantum field theory arising from a finite depth subfactor N⊂M has a natural basis labeled by certain M∞- M∞ bimodules of the asymptotic inclusion M∨(M'∩M∞)⊂M∞, and moreover that all these bimodules are given by the basic construction from M∨(M'∩M∞)⊂M∞ if the fusion graph is connected. This Hilbert space is an analogue of the space of conformal blocks in conformal field theory. It is also shown that after passing to the asymptotic inclusions we have S- and T-matrices, analogues of the Verlinde identity and Vafa's result on roots of unity. It is explained that the asymptotic inclusions can be regarded as a subfactor analogue of the quantum double construction of Drinfel'd. These claims were announced by Λ. Ocneanu in several talks, but he has not published his proofs, so details are given here along the lines outlined in his talks.

scholarly journals CFT Correlators for Cardy Bulk Fields via String-Net Models

2021 ◽  
Author(s):  
Christoph Schweigert ◽  
◽  
Yang Yang ◽  
◽  
◽  
...  
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Mapping Class Group ◽  
Class Group ◽  
Tensor Category ◽  
Geometric Method ◽  
Mapping Class ◽  
Frobenius Algebra ◽  
Conformal Field ◽  
Modular Tensor Category

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category C. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center Z(C) gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

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