New Lagrangians for Bosonic m-branes with vanishing cosmological constant

A class of conformally invariant σ model actions in 2n dimensions is shown to be classically equivalent to the Nambu-Goto action for an extended object, an m-brane (m+1=2n), embedded in a higher dimensional space-time (d⪖m+1). when m is even, a (2n + 1)-dimensional σ model action is also constructed, which is classically equivalent to the Nambu-Goto action, but in this case there is no conformal invariance. In both cases the cosmological constant can be set to zero.     

Supersymmetry among free fermions and superstrings

A complete classification is given of all supersymmetric theories of free massless two-dimensional fermions. This, in particular, implies a classification of all free-fermion representations of super Kac-Moody algebras. It is shown that these cannot be used to construct new string theories with unbroken supersymmetry in Minkowski space-time, other than the torus-compactifications of the known ten-dimensional superstrings. Assuming anti-de-Sitter space-time could restore conformal invariance, it is shown how one could construct a string theory whose low-lying excitations form a multiplet of gauged N = 8 supergravity.     

Conformal invariance,the KdV equation and coadjoint orbits of the Virasoro algebra

We use the transformation properties of the energy-momentum tensor of 2-dimensional conformal field theories to study the symplectic geometry of DiffS¹/S¹ together with the other coadjoint orbits of the Virasoro algebra in the space of quadratic differentials. Also, we examine orbits of the N = 1 and N = 2 super-Virasoro algebras and investigate the connections with nonlinear differential equations of KdV type. The importance that the dual of the Virasoro algebra may have in the geometric formulation of string theory is discussed as well.     

Singular vectors and invariant equations for the Schrödinger algebra in <Emphasis Type="Italic">n</Emphasis> ≥ 3 space dimensions. The general case

The singular vectors in Verma modules over the Schrödinger algebra ?(n) in (n + 1)-dimensional space-time are found for the case of general representations. Using the singular vectors, hierarchies of equations invariant under Schrödinger algebras are constructed.     

The Klein-Gordon operator in conformal space

We consider the Minkowski space M₄ as a local chart of a compact differentiable pseudo-Riemannian manifold M⁴_c, on which the whole conformal group $\text{O(2, 4)}\text{Z}{}_2$ acts continuously. We investigate the conditions under which functions or differential operators on the space M₄ can be uniquely continued to the conformal manifold M⁴_c. This is done by using methods well-known in the theory of differentiable manifolds. In particular, we show that the Klein-Gordon operator □+m² can be uniquely continued to the space M⁴_c and we discuss the conformal invariance of the Klein-Gordon equation on the manifold M⁴_c.     

Conformal invariance in quantum gravity

Trace anomalies in a conformal invariant theory do not arise when its conformal invariance in four dimensions is extended to an arbitrary number n of space-time dimensions: the theory can be made finite in any order of perturbation theory by conformal invariant counterterms in n dimensions. Such an extension of conformal invariance is possible provided one works in the framework of spontaneously broken conformal invariance. This is shown explicitly by working out several examples at the one-loop level and by examining the Ward identities which lead to a general proof.We speculate upon possible consequences of these results on the nature of gravitation and other fundamental interactions.     

On the identification of finite operator algebras in two-dimensional conformally invariant field theories

We investigate the structure of the linear differential operators whose solutions determine the four-point correlations of primary operators in the d = 2 conformally invariant SU(2) α-model with Wess-Sumino term and the d = 2 critical statistical systems with central Virasoro charge smaller than one. Factorisation properties of the differential operators are related to a finite closure of the operator algebras. We recover the selection and fusion rules of Fateev, Zamolodchikov and Gepner, Witten for the SU(2) α-model. It is outlined how the results of the SU(2) model can be used for the identification of closed operator algebras in the statistical model.     

Dimensional renormalization of scalar field theory in curved space-time

The regularization and renormalization of an interacting scalar field φ in a curved spacetime background is performed by the method of continuation to n dimensions. In addition to the familiar counter terms of the flat-space theory, c-number, “vacuum” counter terms must also be introduced. These involve zero, first, and second powers of the Reimann curvature tensor R_αβψδ. Moreover, the renormalizability of the theory requires that the Lagrange function couple φ² to the curvature scalar R with a coupling constant η. The coupling η must obey an inhomogeneous renormalization group equation, but otherwise it is an arbitrary, free parameter. All the counter terms obey renormalization group equations which determine the complete structure of these quantities in terms of the residues of their simple poles in n ? 4. The coefficient functions of the counter terms determine the construction of φ² and φ⁴ in terms of renormalized composite operators 1, [φ²], and [φ⁴]. Two of the counter terms vanish in conformally flat space-time. The others may be computed from the theory in purely flat space-time. They are determined, in a rather intricate fashion, by the additive renormalizations for two-point functions of [φ²] and [φ⁴] in Minkowski space-time. In particular, using this method, we compute the leading divergence of the R² interaction which is of fifth order in the coupling constant λ.     

A local approach to dimensional reduction: II. Conformal invariance in Minkowski space

We consider the problem of obtaining conformally invariant differential operators in Minkowski space. We show that the conformal electrodynamics equations and the gauge transformations for them can be obtained in the frame of the method of dimensional reduction developed in the first part of the paper. We describe a method for obtaining a large set of conformally invariant differential operators in Minkowski space.     

Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.     

Generalized Drinfel'd-Sokolov hierarchies

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct theW _n ^(l) algebras, first discussed for the casen=3 andl=2 by Polyakov and Bershadsky.     

Mass spectrum of the M4×SD solution in Euler invariant type higher derivative gravity

The mass spectrum is computed in Euler invariant type higher derivative gravity theory in the case that the space-time is dimensionally reduced to the four-dimensional Minkowski space × D-dimensional sphere. It is shown at the linearized level that after the compactification there appear massless gravitons, massive spin-two particles, massless vectors and one massive scalar mode. All the vectors are massless and the masses of massive spin-two particles are proportional to the S^D eigenvalues of the laplacian. Classical stability is shown to depend only on three parameters.     

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.     

Nilpotent invariants in N=4 SYM

It is shown that there are no nilpotent invariants in N=4 analytic superspace for n≤4 points. It is argued that there is (at least) one such invariant for n=5 points which is not invariant under U(1)_Y. The consequences of these results are that the n=2 and 3 point correlation functions of the N=4 gauge-invariant operators which correspond to KK multiplets in AdS supergravity are given exactly by their tree level expressions, the 4 point correlation functions of such operators are invariant under U(1)_Y and correlation functions with n≥4 points have non-trivial dependence on the Yang-Mills coupling constant.     

Finite energy solutions for interacting Yang-Mills and Dirac fields on Minkowski space

The Lagrangian for a SU(2) Yang-Mills field interacting with a massless isospin 1/2 Dirac field is conformally invariant. Finite energy solutions are obtained by a conformal mapping of Minkowski space onto the compact manifoldS ¹×S³ with pseudo-Riemannian metric. They are symmetric with respect to the isometric group SO(2)×SO(4) ofS ¹×S³.     

Theory of the conformally invariant mass

By interpreting the conformal transformations as space-time-dependent change of units and introducing the concept of the conformally invariant mass and charge, we develop new conformally invariant Maxwell equations with source terms and equations of motion for massive particles. Although the usual equations of motion with mass terms break the conformal symmetry, it is shown that the Minkowski space is not the most general framework to describe physical processes and there exists a wider consistent dynamics in which conformal invariance is exact. New results also include the general transformation laws of the electromagnetic fields, of currents and force densities. The theory leads naturally to an affine connection and to the 21-parameter inhomogeneous conformal group, ISO(4, 2).     

Invariant differential operators for non-compact Lie groups: Summary of su(4,4) multiplets

The present paper is part of the project of systematic construction of invariant differential operators of noncompact semisimple Lie algebras. Here we give a summary of all multiplets containing physically relevant representations including the minimal ones for the algebra su(4, 4). Due to the recently established parabolic relations the results are valid also for the algebras sl(8, R) and su*(8)