arXiv subjects

Jürgen Fuchs

Publications and source records attributed to Jürgen Fuchs.

44 records · Page 3Linked to original sources

arXiv2003

Lie algebras, Fuchsian differential equations and CFT correlation functions

Affine Kac-Moody algebras give rise to interesting systems of differential equations, so-called Knizhnik-Zamolodchikov equations. The monodromy properties of their solutions can be encoded in the structure of a modular tensor category on (a subcategory of) the representation category of the affine Lie algebra. We discuss the relation between these solutions and physical correlation functions in two-dimensional conformal field theory. In particular we report on a proof for the existence of the latter on world sheets of arbitrary topology.

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arXiv2002

TFT construction of RCFT correlators I: Partition functions

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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arXiv2000

Correlation functions and boundary conditions in RCFT and three-dimensional topology

We give a general construction of correlation functions in rational conformal field theory on a possibly non-orientable surface with boundary in terms of 3-dimensional topological quantum field theory. The construction applies to any modular category. It is proved that these correlation functions obey modular and factorization rules. Structure constants are calculated and expressed in terms of the data of the modular category.

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arXiv1994

Modular invariants and fusion rule automorphisms from Galois theory

We show that Galois theory of cyclotomic number fields provides a powerful tool to construct systematically integer-valued matrices commuting with the modular matrix S, as well as automorphisms of the fusion rules. Both of these prescriptions allow the construction of modular invariants and offer new insight in the structure of known exceptional invariants.

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arXiv1994

Towards a classification of rational Hopf algebras

Rational Hopf algebras (certain quasitriangular weak quasi-Hopf $^*$-algebras) are expected to describe the quantum symmetry of rational field theories. In this paper methods are developped which allow for a classification of all rational Hopf algebras that are compatible with some prescribed set of fusion rules. The algebras are parametrized by the solutions of the square, pentagon and hexagon identities. As examples, we classify all solutions for fusion rules with two or three sectors, and for the level three affine $A_1$ fusion rules. We also establish several general properties of rational Hopf algebras, and we present a graphical description of the coassociator in terms of labelled tetrahedra which allows to make contact with conformal field theory fusing matrices and with invariants of three-manifolds and topological lattice field theory.

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arXiv1993

The Quantum Symmetry of Rational Field Theories

The quantum symmetry of a rational quantum field theory is a finite- dimensional multi-matrix algebra. Its representation category, which determines the fusion rules and braid group representations of superselection sectors, is a braided monoidal C^*-category. Various properties of such algebraic structures are described, and some ideas concerning the classification programme are outlined. (Invited talk given at the III. International Conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June 1993. To appear in Teor.Mat.Fiz.)

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arXiv1993

Non-Hermitian Symmetric N=2 Coset Models, Poincare Polynomials, and String Compactification

The field identification problem, including fixed point resolution, is solved for the non-hermitian symmetric $N=2$ superconformal coset theories. Thereby these models are finally identified as well-defined modular invariant CFTs. As an application, the theories are used as subtheories in tensor products with $c=9$, which in turn are taken as the inner sector of heterotic superstring compactifications. All string theories of this type are classified, and the chiral ring as well as the number of massless generations and anti-generations are computed with the help of the extended Poincare polynomial. Several equivalences between a priori different non-hermitian cosets show up; in particular there is a level-rank duality for an infinite series based on $C$ type Lie algebras. Further, some general results for generic $N=2$ cosets are proven: a simple formula for the number of identification currents is found, and it is shown that the set of Ramond ground states of any $N=2$ coset model is invariant under charge conjugation.

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arXiv1992

On the Landau-Ginzburg description of $(A_1^{(1)})^{\oplus N}$ invariants

We search for a \Lg\ interpretation of non-diagonal modular invariants of tensor products of minimal $n=2$ superconformal models, looking in particular at automorphism invariants and at some exceptional cases. For the former we find a simple description as \lgo s, which reproduce the correct chiral rings as well as the spectra of various Gepner--type models and orbifolds thereof. On the other hand, we are able to prove for one of the exceptional cases that this conformal field theory cannot be described by an orbifold of a \Lg\ model with respect to a manifest linear symmetry of its potential.

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