A geometrical setting for the Lax representation

A geometrical interpretation of the Lax representation is suggested; it is read as the condition on a section of a fiber bundle based on the phase manifold to be covariantly constant, with respect to a suitable connection, along the dynamics.     

N-Superstring vertex and the vertex operator for the emission of the superstring

The BRST invariant, supersymmetric N-string vertex which applies to both, the Neveu-Schwarz and Ramond sector of the superstring is formulated using the vertex operator for the emission of a superstring. It is shown that the N-superstring vertex thus obtained is cyclic symmetric when GSO projected on-shell states and operators for the external strings are applied. The constraint equations of this vertex and their singularity structure are examined and we show that this vertex also has the required property of a transition operator. We also give a proof of its BRST invariance and supersymmetry.     

Lax representation and Kronecker product

We show how the Kronecker product can be used to find new Lax representations.     

A central extension of DY h(gl2) and its vertex representations

A central extension of DY _h(gl₂) is proposed. The bosonization of level 1 module and vertex operators are also given.     

On the Lax representation for the nonlinear evolution equations

By the classical differential geometry techniques it is shown that a general partial differential equation of the second order with two independent variables can be represented in the Lax operator form [X ₁ X ₂]=0, whereX _i =/x _i – ⁱ ,i=1,2 and ⁱ are the 3×3 matrices. The problem of the introduction of the spectral parameter in this representation is shortly discussed.Presented at the International Symposium Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 14–19, 1981.The author is pleased to thank V. K. Mel'nikov for the discussion of this work.     

Induced modules for vertex operator algebras

For a vertex operator algebraV and a vertex operator subalgebraV which is invariant under an automorphismg ofV of finite order, we introduce ag-twisted induction functor from the category ofg-twistedV-modules to the category ofg-twistedV-modules. This functor satisfies the Frobenius reciprocity and transitivity. The results are illustrated withV being theg-invariants in simpleV orV beingg-rational.The first author was supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.The second author was supported by NSF grant DMS-9401389.     

Lax representation for the systems of S. Kowalevskaya type

We describe a number of dynamical systems that are generalizations of the S. Kowalevskaya system and admit the Lax representation.     

A bosonic representation of the twisted string emission vertex

We present an explicit construction of the twist fields which create states in the twisted sectors of the Hilbert space in twisted bosonic string models. Such vertex operators have a complicated structure, which intertwines between the various sectors in the theory, and therefore represents a bosonic analogue of the fermion emission vertex which intertwines between the two sectors of the spinning string.     

Lax representation with a spectral parameter for the Kowalewski top and its generalizations

We construct a Lax pair with a spectral parameter for the Kowalewski top in two constant force fields combined with magnetic interaction, and for its higher-dimensional generalizations.     

Classification of screening systems for lattice vertex operator algebras

Letters in Mathematical Physics - We study and classify systems of certain screening operators arising in a generalized vertex operator algebra, or more generally an abelian intertwining algebra...     

On the classification of simple vertex operator algebras

Inspired by a recent work of Frenkel-Zhu, we study a class of (pre-)vertex operator algebras (voa) associated to the self-dual Lie algebras. Based on a few elementary structural results we propose thatV, the category of Z₊-graded prevoasV in whichV[0] is one-dimensional, is a proper setting in which to study and classify simple objects. The categoryV is organized into what we call the minimalk ^th types. We introduce a functor —which we call the Frenkel-Lepowsky-Meurman functor—that attaches to each object inV a Lie algebra. This is a key idea which leads us to a (relative) classification of thesimple minimal first type. We then study the set of all Virasoro structures on a fixed minimal first typeV, and show that they are in turn classified by the orbits of the automorphism group Aut((V)) in cent((V)). Many new examples of voas are given. Finally, we introduce a generalized Kac-Casimir operator and give a simple proof of the irreducibility of the prolongation modules over the affine Lie algebras.     

The Lax representation for an integrable class of relativistic dynamical systems

We exhibit the Lax pair for the class of relativistic dynamical systems recently introduced by Ruijsenaars and Schneider, whose equations of motion read, whereB is an arbitrary constant and the Weierstrass elliptic function.For the 3 academic years 1983–1986     

Effective integral equation and vertex operator of the green function

A simple and rigorous formulation of effective integral equations for the Green functions is presented and a general formula for the effective vertex operator (or function) has been derived. As an illustration, this formalism has been applied to obtain (i) the effective Dyson equation and effective mass operator for the single-particle Green function, and (ii) the effective integral equation and effective interaction operator for the two-particle Green function as well as those for the particle-hole Green function.     

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE ₈) and to get various extensions of the vertex operator algebras associated with integrable representations.Supported by NSF grant DMS-9303374 and a research grant from the Committee on Research, UC Santa Cruz.Supported by NSF grant DMS-9401272 and a research grant from the Committee on Research, UC Santa Cruz.     

New representation of two-loop propagator and vertex functions

We present a new method of calculating scalar propagator and vertex functions in the two-loop approximation, for arbitrary masses of particles. It is based on a double integral representation, suitable for numerical evaluation. Real and imaginary parts of the diagrams are calculated separately, so that there is no need to use complex arithmetics in the numerical program.     

Kraus representation for the density operator of a qubit

We show that the time evolution of the density operator of a qubit can always be described in terms of the Kraus representation. A general scheme on how to construct the Kraus operators for an open qubit system is proposed, which can be generalized to open higher dimensional quantum systems.     

A finite sum representation of the angular momentum projection operator

Exact finite sum representations of the angular momentum projection operator in the following two cases are derived: i) when the intrinsic state is axially symmetric but the azimuthal quantum numberK is not equal to zero, ii) when the intrinsic state does not have axial symmetry. Advantages of such representations over projection via exact numerical quadrature are discussed.     

Lax representation with spectral parameter on a torus for integrable particle systems

Complete integrability is proved for the most general class of systems of interacting particles on a straight line with the Hamiltonian including elliptic functions of coordinates, depending on seven arbitrary parameters and having the structure defined by the root systems of the classical Lie algebras. The Lax representation for them depends on the spectral parameter given on a complex torus /, where is the lattice of periods of the Jacobi functions dependent on the Hamiltonian parameters. The possibility of constructing explicit solutions to the equations of motion is discussed.