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.     

Diagonalization of theXXZ Hamiltonian by vertex operators

We diagonalize the anti-ferroelectricXXZ-Hamiltonian directly in the thermodynamic limit, where the model becomes invariant under the action of . Our method is based on the representation theory of quantum affine algebras, the related vertex operators and KZ equation, and thereby bypasses the usual process of starting from a finite lattice, taking the thermodynamic limit and filling the Dirac sea. From recent results on the algebraic structure of the corner transfer matrix of the model, we obtain the vacuum vector of the Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex operators, which act as particle creation operators in the space of eigenvectors. We check the agreement of our results with those obtained using the Bethe Ansatz in a number of cases, and with others obtained in the scaling limit—thesu(2)-invariant Thirring model.Dedicated to Professors Huzihiro Araki and Noboru Nakanishi on the occasion of their sixtieth birthdays     

The Schur function realization of vertex operators

We present a realization of untwisted vertex operators in terms of operations on Schur functions. Calculations of matrix elements and traces of products of vertex operators are performed using results from the classical theory of symmetric functions. The concepts of compound, composite and supersymmetric Schur functions naturally appear in this context. Furthermore, a trace formula for a product of vertex operators turns out to be a generalization of a MacDonald identity reformulated in terms of Schur functions.     

Particle-field duality and form factors from vertex operators

Using a duality between the space of particles and the space of fields, we show how one can compute form factors directly in the space of fields. This introduces the notion of vertex operators, and form factors are vacuum expectation values of such vertex operators in the space of fields. The vertex operators can be constructed explicitly in radial quantization. Furthermore, these vertex operators can be exactly bosonized in momentum space. We develop these ideas by studying the free-fermion point of the sine-Gordon theory, and use this scheme to compute some form-factors of some non-free fields in the sine-Gordon theory. This work further clarifies earlier work of one of the authors, and extends it to include the periodic sector.     

Vacuum expectation values from fusion of vertex operators

The algebra of fused vertex operators for the ABF model is defined and studied in the free fields approach. Vacuum expectation values of local operators in the scaling theory are reproduced from the matrix elements of the fused vertex operators.     

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.     

Differential vertex operations in Lagrangian field theory

A general framework is derived for studying differential operations in renormalized perturbation theory. The method makes possible a simple, unified derivation of the renormalization group and Callan-Symanzik equations, as well as a direct test for broken symmetries (including broken scale invariance), without the necessity of defining currents and deriving their generalized Ward identites. A second-order differential equation of the Callan-Symanzik type is derived using similar methods.Supported in part by the U.S. Atomic Energy Commission under Contract No. AT-30-1-3829.     

Chiral perturbation theory: brief introduction

In four lectures I presented a brief overview of Chiral Perturbation Theory, from its basics and some key applications in the Goldstone boson sector to its extension to the baryon sector.     

Intrinsic normal-ordered vertex operators from the multiloop N-tachyon amplitude

We construct vertex operators for arbitrary mass level states of the closed bosonic string. Starting from a generalization of the Koba-Nielsen amplitude which is suitable for an arbitrary genus Riemann surface, we read the vertex operators from the residues of the poles for the intermediate states. Since the original expression is metric independent and normal ordered without the need of inventing any regularization scheme, our vertex operators also possess these properties. We discuss their general features.     

Fermionic construction of vertex operators for twisted affine algebras

We construct vertex operator representations of the twisted affine algebras in terms of fermionic (or perafermionic in some cases) elementary fields. The folding method applied to the extended Dynkin diagrams of the affine algebras allows us to determine explicitly these fermionic fields as vertex operators.     

Existence and renormalization of the Sine-Gordon field theory in all dimensions

The Euclidean field theory of the Sine-Gordon model (SG) is investigated by using methods of statistical physics. The SG model is shown to be the continuum limit of the ferromagnetic Harmonic Rotator model (HR) which itself is equivalent to a classical Coulomb plasma of unit charges on a lattice. Using our recent results for the latter models in all Euclidean dimensionsD we determine the existence of the SG field theory in terms of the HR parameters, temperaturet and magnetic fieldB. The following results are derived: 1. InD=1 the SG exists neart=B=0; the quantum theory in zero space dimensions is just the Mathieu equation as Schrödinger equation. 2. ForD>2 the SG field theory exists for allt nearB=0. The theory is actually constructed and is equivalent to a free massive scalar theory. 3. In the most interesting case ofD=2 the SG field theory exists for allt<t=8 nearB=0, it does not exist fortt. All necessary renormalizations are performed and all necessary subtractions are obtained in closed form which proves that the theory is superrenormalizable when it exists. We also discuss the relations between the structure of the particle spectrum of the SG, the phase transitions of the HR, and the binding properties of the classical Coulomb charges.     

Construction of singular vertex operators as degenerate primary conformal fields

A general construction of null fields is given through the use of a vertex operator representation of conformal quantum fields. Singular vertex operators are constructed as possessing the conformal properties of a degenerate primary field. A systematic approach to obtaining the partial differential equations for correlation functions is suggested.     

Semiclassical correlation functions of Wilson loops and local vertex operators

We analyze correlation functions of Wilson loop observables and local vertex operators within the strong-coupling regime of the AdS/CFT correspondence. When the local operator corresponds to a light string state with finite conserved charges the correlation function can be evaluated in the semiclassical approximation of large string tension, where the contribution from the light vertex can be neglected. We consider the cases where the Wilson loops are described by two concentric surfaces and the local vertices are the superconformal chiral primary scalar or a singlet massive scalar operator.