Construction Of Conformal Field Theory Research Articles

Braiding Fibonacci anyons

Fibonacci anyons ε provide the simplest possible model of non-Abelian fusion rules: [1] × [1] = [0] ⊕ [1]. We propose a conformal field theory construction of topological quantum registers based on Fibonacci anyons realized as quasiparticle excitations in the ℤ3 parafermion fractional quantum Hall state. To this end, the results of Ardonne and Schoutens for the correlation function of four Fibonacci fields are extended to the case of arbitrary number n of quasi-holes and N = 3r electrons. Special attention is paid to the braiding properties of the obtained correlators. We explain in details the construction of a monodromy representation of the Artin braid group B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{B} $$\\end{document}n acting on n-point conformal blocks of Fibonacci anyons. The matrices of braid group generators are displayed explicitly for all n ≤ 8. A simple recursion formula makes it possible to extend without efforts the construction to any n. Finally, we construct N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} qubit computational spaces in terms of conformal blocks of 2N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ 2\\mathcal{N} $$\\end{document} + 2 Fibonacci anyons.

On conformal field theories based on Takiff superalgebras

We revisit the construction of conformal field theories based on Takiff algebras and superalgebras that was introduced by Babichenko and Ridout. Takiff superalgebras can be thought of as truncated current superalgebras with -grading which arise from taking p copies of a Lie superalgebra and placing them in the degrees . Using suitably defined non-degenerate invariant forms we show that Takiff superalgebras give rise to families of conformal field theories with central charge . The resulting conformal field theories are defined in the standard way, i.e. they lend themselves to a Lagrangian description in terms of a WZW model and their chiral energy momentum tensor is the one obtained naturally from the usual Sugawara construction. In view of their intricate representation theory they provide interesting examples of conformal field theories.

Conformal field theory construction for non-Abelian hierarchy wave functions

The fractional quantum Hall effect is the paradigmatic example of topologically ordered phases. One of its most fascinating aspects is the large variety of different topological orders that may be realized, in particular nonabelian ones. Here we analyze a class of nonabelian fractional quantum Hall model states which are generalizations of the abelian Haldane-Halperin hierarchy. We derive their topological properties and show that the quasiparticles obey nonabelian fusion rules of type su(q)_k. For a subset of these states we are able to derive the conformal field theory description that makes the topological properties - in particular braiding - of the state manifest. The model states we study provide explicit wave functions for a large variety of interesting topological orders, which may be relevant for certain fractional quantum Hall states observed in the first excited Landau level.

From Luttinger liquid to non-Abelian quantum Hall states

We formulate a theory of non-Abelian fractional quantum Hall states by considering an anisotropic system consisting of coupled, interacting one dimensional wires. We show that Abelian bosonization provides a simple framework for characterizing the Moore Read state, as well as the more general Read Rezayi sequence. This coupled wire construction provides a solvable Hamiltonian formulated in terms of electronic degrees of freedom, and provides a direct route to characterizing the quasiparticles and edge states in terms of conformal field theory. This construction leads to a simple interpretation of the coset construction of conformal field theory, which is a powerful method for describing non Abelian states. In the present context, the coset construction arises when the original chiral modes are fractionalized into coset sectors, and the different sectors acquire energy gaps due to coupling in "different directions". The coupled wire construction can also can be used to describe anisotropic lattice systems, and provides a starting point for models of fractional and non-Abelian Chern insulators. This paper also includes an extended introduction to the coupled wire construction for Abelian quantum Hall states, which was introduced earlier.

Domain Walls, Fusion Rules, and Conformal Field Theory in the Quantum Hall Regime

We provide a simple way to obtain the fusion rules associated with elementary quasiholes over quantum Hall wave functions, in terms of domain walls. The knowledge of the fusion rules is helpful in the identification of the underlying conformal field theory describing the wave functions. We show that, for a certain two-parameter family (k,r) of wave functions, the fusion rules are those of su(r)k. In addition, we give an explicit conformal field theory construction of these states, based on the Mk(k+1,k+r) "minimal" theories. For r=2, these states reduce to the Read-Rezayi states. The "Gaffnian" wave function is the prototypical example for r>2, in which case the conformal field theory is nonunitary.

Generalised N=2 permutation branes

Generalised permutation branes in products of N=2 minimal models play an important role in accounting for all RR charges of Gepner models. In this paper an explicit conformal field theory construction of these generalised permutation branes for one simple class of examples is given. We also comment on how this may be generalised to the other cases.

QUASISYMMETRIC SEWING IN RIGGED TEICHMÜLLER SPACE

One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space".By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmüller space of a genus-g surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them; (2) With this complex structure, the sewing operation is holomorphic; (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent.These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.

The Partition Function of the Two-Dimensional Black Hole Conformal Field Theory

We compute the partition function of the conformal field theory on the two-dimensional euclidean black hole background using path-integral techniques. We show that the resulting spectrum is consistent with the algebraic expectations for the SL(2,R)/U(1) coset conformal field theory construction. In particular, we find confirmation for the bound on the spin of the discrete representations and we determine the density of the continuous representations. We point out the relevance of the partition function to all string theory backgrounds that include an SL(2,R)/U(1) coset factor.

A Note on a Proposal for the Tachyon State in Vacuum String Field Theory

We discuss the proposal of Hata and Kawano for the tachyon fluctuation around a solution of vacuum string field theory representing a D25 brane. We give a conformal field theory construction of their state -- a local insertion of a tachyon vertex operator on the sliver surface state, and explain why the on-shell momentum condition emerges correctly. We also show that a naive computation of the D25-brane tension using data for the three point coupling of this state gives an answer that is $(\pi^2/3)(16/27\ln 2)^3 \simeq 2.0558$ times the expected answer. We demonstrate that this problem arises because the HK state does not satisfy the equations of motion in a strong sense required for the computation of D-brane tension from the on-shell 3-tachyon coupling.

A general construction of conformal field theories from scalar anti-de Sitter quantum field theories

We provide a new general setting for scalar interacting fields on the covering of a ( d+1 )-dimensional AdS spacetime. The formalism is used at first to construct a one-parameter family of field theories, each living on a corresponding spacetime submanifold of AdS, which is a cylinder R×Sd−1 . We then introduce a limiting procedure which directly produces Luscher–Mack CFT's on the covering of the AdS asymptotic cone. Our generalized AdS → CFT construction is nonperturbative, and is illustrated by a complete treatment of two-point functions, the case of Klein–Gordon fields appearing as particularly simple in our context. We also show how the Minkowskian representation of these boundary CFT's can be directly generated by an alternative limiting procedure involving Minkowskian theories in horocyclic sections (nowadays called ( d− 1)-branes, 3-branes for AdS5 ). These theories are restrictions to the brane of the ambient AdS field theory considered. This provides a more general correspondence between the AdS field theory and a Poincare invariant QFT on the brane, satisfying all the Wightman axioms. The case of two-point functions is again studied in detail from this viewpoint as well as the CFT limit on the boundary.

Global vertex operators on Riemann surfaces

We develop an approach towards construction of conformal field theory starting from the basic axioms of vertex operator algebras.

Poincar� polynomials and level rank dualities in the N=2 coset construction

We review the coset construction of conformal field theories; the emphasis is on the construction of the Hilbert spaces for these models, especially if fixed points occur. This is applied to the $N=2$ superconformal cosets constructed by Kazama and Suzuki. To calculate heterotic string spectra we reformulate the Gepner con- struction in terms of simple currents and introduce the so-called extended Poincar\'e polynomial. We finally comment on the various equivalences arising between models of this class, which can be expressed as level rank dualities. (Invited talk given at the III. International Conference on Mathematical Physics, String Theory and Quantum Gravity, Alushta, Ukraine, June 1993. To appear in Theor. Math. Phys.)

Sewing Construction in KN Operator Formalism of Conformal Field Theory11The project supported in part by NSF of Educational Commission of Henan.

It is shown that the KN operator formalism for a nonchiral bosonic system can be extended to higher genus Riemann surfaces with punctures by using the construction of conformal field theories. This completes the full formulation of the KN opera tor formalism and preparation of describing bosonic string scattering amplitudes. We also elaborate on the sewing prescription further.

Symmetries of integrable hierarchies and matrix model constraints

The orbit construction associates a soliton hierarchy to every level-one vertex realization of a simply laced affine Kac-Moody algebra g. We show that the τ-function of such a hierarchy has the (truncated) Virasoro algebra as an algebra of infinitesimal symmetry transformations. To prove this we use an appropriate bilinear form of these hierarchies together with the coset construction of conformal field theory. For A l (1) the orbit construction gives either the Toda or the KdV hierarchy. These both occur in the one-matrix model of two-dimensional quantum gravity, before and after the double scaling limit respectively. The truncated Virasoro symmetry algebra is exactly the algebra of constraints of the one-matrix model. The partition function of the one-matrix model is therefore an invariant τ-function. We also consider the case of A l (1) with l > 1. Surprisingly, the symmetry algebra in that case is not simply a truncated Casimir algebra. It appears that again only the Virasoro symmetry survives. We speculate on the relation with multi-matrix models.

Free boson realization of vector model scaling limits

It is shown that the theories in the double scaling limits of the O( N) vector models are closely related to the Feigin-Fuchs construction of conformal field theories. In particular the partition functions are written in terms of the charge screening operators. The boson is related to collective fields of the vector models.

Non-critical orbifolds

We generalize the notion of orbifold to statistical mechanical models away from their critical points. In the simplest models, our non-critical orbifold construction reduces to Kramers-Wannier duality. More generally, it gives a new model whose toroidal partition function can be expressed in terms of that of the original model. The construction also preserves the star-triangle relations, so that the “orbifold” of an exactly solvable model remains exactly solvable. At a critical point, our construction reduces to the conventional orbifold construction of conformal field theory.

A new approach to the construction of conformal field theories

We propose a generalization of Coulomb gas approach to rational conformal field theories to higher numbers of bosons. This is achieved by letting the screening charges live on rational lattices and not just on lattices associated with Lie groups. The characters are constructed using a certain “Weyl” group associated with these rational lattices and they form finite-dimensional representations of the modular group. We further propose a method to construct correlation functions. We study in particular a few examples for the 2-boson case and obtain a bosonic realization of the spin- 4 3 parafermionic series. Several non-trivial checks are also made in order to show that our prescription for computing correlation functions indeed gives correct results.

Cohomology and the operator formalism

This letter points a close parallel between the operator formalism for string theory and the action of a Lie algebra on a differential complex. The construction of conformal field theories can then be regarded as a cohomology problem; we suggest that this viewpoint may survive the generalization beyond finite genus Riemann surfaces.

Extended current algebras and the coset construction of conformal field theories

We investigate extended algebras associated with coset construction. We associate a Verlinde type algebra A with the branching functions. Each allowed choice of a subalgebra B corresponds to an extended algebra whose classes of currents are in one to one correspondence with the elements of B. The subalgebra B encodes the fusion rules of the currents. The elements of A/B are in one to one correspondence with the primary fields. Each coset element corresponds to one primary field and its descendants. The fusion rules of the operator algebra are encoded in A.

GN ⊗ GL/GN+L CONFORMAL FIELD THEORIES AND THEIR MODULAR INVARIANT PARTITION FUNCTIONS

We study a Feigin-Fuchs construction of conformal field theories based on a G ⊗ G/G coset space, in terms of screened bosons and parafermions. This allows us to get the formula for the conformal dimensions of primary operators. Lists of modular invariant partition functions for the SU(3), SO(5) and G2 Wess-Zumino-Witten models are given. Besides the principal series of diagonal invariants, a complementary series exists for SU(3) and SO(5), which is due to the outer automorphism of the Kac-Moody algebra. Moreover, exceptional solutions appear at levels 5, 9, 21 for SU(3), at levels 3, 7, 12 for SO(5) and at levels 3, 4 for G2. From these modular invariants, those for the corresponding GN ⊗ GL/GN+L models are constructed.