Superprojectors

We present a simple algorithm for constructing the N-extended superfield projection operators for irreducible representations of supersymmetry, and explicitly perform all simplifications due to spinor derivative algebra. The method is based on covariant expansion of a general superfield in terms of chiral superfields, and requires no knowledge of Casimir operators. We list these superprojectors for various N = 1, 2, and 4 superfields, and apply our results to quantized the linearized N = 2 vector multiplet in a supersymmetric gauge.     

Chiral deformations of conformal field theories

We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W_1+∞ algebra, that is treated in detail.     

Wk structure of A SU(2)-level k from covariant construction of Kac-Moody algebras

The W_k structure underlying the transverse realization of affine SU(2) at level k is analyzed. The extension of the equivalence existing between the covariant and light-cone gauge realization of an affine Kac-Moody algebra to W_k algebras is given. Higher spin generators are extracted by the less singular terms in the operator product expansion of the parafermions constructed by means of the projection of the covariant on the light-cone gauge. These fields can be written in terms of only one free boson compactified on a circle.     

Tensor methods for the exceptional group E6

We present a tensor formalism to describe irreducible representations of the exceptional group E₆. Irreducible tensors are characterized by covariant and contravariant indices associated with the irreducible representation 27, and a third (orthogonal-type) index associated with the 78; contractions of these indices with a set of invariant tensors are required to vanish for irreducibility. The formalism is applied to the reduction of Kronecker products of E₆ irreducible representations. As a further illustration of the method, we construct explicitly the Higgs potential for scalar fields in the E₆ representations 27, 78, 351, 351′.     

Conformal blocks and generalized Selberg integrals

Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special “conservation” relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko–Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an n-point conformal block on Riemann sphere, one reproduces the earlier conjectured β-ensemble representation of conformal blocks. The statement can also be regarded as a relation between the 3j  -symbols of the Virasoro algebra and the slightly generalized Selberg integrals IY$I_Y$, associated with arbitrary Young diagrams. The conformal blocks are multilinear combinations of such integrals and the AGT conjecture relates them to the Nekrasov functions which have exactly the same structure.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.     

Super-gauge transformations on the six-dimensional hypercone

Super-gauge transformations on physical space-time are defined on the projective six-dimensional hypercone. The basic observation which motivates the present approach is that the product of any two super-gauges is a general conformal (+ γ₅) transformation and indeed these latter transformations are simply realized on the six-dimensional hypercone. It turns out that also super-gauge transformations have a natural action on this projective space where the parameters of such transformations are (totally anticommuting) spinors which transform according to the fundamental (eight-dimensional) representation of the spinor group SU (2,2), isomorphic to the conformal group. The enlarged algebra of super-gauges, conformal and chiral transformations and its most relevant representations are discussed in the present formalism.     

Six-dimensional formulation of meson equations

The structure of the conformal group is studied by a generalisation of quaternion methods to six dimensions. Some simpleSO(4,2) covariant equations are shown to correspond to the Kemmer formulation for pseudoscalar and vector mesons, and the matrix elements of the irreducible representations of the Kemmer algebra are expressed as traces of products of Dirac matrices.     

On the two-vector invariants of point groups

The paper is devoted to group-theoretical analysis of a function of two vectors, invariant under some point group (special attention is paid to the O_h group). In particular, the invariants in the direct product of spaces transforming according to the lth and l'th irreducible representations of the rotation group are studied. A compact formula determining the number of such invariants for the group O_h is found. It is shown that all possible invariants in the considered product space can be constructed from all possible scalar products of vector functions of both vectors transforming according to complex conjugate irreducible representations. The addition theorem for these functions is proved.     

The algebraic structure of the fermionic string

The local structure of the product expansion algebra of the covariant NSR string is analyzed. An “on-shell” Kac-Moody like algebra is found to generate the BRST invariant part of the covariant lattice Γ_5,1. This algebra is a local version of ten-dimensional SUSY.     

Covariant representations in a fibre bundle framework

Canonical and covariant representations of Lie groups of the semidirect product form G = N⊙K with N Abelian, are analyzed in a fibre bundle framework. We exhibit first the relationship between both kinds of representations in such framework. Two complementary methods of selecting irreducible representations from the covariant ones are developed. The first one proceeds by restriction to an invariant subspace and is exemplified in the case of massive integer spin representations of the Poincaré group. The second method takes quotients and is particularly useful when we deal with reducible but indecomposable representations. A family of stepped gauge transformations is generated when the method is used to obtain the covariant massless integer helicity representations of the Poincaré group; the electromagnetic and gravitational gauge transformations are just the first two cases of such a family.     

Lifting a conformal field theory from d-dimensional flat space to (d+1)-dimensional AdS space

A quantum field theory on anti-de Sitter space can be constructed from a conformal field theory on its boundary Minkowski space by an inversion of the holographic mapping. The resulting theory is defined by its Green functions and is conformally covariant. The structure of operator product expansions is carried over to AdS space. We show that this method yields a higher spin field theory HS(4) from the minimal conformal O(N) sigma model in three dimensions.     

Recent developments in conformal invariant quantum field theory

A review of the recent results concerning the kinematics of conformal fields, the analysis of dynamical equations and dynamical derivation of the operator product expansion is given.The classification and transformational properties of fields which are transformed according to the representations of the universal covering group of the conformal group have been considered. A derivation of the partial wave expansion of Wightman functions is given. The analytical continuation to the Euclidean domain of coordinates is discussed. As shown, in the Euclidean space the partial wave expansion can be applied either to one-particle irreducible vertices or to the Green functions, depending on the dimensions of the fields.The structure of Green functions, which contain a conserved current and the energy-momentum tensor, has been studied. Their partial wave expansion has been obtained. A solution of the Ward identity has been found. Special cases are discussed.The program of the construction of exact solution of dynamical equations is discussed. It is shown, that integral dynamical equations for vertices (or Green's functions) can be diagonalized by means of the partial wave expansion. The general solution of these equations is obtained. The equations of motion for renormalized fields are considered. The way to define the product of renormalized fields at coinciding points (arising on the right-hand side) is discussed. A recipe for calculating this product is presented. It is shown, that this recipe necessarily follows from the renormalized equations.The role of bare term and of canonical commutation relations (for unrenormalized fields) is discussed in connection with the problem of the field product determination at coinciding points. As a result an exact relation between fundamental field dimensions is found for various three-linear interactions (section 16 and Appendix 6). The problem of closing the infinite system of dynamical equations is discussed.Al above said results are demonstrated using Thirring model as an example. A new approach to its solving is developed.The program od closing the infinite system of dynamical equations is discussed. The Thirring model is considered as an example. A new approach to the solution of this model is discussed.Methods are developed for the approximate calculation of dimensions and coupling constants in the 3-vertex and 5-vertex approximations. The dimensions are calculated in the γ_?³ theory in 6-dimensional space.The problem of calculating the critical indices in statistics (3-dimensional Euclidean space) is considered. The calculation of the dimension is carried out in the framework of the γ_?⁴ model. The value of the dimension and the critical indices thus obtained coincide with the experimental ones.     

Conformal field theories and modular invariant partition functions for parafermion field theories

Parafermion conformal field theories with D_2N discrete symmetry are examined in detail. The structure of field space of parafermion field theories is studied with the help of a projection operator G. Characters of the representations of the twist sector of parafermion algebra and projected characters are given. A new class of modular invariant partition functions, therefore conformal field theories, for parafermion theories are found. We argue that the principal theories correspond to the generic critical SOS models of Andrew, Baxter and Forrest.     

On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

A duality theorem for the automorphism group of a covariant system

In this paper we examine the covariant representation theory of a covariant system (A, G) introduced by Doplicher, Kastler and Robinson. (A is aC*-algebra andG is a locally compact group of automorphisms ofA.) We define the concept of left tensor product of two covariant representations. Loosely stated, our duality theorem says thatG is canonically isomorphic to the set of bounded operator valued maps on the set of covariant representations of the covariant system (A, G) which preserve direct sums, unitary equivalence and left tensor products. We further show that the enveloping von Neumann algebraA(A, G) of the covariant system (A, G) admits a (not necessarily injective) comultiplicationd such that (A(A, G),d) is a Hopf von Neumann algebra. The intrinsic group of this Hopf von Neumann algebra is canonically isomorphic and (strong operator topology) homeomorphic toG.     

Operator product expansions in conformally covariant quantum field theory

Operator products in quantum field theory on two-dimensional Minkowski space are expanded into a series of local operators by means of the tensor product decomposition theorem for representations of the conformal group. The Thirring model is used as an explicit example. Two types of expansions result. If the operator product acts on the vacuum state, we obtain strictly covariant expansions. In general however, each term in the expansion is only semicovariant.     

Group representation theory in relativistic electrodynamics

Invariant expansions of the electromagnetic field tensor F and its covariant derivative ℬF in the presence of gravity are obtained on the basis of the classical problem of elementary orthogonal group representation theory regarding the expansion of the tensor product of representations in terms of irreducible components. Vladimirsk State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 90–93, May, 1996.