Construction Of Field Theory Research Articles

Anyon Field Theory and Fractional Quantum Hall States

We analyze the fractional quantum Hall effect by using an anyon field theory defined perturbatively in terms of a boson field. It is shown explicitly that there exist fractional quantum Hall (FQH) states which are condensed phases of bosonized electrons at zero momentum. When the Coulomb interaction is switched off, these states are degenerate with other states which are obtained by solving a certain self-dual equation. Taking into account the electron spin degrees of freedom we also show the existence of spin-singlet FQH states at the half-filling (ν = 1/2) and find vortex soliton excitations carrying the electric charge e/4. Furthermore, by using singular gauge transformations, we construct field theories of vortex solitons and derive the hierarchy of hte FQH states.

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.

Heterotic conformal field theory and Gepner's construction.

We discuss some general properties of heterotic conformal field theory in which conformal anomalies {ital c} are different for the left-moving and right-moving sectors. It is precisely this type of theory that can be applied immediately to the construction of heterotic string theory. We discuss a general way of constructing such a theory using free fermions. The construction is then applied to generalize Gepner's construction of superstring solutions using the tensor products of {ital N}=2 superconformal field theories.

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 path integral construction of superconformal field theories from a gauged supersymmetric Wess-Zumino-Witten action

A path integral construction of superconformal field theories is presented based on gauged supersymmetric Wess-Zumino-Witten actions. The conformal charge of the model is identical to that of a GKO construction. Ghost modes appear in the formalism as a result of gauge fixing. As a consequence, the holomorphic stress-tensor T( z) and superconformal generator G( z) obtained in the construction have ghost contributions. Weak operator first-class constraints restrict the theory to physical states which satisfy the constraints. The superconformal algebra closes when evaluated between ghost-free physical states.

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.

The Φ4 equations of motion. a new method for the construction of a Wightman quantum field theory

A new method for the construction of a nontrivial Φ 4 model consistent with the general principles of a Wightman Quantum Field Theory is proposed. The renormalized equations of motion for the connected Green's functions in the Euclidean momentum space are considered. Using a fixed-point method in an appropriate Banach space, the existence of a unique nontrivial solution of these equations is proved when the coupling constant is fixed positive and smaller than a finite value.

Toward construction of string field theory

Attempts to construct string field theory for closed string is given covering use of gauge techniques of BRST and extended BFV Hamiltonian approach.

Supersymmetric embedding of a model of a quantum harmonic oscillator interacting with infinitely many bosons

A mathematically rigorous analysis is given on supersymmetric embedding of a model of a one-dimensional quantum harmonic oscillator interacting with infinitely many bosons moving in the s-dimensional space Rs. The model is exactly soluble. By a rigorous explicit construction of supersymmetric quantum field theories, it was proven that the Hamiltonian of the model is supersymmetrically embeddable if it is non-negative. The index of the Dirac–Kähler-type operators associated with the supercharges of the supersymmetric quantum field theories is computed.

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.

The Φ4 equations of motion. II. The zero-, one-, and two-dimensional solutions

This is the second paper of the series concerning the solution of the system of Φ4 equations of motion for the Schwinger functions by a fixed-point method. These works constitute a part of a general program towards the construction of a Φ44 Wightman quantum field theory (QFT). In the previous paper [J. Math. Phys. 29, 2092 (1988)] (Paper I), a general outline of the program has been presented. Moreover, the ‘‘nice’’ properties of signs, splitting, and norms revealed ‘‘experimentally’’ by the Φ iteration have been analyzed. We have shown how this iterative procedure converges to the solution (if the coupling constant is fixed positive and smaller than a finite value) thanks to the conservation of these properties, which constitute in fact a complete system of self-consistent conditions. Taking into account this information, in the present paper, the answer to the zero-, one-, and two-dimensional problems is given. To be precise, the fixed-point method is constructed by formulating the properties of signs and splitting of the Φ iteration in terms of particular subsets Φ0Λ ⊆ℬ0 (zero dimensions) and ΦΛ⊆ℬ (one and two dimensions) of the appropriated Banach spaces ℬ0 and ℬ, defined exactly with the norms provided by the Φ iteration. The basic ingredients, both introduced already in I, are, on the one hand, the bounded positive sequences of the splitting constants and, on the other hand, the sweeping factors, which carry all the combinatorial information for the global terms. Their absolute and relative bounds yield the stability of the corresponding subsets and the conservation of the norms. F o r the zero-dimensional problem a simpler equivalent system (nonlinear map) of equations in the space of the splitting sequences is solved by ensuring the existence of the solution (resp. contractivity) inside the corresponding subset when the coupling constant Λ satisfies 0<Λ≲0.1 (resp. 0<Λ≲0.01). For the two- (or one-) dimensional problem it is shown that when 0<Λ≲0.006, the subset ΦΛ is stable under the nonlinear mapping (represented by the equations of motion), which in turn is contractive inside ΦΛ under weaker conditions on Λ. By this last result the convergence of the Φ iteration to the unique fixed point (presented in I) is reobtained in a direct way.

Systems with outer constraints. Gupta-Bleuler electromagnetism as an algebraic field theory

Since there are some important systems which have constraints not contained in their field algebras, we develop here in aC*-context the algebraic structures of these. The constraints are defined as a groupG acting as outer automorphisms on the field algebra ℱ, α:G ↦ Aut ℱ, α G ⊄ Inn ℱ, and we find that the selection ofG-invariant states on ℱ is the same as the selection of states ω onM(G \(M(G\mathop \times \limits_\alpha F)\) ℱ) by ω(U g)=1∨g∈G, whereU g ∈M (G \(M(G\mathop \times \limits_\alpha F)\) ℱ)/ℱ are the canonical elements implementing α g . These states are taken as the physical states, and this specifies the resulting algebraic structure of the physics inM(G \(M(G\mathop \times \limits_\alpha F)\) ℱ), and in particular the maximal constraint free physical algebra ℛ. A nontriviality condition is given for ℛ to exist, and we extend the notion of a crossed product to deal with a situation whereG is not locally compact. This is necessary to deal with the field theoretical aspect of the constraints. Next theC*-algebra of the CCR is employed to define the abstract algebraic structure of Gupta-Bleuler electromagnetism in the present framework. The indefinite inner product representation structure is obtained, and this puts Gupta-Bleuler electromagnetism on a rigorous footing. Finally, as a bonus, we find that the algebraic structures just set up, provide a blueprint for constructive quadratic algebraic field theory.

Phase space analysis and particle structure in field theory

Progress in the last few years in constructive field theory has included (1) the construction of three-dimensional, non-Abelian gauge theories in a finite volume, (2) the construction of models that are asymptotically free or have a nontrivial fixed point, (3) a new presentation of perturbation theory yielding old and new large-order estimates, and (4) asymptotic completeness for renormalizable models in the two-particle region; the existence of multiparticle structure equations in the Euclidean region, provided that the coupling constant goes to zero as the energy increases; these equations yield formally the asymptotic completeness. In obtaining most of these results the phase space expansion pioneered by Glimm and Jaffe plays a crucial role. The first part of this review is a description of the phase space expansion in the “Ecole Polytechnique” version; the second part is devoted to the construction of models, the third part treats perturbation theory; and the last part deals with asymptotic completeness.

Integral representations of the Schwinger functions for wick polynomials in the free field

Integral representations for the Schwinger functions of products of Wick polynomials in the free field are obtained, i.e., Euclidean realizations of Wightman quantum fields specified by Wick polynomials in the free field are obtained. These Euclidean realizations are used to propose a new mathematically rigorous approach to the construction of quantum field theory with polynomial interaction in a finite volume of d-dimensional space-time even without ultraviolet cutoff. In particular, the generating functional of the Schwinger functions is constructed for purely imaginary values of the coupling constant. The theory constructed in this manner takes into account explicitly the presence of ultraviolet divergences and if expanded with respect to the coupling constant gives a renormalized perturbation-theory series.

Superconformal and super KAC-moody invariant quantum field theories in two dimensions

Two-dimensional quantum field theories invariant under the superconformal and super Kac-Moody transformations are constructed, generalizing the construction of conformal field theories following Belavin-Polyakov-Zamolodchikov. The linear differential equation for the multipoint correlation is constructed and four-point functions are explicitly calculated using the global OSp(2,1) invariance.

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q 2. In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ +=P − QP +, whereP ±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q +) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q +)=n−1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.

Vertex operator construction of superconformal ghosts and string field theory

Superconformal ghosts in string theories are characterized by the SU(1,1) Kac-Moody algebra with central charge − 1 2 . These ghost fields are constructed as the vertex operators realizing spinor representations of the Kac-Moody algebra. Representations of the canonical commutation relations of the superconformal ghosts are analyzed extensively. All irreducible representations are found to possess only the trivial inner product but for one exceptional case. Consequently, in superstring field theory it is necessary to consider reducible representations in general. Hilbert spaces with a non-trivial inner product are explicitly obtained upon which second quantization of superstring may be carried out.

Positive bare action for φ4 theory in 2, 3, and 4 euclidean dimensions

By adding appropriate auxiliary fields we exhibit a positive definite bare action for φ 4 theory in 2, 3, and 4 dimensions. As the zeta function regulator is removed the auxiliary fields decouple, leaving a perturbation theory identically that of φ 4 theory renormalized by zeta function regularization with minimal prescription for removing pole parts. We expect this method may allow a rigorous, nonperturbative construction of various field theories.