Conformal invariance,current algebra and modular invariance

A large class of conformally invariant models in two dimensions is realised by constraining free fermion theories. The Fock spaces of the constrained theories are described, using the representation theory of affine Kac-Moody algebras. The results are extended to superconformally invariant theories. Projections of the models, producing consistent two-dimensional field theories, are discussed.     

Virasoro and KdV

We investigate the structure of representations of the (positive half of the) Virasoro algebra and situations in which they decompose as a tensor product of Lie algebra representations. As an illustration, we apply these results to the differential operators defined by the Virasoro conjecture and obtain some factorization properties of the solutions as well as a link to the multicomponent KP hierarchy.     

Virasoro algebra in the solution space of the Ernst equation

In this Letter, a new transformation given in the solution space of the Ernst equation is shown to keep the Ernst equation invariant. It is also proved that the transformation enlarges the Cosgrove symmetry and constructs the Virasoro algebra.     

Kepler方程的共形不变性、Mei对称性与守恒量

研究Kepler系统在无限小变换下的共形不变性、Mei对称性.给出该系统与总能量、角动量不同的新守恒量.并在广义坐标和广义速度构成的空间中讨论这些守恒量的独立性.     

Kac-Moody algebra,nonlocal symmetries,and backlund transformation for KdV equation

Gauge transformation of the Lax eigenfunction through the explicit use of Lie group generators is seen to generate a two-parameter Backlund transformation. Explicit integration of this in two particular cases leads to sech² type and rational solutions starting from the trivial one. A method is indicated to generate infinitesimal transformations aroundu in the sense of Steudel, which in this case leads to a nonlocal structure of transformations.     

Graph theory,SO(n) current algebra and the Virasoro master equation

We announce an isomorphism between a set of generically irrational affine-Virasoro constructions onSO(n) and the unlabelled graphs of ordern. On the one hand, the conformal constructions are classified by the graphs, while, conversely, a group-theoretic and conformal field-theoretic identification is obtained for every graph of graph theory. High-level expansion provides a strong argument that each construction is unitary down to some finite critical level.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY85-15857     

Multidimensional analogue of Virasoro algebra and quantum higher-order water wave equation

The bi-Hamiltonian structure of higher-order water wave equation is seen to give rise to a multidimensional analogue of the Virasoro algebra. The quantum version of the equation can be constructed from the corresponding operator product expansion. Possible forms of some conserved quantities are given.     

Symplectic manifolds,coadjoint orbits,and mean field theory

Mean field theory is given a geometrical interpretation as a Hamiltonian dynamical system. The Hartree-Fock phase space is the Grassmann manifold, a symplectic submanifold of the projective space of the full many-fermion Hilbert space. The integral curves of the Hartree-Fock vector field are the time-dependent Hartree-Fock solutions, while the critical points of the energy function are the time-independent states. The mean field theory is generalized beyond determinants to coadjoint orbit spaces of the unitary group; the Grassmann variety is the minimal coadjoint orbit.     

The lie algebra of invariant group of the KdV,MKdV, or Burgers equation

Let (E): u _t=H(u) denote the KdV, MKdV or Burgers equation, and U(s)=(D^j _s)/u _j, where D=d/dx, u _i=Dⁱ _u, s=s(u, u ₁, ..., u _n) is a polynomial of u _i with constant coefficients, be the generator of invariant group of equation (E). We prove in this paper that all such generators form a commutative Lie algebra, from which it follows that for any symmetry s(u, ..., u _n) of (E), the evolution equation u _t=s(u, ..., u _n) possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations).     

Deformation of a Virasoro algebra and the integrals of motion of the quantum sine-Gordon equation

A special deformation of a Virasoro algebra such that the screening operator is not deformed (the space where it operates is deformed) is studied. This deformation leads to a 3-index algebra. The residue of the generating function of the generators of this algebra is a generating function of the integrals of motion for the quantum sine-Gordon model. The algebra of generating functions is calculated. Explicit formulas are presented for the first few integrals of motion. Pis’ma Zh. éksp. Teor. Fiz. 63, No. 5, 375–380 (10 March 1996)     

Yangian construction of the Virasoro algebra

We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algera.     

Singular vectors of the Virasoro algebra

We present an explicit construction of the singular (or null) vectors in highest weight Verma modules.     

Modified Virasoro and super-Virasoro algebra

Applying the paraquantization of order Q to an open bosonic and spining strings, modified Virasoro and super-Virasoro algebra are obtained. It is shown that the anomaly c-number term is a linear function of Q.     

On the quantum super Virasoro algebra

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.