We review how modular categories, and commutative and non-commutative
Frobenius algebras arise in rational conformal field theory. For Euclidean CFT
we use an approach based on sewing of surfaces, and in the Minkowskian case we
describe CFT by a net of operator algebras.Comment: 21 pages, contribution to proceedings for "Non-commutative Structures
  in Mathematics and Physics" (Brussels, July 2008

Kong, Liang

Runkel, Ingo

arXiv.org e-Print Archive

CiteSeerX

Caltech Authors

English

A Functional-Analytic Theory of Vertex (Operator) Algebras, I, II,

A local logarithmic conformal field theory,

A local logarithmicconformal ﬁeld theory,

Algebraic quantum ﬁeld theory,

An Algebraic approach to quantum ﬁeld theory,

AQFT from n-functorial QFT,

AQFT from n-functorialQFT,

Axiomatic conformal ﬁeld theory,

Boundary conditions, fusion rules and the Verlinde formula,

Boundary conformal fields and Tomita-Takesaki theory,

Boundary S matrix and boundary state in two-dimensional integrable quantum field theory,

Boundary S matrix and boundary state in two-dimensional integrablequantumﬁeldtheory,Int.J.Mod.Phys.A9(1994)3841–3885[Erratum-ibid.A 9(1994) 4353] [hep-th/9306002].

Canonical tensor product subfactors,

Cardy algebras and sewing constraints, I,

Cardy algebras and sewing constraints,I,

Cardy condition for open-closed field algebras,

Cardy condition for open-closed ﬁeld algebras,Commun.

Cate´gories pre´modulaires, modularisations et invariants des varie´te´s de dimension 3,

Category theory for conformal boundary conditions,

Chiral Observables and Modular Invariants,

Chiral Observables and ModularInvariants,Commun.

Classical and Quantum Conformal Field Theory,

Classification of local conformal nets:

Conformal boundary conditions and threedimensional topological field theory,

Conformal boundary conditions and threedimensionaltopologicalﬁeld theory,

Conformal correlation functions, Frobenius algebras and triangulations,

Conformal correlation functions, Frobenius algebras and triangulations,Nucl.

Conformal field theory and elliptic cohomology,

Conformal field theory,

Conformal theories and punctured surfaces,

Correspondences of ribbon categories,

D-branes and K-theory in 2D topological field theory,

Differential equations, duality and modular invariance,

Differential equations, dualityand modular invariance,

dzki, Conformal ﬁeld theory,

eres, Cat´ egories pr´ emodulaires,modularisationset invariants des vari´ et´ es de dimension 3,

Frobenius Algebras and 2D Topological Quantum Field Theories,

From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories,

From Subfactorsto Categoriesand TopologyI. Frobeniusalgebrasin and Moritaequivalence of tensor categories,

Full ﬁeld algebras,

Full ﬁeld algebras, operads and tensor categories,

How to remove the boundary in CFT - an operator algebraic procedure,

Inﬁnite conformal symmetry in twodimensional quantum ﬁeld theory,

Introduction to Vertex Operator Algebras and Their Representations,

Introduction to Vertex Operator Algebras and Their Representations, Birkh¨ auser,

Invariants of 3-manifolds via link polynomials and quantum groups,

J.Fuchs,I.Runkeland C.Schweigert,TFT constructionofRCFTcorrelators.I:Partitionfunctions,

Kriz,Conformalﬁeld theoryand ellipticcohomology,Adv.

Liouville theory revisited,

Local ﬁelds in boundary conformal

Local Quantum Physics: Fields, Particles, Algebras, 2nd edition,

Longo,Classiﬁcationof local conformal nets: Case c

Mathematical Theory of Quantum Fields,

Modular categories of types

Modular invariance for conformal full field algebras,

Modular invariance of vertex operator algebras,

Modular Structure And Duality In Conformal Quantum Field Theory,

Modularinvarianceof vertex operatoralgebras,J.

Morita classes of algebras in modular tensor categories,

Multi-interval subfactors and modularity of representations in conformal field theory,

Multi-interval subfactors and modularityof representations in conformal ﬁeld theory,

Nets of subfactors,

Noncommutative two-dimensional topological ﬁeld theories and Hurwitz numbers for real algebraic curves,

On a-induction, chiral generators and modular invariants for subfactors,Commun.

On Axiomatic Approaches to Vertex Operator Algebras and Modules,

On fusion categories,

On q-analog of McKay correspondence and ADE classification of sl(2) conformal field theories,

On q-analog of McKay correspondenceand ADE classiﬁcationof sl(2) conformal ﬁeld theories,

On the structure of open-closed topological field theory in two dimensions,

On α-induction, chiral generators and modular invariants for subfactors,

Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras,

Open-string vertex algebras, tensor categories and operads,

Operator algebras and conformal ﬁeld theory,

Operator content of two-dimensional conformal invariant theories,

Quantum field theory and the Jones polynomial,

Quantum Invariants of Knots and 3-manifolds, de Gruyter,

Quantumﬁeld theoryand theJones polynomial,Commun.

Rigidity and modularity of vertex tensor categories,

Sewing conformal ﬁeld theories I, II,

Sewing constraints for conformal ﬁeld theories on surfaces with boundaries,

Tensor products of modules for a vertex operator algebra and vertex tensor categories, in: Lie Theory and Geometry,

TFT construction of RCFT correlators. I: Partition functions,

TFT construction of RCFT correlators. V: Proof of modular invariance and factorisation,

TFT construction of RCFT correlators.V: Proof of modular invariance and factorisation,Theo.

The Analytic Geometry of Two-Dimensional Conformal Field Theory,

The Chern-Simons Theory And Knot Polynomials,

The deﬁnition of conformal ﬁeld theory, preprint 1988; also

Topological and conformal field theory as Frobenius algebras,

Topologicaland conformalﬁeld theoryas Frobenius algebras,

Two-dimensional Conformal Geometry and Vertex Operator Algebras, Birkh¨ auser,

Two-dimensional Conformal Geometry and Vertex Operator Algebras, Birkha¨user,

Uniqueness of open/closed rational CFT with given algebra of open states,

Vertex Algebras and Algebraic Curves, 2nd edition,

Algebraic Structures in Euclidean and Minkowskian Two-Dimensional
  Conformal Field Theory