Free Fermion Fields Research Articles

Conformal four point functions and the operator product expansion

Various aspects of the four point function for scalar fields in conformally invariant theories are analysed. This depends on an arbitrary function of two conformal invariants u,v. A recurrence relation for the function corresponding to the contribution of an arbitrary spin field in the operator product expansion to the four point function is derived. This is solved explicitly in two and four dimensions in terms of ordinary hypergeometric functions of variables z,x which are simply related to u,v. The operator product expansion analysis is applied to the explicit expressions for the four point function found for free scalar, fermion and vector field theories in four dimensions. The results for four point functions obtained by using the AdS/CFT correspondence are also analysed in terms of functions related to those appearing in the operator product discussion.

On the trace anomaly as a measure of degrees of freedom

Recent conjectures of the c-theorem in four and higher dimensions have suggested that the coefficient of the Euler characteristic in the trace anomaly could measure the degrees of freedom in field theory and decrease along the renormalization-group flow. We compute this quantity for free massless scalar, fermion and antisymmetric tensor fields in any dimension, and analyse its dependence on spin and space-time dimension. In the limit of large number of dimensions, where the theories become semiclassical, we find that this quantity does not approach the classical number of field components, but is enhanced for spinful particles. This seemingly strange behaviour is found to be consistent with known renormalization-group patterns and a specific c-theorem conjecture.

Free strings and superstrings on the [formula omitted] spaces

We construct a free bosonic, fermionic and supersymmetric field theories on the M C n spaces for critical dimensions, i.e., n=2 p . This procedure allows us to exhibit the underlying geometry of these spaces. In particular, these spaces are shown to be in correspondence with the n real spheres S n−1 .

Symplectic fermions

We study two-dimensional conformal field theories generated from a ``symplectic fermion'' - a free two-component fermion field of spin one - and construct the maximal local supersymmetric conformal field theory generated from it. This theory has central charge c=-2 and provides the simplest example of a theory with logarithmic operators. Twisted states with respect to the global SL(2,C)-symmetry of the symplectic fermions are introduced and we describe in detail how one obtains a consistent set of twisted amplitudes. We study orbifold models with respect to finite subgroups of SL(2,C) and obtain their modular invariant partition functions. In the case of the cyclic orbifolds explicit expressions are given for all two-, three- and four-point functions of the fundamental fields. The C_2-orbifold is shown to be isomorphic to the bosonic local logarithmic conformal field theory of the triplet algebra encountered previously. We discuss the relation of the C_4-orbifold to critical dense polymers.

Decoupling transformations in path-integral bosonization

We construct non-anomalous transformations that decouple fermionic fields in interaction with a gauge field, in the path-integral representation of the generating functional. Those transformations express the original fermionic fields in terms of non-interacting ones, through non-local functionals depending on the gauge field. This procedure, holding true in any number of space-time dimensions both in the Abelian and non-Abelian cases, is then applied to the path-integral bosonization of the Thirring model in three dimensions. Knowledge of the decoupling transformations allows us, contrarily to previous bosonizations, to obtain the bosonization with an explicit expression of the fermion fields in terms of bosonic ones and free fermionic fields. We also explain the relation between our technique, in the two-dimensional case, and the usual decoupling in two dimensions.

Fermionic fields in the d-dimensional anti-de Sitter spacetime

Arbitrary spin free massless fermionic fields corresponding to mixed symmetry representations of the SO ( d −1) compact group and propagating in even d -dimensional anti-de Sitter spacetime are investigated. Free wave equations of motion, subsidiary conditions and the corresponding gauge transformations for such fields are proposed. The lowest eigenvalues of the energy operator for the massless fields and the gauge parameter fields are derived. The results are formulated in SO ( d −1,2) covariant form as well as in terms of intrinsic coordinates.

Lattice bosonization

A free lattice fermion field theory in 1 + 1 dimensions can be interpreted as SOS-type model, whose spins are integer-valued. We point out that the relation between these spins and the fermion field is similar to the abelian bosonization relation between bosons and fermions in the continuum. Though on the lattice the connected 2n-point correlation functions of the integer-valued spins are not zero for any n ≥ 1, the two-point correlation function of these spins is that of free bosons in the infrared. We also conjecture the form of the Wess-Zumino-Witten chiral field operator in a nonabelian lattice fermion model. These constructions are similar in spirit to the “twistable string” idea of Krammer and Nielsen.

Gravity in 2 + 1 dimensions as a Riemann - Hilbert problem

In this paper we consider (2 + 1)-dimensional gravity coupled to N point particles. We introduce a gauge in which the z and components of the dreibein field become holomorphic and anti-holomorphic, respectively. As a result we can restrict ourselves to the complex plane. Next we show that solving the dreibein field is equivalent to solving the Riemann - Hilbert problem for the group SO(2,1). We give the explicit solution for two particles in terms of hypergeometric functions. In the N-particle case we give a representation in terms of conformal field theory. The dreibeins are expressed as correlators of two free fermion fields and twist-operators at the position of the particles.

Vertex operators in 2Kdimensions

A formula is proposed which expresses free fermion fields in 2K dimensions in terms of the Cartan currents of the free fermion current algebra. This leads, in an obvious manner, to a vertex operator construction of non-Abelian free fermion current algebras in arbitrary even dimensions. It is conjectured that these ideas may generalize to a wide class of conformal field theories.

Hamiltonian superoperators

We present a theory of Hamiltonian superoperators associated with a Lie superalgebra and its modules. By using the free fermionic fields from physics, we establish a super version of the formal variational calculus introduced by Gel'fand and Dikii (1975, 1976). Moreover, we prove that a super skew-symmetric matrix differential operator in our super formal variational calculus is a Hamiltonian operator if and only if its Schouten-Nijenhuis super-bracket is zero, when the characteristic of the base field is not two. Some interesting examples of Hamiltonian superoperators are also given.

RECENT DEVELOPMENTS IN D=2 STRING FIELD THEORY

We review the recent developments in the construction of string field theory in two dimensions. We analyze the bewildering number of string field theories that have been proposed, all of which correctly reproduce the correlation functions of two-dimensional string theory. These include (1) free fermion field theory, (2) collective string field theory, (3) temporal gauge string field theory and (4) nonpolynomial string field theory. We will analyze discrete states, the ω(∞) symmetry, and correlation functions in terms of these different string field theories. We will also comment on the relationship between these field theories, which is still not well understood.

String model in D=1+3 dimensions: the non-standard approach to Hamiltonian dynamics and quantization

The open spinning string is investigated in Minkowski space E1.3. By means of a reduction to the Wess-Zumino-Witten-Novikov (WZWN) model a new Hamiltonian formalism is constructed. The Poisson bracket structure of the theory is given in terms of the algebra ((sl(2,C)(X)(t-1, t))(+)Cz)(X)P, where P is the Poincare algebra. The covariant quantization is fulfilled in D=1+3 dimensions with help of 'bosonization' methods. The resulting theory is a combination of the theory of a free fermionic field in two dimensions and the theory of a free particle in Minkowski space. Non-triviality is conditioned by the presence of a finite number of constraints. The formula for the mass spectrum is discussed.

Energy-momentum tensor in conformal field theories near a boundary

The requirements of conformal invariance for the two-point function of the energy-momentum tensor in the neighbourhood of a plane boundary are investigated, restricting the conformal group to those transformations leaving the boundary invariant. It is shown that the general solution may contain an arbitrary function of a single conformally invariant variable ν, except in dimension 2. The functional dependence on ν is determined for free scalar and fermion fields in arbitrary dimension d and also to leading in the ϵ-expansion about d = 4 for the nongaussian fixed point in φ 4 theory. The two-point correlation function of the energy-momentum tensor and a scalar field is also shown to have a unique expression in terms of ν and the overall coefficient is determined by the operator product expansion. The energy-momentum tensor on a general curved manifold is further discussed by considering variations of the metric. In the presence of a boundary this procedure naturally defines extra boundary operators. By considering diffeomorphisms these are related to component of the energy-momentum tensor on the boundary. The implications of Weyl invariance in this framework are also derived.

Higher-derivative Schwinger model.

Using the operator formalism, we obtain the bosonic representation for the free fermion field satisfying an equation of motion with higher-order derivatives. Then, we consider the operator solution of a generalized Schwinger model with higher-derivative coupling. Since the increasing of the derivative order implies the introduction of an equivalent number of extra fermionic degrees of freedom, the mass acquired by the gauge field is bigger than the one for the standard two-dimensional QED. An analysis of the problem from the functional integration point of view corroborates the findings of canonical quantization, and corrects certain results previously announced in the literature on the basis of Fujikawa's technique.

Level-one representations of the affine lie algebraB n (1)

We give an explicit description of the Kac-Peterson-Lepowsky construction of the basic representation for the affine Lie algebraB n (1). Using the conjugacy classes of the Weyl group ofB n, we describe all inequivalent maximal Heisenberg subalgebras of the corresponding affine Lie algebra. We associate to these Heisenberg subalgebras multicomponent charged and neutral free fermionic fields. The boson-fermion correspondence for these fields provides us with fermionic vertex operators, whose ‘normal ordered products’ give the (twisted) vertex operators of the Kac-Peterson-Lepowsky construction.

The two-dimensional Coulomb gas on a sphere: Exact results

At the special value of the reduced inverse temperatureΓ=2, the equilibrium statistical mechanics of a two-dimensional Coulomb gas confined to the surface of a sphere is an exactly solvable problem, just as it was for the Coulomb gas in a plane. The thermodynamic quantities and all the correlation functions can be calculated. Use is made of an isomorphism between the classical Coulomb gas and the free Fermi field theory associated with the Dirac operator on the sphere.

Level one representations of the twisted affine algebrasa n (2) andd n (2)

We give an explicit description of the Kac-Peterson-Lepowsky construction of the level one representations for the twisted affine Lie algebrasan(2) anddn(2). We assign to any conjugacy class of the Weyl group ofan anddn 1) an equivalent class of maximal Heisenberg subalgebras of the corresponding twisted affine Lie algebras, and 2) multicomponent charged and neutral free fermionic fields. The boson-fermion correspondence for these fields provides us with fermionic vertex operators, whose ‘normal ordered products’ give the (twisted) vertex operators of the Kac-Peterson-Lepowsky construction.

Alternative bosonization of the Thirring model.

We introduce a construction of the interacting fermion field in the Thirring model in terms of three independent fields: a free fermion and two free boson fields, one of which is a ghost. One attractive feature of this construction is that all of the fermionic properties can be isolated in the free fermion field. A second attractive aspect is that the dropping of anomalous factors in the point-splitting definition of current densities can be accomplished quite naturally in terms of limiting procedures. Unwelcome features include the necessity of projecting out unphysical states and the difficulty of maintaining this projection in the presence of an ultraviolet cutoff.

Geometry of the string equations

The string equations of hermitian and unitary matrix models of 2D gravity are flatness conditions. These flatness conditions may be interpreted as the consistency conditions for isomonodromic deformation of an equation with an irregular singularity. In particular, the partition function of the matrix model is shown to be the tau function for isomonodromic deformation. The physical parameters defining the string equation are interpreted as moduli of meromorphic gauge fields, and the compatibility conditions can be interpreted as defining a “quantum” analog of a Riemann surface. In the latter interpretation, the equations may be viewed as compatibility conditions for transport on “quantum moduli space” of correlation functions in a theory of free fermions. We discuss how the free fermion field theory may be deduced directly from the matrix model integral. As an application of our formalism we discuss some properties of the BMP solutions of the string equations. We also mention briefly a possible connection to twistor theory.

The super W ∞ algebra

We construct a higher-spin N = 2 superalgebra whose bosonic sector is W ∞ ⊕ W 1+∞. It can be realised in terms of bilinear currents involving free complex bosonic and fermionic fields. We also discuss the spectral flow, and various truncations.