Introduction to Conformal Field Theory

Schellekens, A.N.

doi:10.1002/prop.2190440802

Cited by 25 publications

(28 citation statements)

References 47 publications

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“…This follows from a general treatment of algebra objects containing only simple currents as simple subobjects, which will be presented in a separate paper. For now we only remark that simple current theory [93,94,27,95] tells us that to obtain a modular invariant, a simple current J must be [96] in the so-called effective center , i.e. the product of its order (the smallest natural number ℓ such that J ⊗ℓ ∼ = 1) and its conformal weight must be an integer.…”

Section: Example: Free Bosonmentioning

confidence: 99%

TFT construction of RCFT correlators I: partition functions

2002

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We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore--Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A-A-bimodules. The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties. We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore--Seiberg data.

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Section: Example: Free Bosonmentioning

confidence: 99%

TFT construction of RCFT correlators I: partition functions

2002

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“…The model is critical for all points on the self-dual line with x 1 ≥ . At the point x 1 = sin(π/16) sin(3π/ 16) , called Fateev-Zamolodchikov (FZ) point, the model is fully integrable and can be described by Z 4 parafermionic CFT [11]. By an easy change of variables, see (6), one can show that the above partition function corresponds to the Hamiltonian of two coupled Ising models which decouple for x 2 = x 2 1 , so for x 1 = √ 2 − 1 we have the Ising universality class.…”

Section: Introductionmentioning

confidence: 99%

Critical domain walls in the Ashkin–Teller model

Caselle¹,

Lottini²,

Rajabpour³

2011

J. Stat. Mech.

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Abstract. We study the fractal properties of interfaces in the 2d Ashkin-Teller model. The fractal dimension of the symmetric interfaces is calculated along the critical line of the model in the interval between the Ising and the four-states Potts models. Using Schramm's formula for crossing probabilities we show that such interfaces can not be related to the simple SLE κ , except for the Ising point. The same calculation on non-symmetric interfaces is performed at the four-states Potts model: the fractal dimension is compatible with the result coming from Schramm's formula, and we expect a simple SLE κ in this case.

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“…For instance, the automorphism group of the H 4 root system is 2I × 2I; in the spinor picture, it is not surprising that 2I yields both the root system and the two factors in the automorphism group. We noted earlier that the binary polyhedral spinor groups and the ADE-type affine Lie algebras are connected via the McKay correspondence [28], for instance the binary polyhedral groups (2T, 2O, 2I) and the Lie algebras (E 6 , E 7 , E 8 ) -for these (12,18,30) is both the Coxeter number of the respective Lie algebra and the sum of the dimensions of the irreducible representation of the polyhedral group.…”

Section: φ Is Invariant Under All Reflections With Respect To the Innmentioning

confidence: 99%

“…However, the connection between (A 3 , B 3 , H 3 ) and (E 6 , E 7 , E 8 ) via Clifford spinors does not seem to be known. In particular, we note that (12,18,30) is exactly the number of roots Φ in the 3D root systems (A 3 , B 3 , H 3 ), which feeds through to the binary polyhedral groups and via the McKay correspondence all the way to the affine Lie algebras. Our construction therefore makes deep connections between trinities and puts the McKay correspondence into a wider framework, as shown in Table 2 and Figure 3.…”

Section: φ Is Invariant Under All Reflections With Respect To the Innmentioning

confidence: 99%

A 3D Spinorial View of 4D Exceptional Phenomena

Dechant

2016

Symmetries in Graphs, Maps, and Polytopes

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We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description of reflections via 'sandwiching'. This extends to a description of orthogonal transformations in general by means of 'sandwiching' with Clifford algebra multivectors, since all orthogonal transformations can be written as products of reflections by the Cartan-Dieudonné theorem. We begin by viewing the largest non-crystallographic reflection/Coxeter group H 4 as a group of rotations in two different ways -firstly via a folding from the largest exceptional group E 8 , and secondly by induction from the icosahedral group H 3 via Clifford spinors. We then generalise the second way by presenting a construction of a 4D root system from any given 3D one. This affords a new -spinorial -perspective on 4D phenomena, in particular as the induced root systems are precisely the exceptional ones in 4D, and their unusual automorphism groups are easily explained in the spinorial picture; we discuss the wider context of Platonic solids, Arnold's trinities and the McKay correspondence. The multivector groups can be used to perform concrete group-theoretic calculations, e.g. those for H 3 and E 8 , and we discuss how various representations can also be constructed in this Clifford framework; in particular, representations of quaternionic type arise very naturally.

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Introduction to Conformal Field Theory

Abstract: An elementary introduction to conformal field theory is given. Topics include free bosons and fermions, orbifolds, affine Lie algebras, coset conformal field theories, superconformal theories, correlation functions on the sphere, partition functions and modular invariance.

Cited by 25 publications

References 47 publications

TFT construction of RCFT correlators I: partition functions

TFT construction of RCFT correlators I: partition functions

Critical domain walls in the Ashkin–Teller model

A 3D Spinorial View of 4D Exceptional Phenomena

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