Exchange algebra and the Drinfeld-Sokolov theorem

We analyse the relation between the exchange algebra and the separation of the chiralities in classical Toda field theory. We show that there exists a conformally covariant Bloch wave basis such that the two chiralities commute. In terms of this basis we then reconstruct the periodic and local solution of Toda field theory.Work supported by Fondazione Angelo Della Riccia     

Drinfeld-Sokolov Construction of Heterotic Toda Model

A free field representation for the left-right asymmetric conformal Toda theory based on simplf-laced even-rank Lie algebras is given. It is shown that the classical chiral exchange algebra for such theories can be reconstructed from free chiral bosons via Drinfeld-Sokolov linear systems, and is a bit more complicated than that of the standard Toda due to some additional δ-function terms and extra degrees of freedom.     

Generalized Drinfeld-Sokolov reductions and KdV type hierarchies

Generalized Drinfeld-Sokolov (DS) hierarchies are constructed through local reductions of Hamiltonian flows generated by monodromy invariants on the dual of a loop algebra. Following earlier work of De Groot et al., reductions based upon graded regular elements of arbitrary Heisenberg subalgebras are considered. We show that, in the case of the nontwisted loop algebra (gl _n ), graded regular elements exist only in those Heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn=pr or to equal numbers plus onen=pr+1. We prove that the reduction belonging to the grade 1 regular elements in the casen=pr yields thep×p matrix version of the Gelfand-Dickeyr-KdV hierarchy, generalizing the scalar casep=1 considered by DS. The methods of DS are utilized throughout the analysis, but formulating the reduction entirely within the Hamiltonian framework provided by the classical r-matrix approach leads to some simplifications even forp=1.     

Exact and numerical solutions of generalized Drinfeld-Sokolov equations

In this Letter, we consider a system of generalized Drinfeld-Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM).     

Universal Drinfeld-Sokolov reduction and matrices of complex size

We construct affinization of the algebra of complex size matrices, that contains the algebras for integral values of the parameter. The Drinfeld-Sokolov Hamiltonian reduction of the algebra results in the quadratic Gelfand-Dickey structure on the Poisson-Lie group of all pseudodifferential operators of complex order.This construction is extended to the simultaneous deformation of orthogonal and symplectic algebras which produces self-adjoint operators, and it has a counterpart for the Toda lattices with fractional number of particles.Partially supported by NSF grant DMS 9307086.Partially supported by NSF grant DMS 9401215.     

Equivalence of the Drinfeld-Sokolov reduction to a bi-Hamiltonian reduction

We show that the Drinfeld-Sokolov reduction is equivalent to a bi-Hamiltonian reduction, in the sense that these two reductions, although different, lead to the same reduced Poisson (more correctly, bi-Hamiltonian) structure. In order to do this, we heavily use the fact that they are both particular cases of a Marsden-Ratiu reduction.This work has been supported by the Italian MURST and by the GNFM of the Italian CNR.     

Characters and fusion rules forW-algebras via quantized Drinfeld-Sokolov reduction

Using the cohomological approach toW-algebras, we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized Drinfeld-Sokolov reduction.Supported in part by Junior Fellowship from Harvard Society of FellowsSupported in part by NSF grants DMS-8802489 and DMS-9103792Supported in part by RIMS-91 ProjectCommunicated by A. Jaffe     

Drinfeld-Sokolov reduction on a simple lie algebra from the bihamiltonian point of view

We show that the Drinfeld-Sokolov reduction can be framed in the general theory of bihamiltonian manifolds, with the help of a specialized version of a reduction theorem for Poisson manifolds by Marsden and Ratiu.This work has been supported by the Italian MURST and by the GNFM of the Italian C.N.R.     

Extrinsic hermitian geometry of functional determinants for vector subbundles and the Drinfeld-Sokolov ghost system

In this paper, a novel method is presented for the study of the dependence of the functional determinant of the Laplace operator associated to a subbundleE of a hermitian holomorphic vector bundleE ⁰ over a Riemann surface on the hermitian structure (h, H) ofE ⁰. The generalized Weyl anomaly of the effective action is computed and found to be expressible in terms of a suitable generalization of the Liouville and Donaldson actions. The general techniques worked out are then applied to the study of a specific model, the Drinfeld-Sokolov (DS) ghost system arising inW-gravity. The expression of the generalized Weyl anomaly of the DS ghost effective action is found. It is shown that, by a specific choice of the fiber metricH _h depending on the base metrich, the effective action reduces into that of a conformal field theory. Its central charge is computed and found to agree with that obtained by the methods of hamiltonian reduction and conformal field theory. It is also shown that the dependence of the effective action onH is local. The DS holomorphic gauge group and the DS moduli space are defined and their dimensions are computed.     

Einstein gravity emerging from quantum Weyl gravity

We advocate a conformal invariant world described by the sum of the Weyl, Dirac, and Yang-Mills action. Quantum fluctuations bring back Einstein gravity so that the long-distance phenomenology is as observed. Formulas for the induced Newton's constant and Eddington's constant are derived in quantized Weyl gravity. We show that the analogue of the trace anomaly for the Weyl action is structurally similar to that for the Yang-Mills action.     

Einstein gravity as spontaneously broken Weyl gravity

We study the locally conformal invariant Weyl theory of gravitation and introduce a conformally coupled scalar field. Einstein gravity is induced by spontaneous breaking of the local conformal symmetry in an effective long range approximation. The effective potential for the scalar field is calculated at the one-loop level up to curvature squared in order in an arbitrary curved background. The non-zero vacuum expectation value of the scalar field induces the dimensional Einstein's gravitational coupling constant stably in case ofR > 0. ForR < 0, the phase transition occurs from the symmetric phase to the broken phase as the curvature decreases. This theory may be an attractive candidate for the primordial inflationary universe scenario.     

Stochastic gravity

We investigate stochastic gravity as a potentially fruitful avenue for studying quantum effects in gravity. Following the approach of stochastic electrodynamics (sed), as a representation of the quantum gravity vacuum we construct a classical state of isotropic random gravitational radiation, expressed as a spin-2 field,h _µ (x), composed of plane waves of random phase on a flat spacetime manifold. Requiring Lorentz invariance leads to the result that the spectral composition function of the gravitational radiation,h(), must be proportional to 1/ ². The proportionality constant is determined by the Planck condition that the energy density consist of/2 per normal mode, and this condition sets the amplitude scale of the random gravitational radiation at the order of the Planck length, giving a spectral composition functionh() =16c ²L_p/². As an application of stochastic gravity, we investigate the Davies-Unruh effect. We calculate the two-point correlation function (R _iojo(O–/2)R _kolo(O,+/2)) of the measureable geodesic deviation tensor field,R _iojo, for two situations: (i) at a point detector uniformly accelerating through the random gravitational radiation, and (ii) at an inertial detector in a heat bath of the random radiation at a finite temperature. We find that the two correlation functions agree to first order ina/c provided that the temperature and acceleration satisfy the relationkT=a/2c.     

Topological gravity

A version of conformal gravity is formulated with a local fermionic symmetry that is reminiscent of BRST invariance. It may have mathematical applications (gravitational counterpart of Donaldson theory) or physical ones (unbroken phase of general relativity).     

Cohomological gravity

We construct a theory of cohomological gravity in arbitrary dimensions based upon a local vector supersyrnmetry algebra. The observables in this theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occurring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.This essay received an honorable mention from the Gravity Research Foundation, 1992-Ed.     

Higgs-field gravity

It is shown that any excited Higgs field mediates an attractive scalar gravitational interaction of Yukawa type between the elementary particles, which become massive by the ground state of the Higgs field.