The linear system for self-dual gauge fields in a spacetime of signature 0

The overdetermined linear system for the self-dual Yang—Mills (SDYM) equations is examined in a flat four-dimensional space whose metric has signature 0. There are three different domains for the system, and correspondingly three (essentially) different solutions to the linear system for a given gauge field. If the gauge potential is real analytic, two of the solutions patch together to give a holomorphic function in an annular region of projective twistor space. Conversely, an arbitrary holomorphic GL(n, )-valued function in such a domain can be uniquely factored (on the real lines) to give a solution to SDYM with gauge group U(n). The set of all real analytic u(n)-valued gauge fields can thus be parametrized by the points of a certain double coset space.     

Loop Group-Valued CS and WZNW models,integrable systems,and self-dual Yang-Mills theory

Affine CS and WZNW theories with values in infinite-dimensional (loop) groups are proposed. It appears that the affine CS theory naturally introduces a spectral parameter into a CS theory. The Sinh-Gordon, KdV, and nonlinear Schrödinger equations are obtained, via Hamiltonian reductions, from the affine WZNW. It is shown that the self-dual Yang-Mills (SDYM) equation is related to the equation of motion of the affine WZNW and, thus, symmetry algebra underlying the SDYM can be identified with the affine two-loop Kac-Moody algebra of the affine WZNW.K. C. Wong Research Award Winner, address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

SELF-DUAL YANG-MILLS FIELDS IN STATIC AXIALLY SYMMETRIC CASE AND RELATED TOPICS

In this article we will present an explicit geometric picture about the complete integrability of the static axially symmetric SDYM equation and the gravitational Ernst equation, interpret the correspondence between their Bäcklund transformation formulae and the transformations from one focal surface of Weingarten congruence to the integrability of the B.T.will be proved. It is shown that for the axially symmetric SDYM equation and gravitational Ernst equation the adjoint space of the group (SL(2r)) is a 3-dimensional Minkowski space, and the corresponding soliton surfaces have negative variable curvature. After introducing the generator R we can explain the B.T. as the rotation around the common tangent between two surfaces of solitons. Using Roccato eqiatopm we will confirm in this paper the integrability of B.T. amd prpve that the B.T. is strong, i. e. , the nwe and old solutions satisfy equations of motion separately. Some related topics are also discussed.     

Explicit quantization of the Chern-Simons action

We quantize the three-dimensional Chern-Simons action explicitly. We found that the geometric quantization of the action strongly depends on the topology of the (fixed-time) Riemann surface. On the disk the phase space and the symplectic structure are the same as those of the (chiral) Wess-Zumino-Witten model. On the torus the Hilbert space is the vector space of characters of Kac-Moody algebras. The fusion rules of the primary fields are derived from theclassical matching condition of the holonomy. In general case, the wave-functional of the theory is the generating function of the current insertion in Wess-Zumino-Witten model.     

Extended non-abelian gauge symmetries in the classical WZNW model

Higher spin extensions of the non-Abelian gauge symmetries for the classical WZNW model are considered. Both linear and nonlinear realizations of the extended affine Kac-Moody algebra are obtained. It is a characteristic property of the WZNW model that it admits a higher spin linear realization of the extended affine Kac-Moody algebra which is equivalent to the corresponding higher spin nonlinear realization of the same algebra. However, in both cases, the higher spin Noether currents do not span an invariant space with respect to their generating transformations. Here, the current space is extended to an invariant space which allows us to gauge the symmetry.Supported by Bulgarian Foundation on Fundamental Research under contract Ph-318/93-95.     

Solitons and vertex operators in twisted affine Toda field theories

Affine Toda field theories in two dimensions constitute families of integrable, relativistically invariant field theories in correspondence with the affine Kac-Moody algebras. The particles which are the quantum excitations of the fields display interesting patterns in their masses and coupling which have recently been shown to extend to the classical soliton solutions arising when the couplings are imaginary. Here these results are extended from the untwisted to the twisted algebras. The new soliton solutions and their masses are found by a folding procedure which can be applied to the affine Kac-Moody algebras themselves to provide new insights into their structures. The relevant foldings are related to inner automorphisms of the associated finite dimensional Lie group which are calculated explicitly and related to what is known as the twisted Coxeter element. The fact that the twisted affine Kac-Moody algebras possess vertex operator constructions emerges naturally and is relevant to the soliton solutions.     

GROUP PROPERTY AND ALGEBRA STRUCTURE FOR A KIND OF BÄCKLUND TRANSFORMATIONS

We explain further the group property and algebra struc-tore for a kind of Bäcklund transformations. If the Lax repre- sentation of a given nonlinear (evolution) system is dependent on a complex spectral parameter fractional Zinearly, the Back-land transformations with respect to the principal pseudopoten- tials (fundamental solutions) of the Lax representation can be identified with solutions of the regular matrix Riemann-Hilbert problem. It is shown that this kind of Riemann-HIbert probl em constructs an infinite dimensional Lie group and its infinitesi- mal form presents itself the infinite dimensional algebra of Kac-Moody type.     

Characteristic identities for Kac-Moody algebras

Characteristic identities are derived for the generators of a simple Kac-Moody algebra in any highest-weight unitary representation. Entries of powers of the characteristic matrix are rigorously defined on such modules. The eigenvalues of the characteristic matrix are shifted by generators of the Virasoro algebra which commutes with the diagonal action of the Kac-Moody algebra on a tensor product module. The characteristic identity can be cast as a product of a finite number of factors linear in the sine of the characteristic matrix, and the corresponding projection operators project on to modules of the diagonal Kac-Moody algebra.     

NONLINEAR σ-MODEL ON MULTID IMENSI CNAL CURVED SPACE WITH CERTAIN CYLINDRICAL SYMMETRY

A dual transformation is found for a class of nonlinear σ-model defined on a multidimensional curved space with cylindrical symmetry. The system is invariant under a proper combination of the dual transformation and the general coordinate transformation. An infinite number of nonlocal conservation laws as well as the Kac-Moody algebra follow directly from the dual transformation. A Backlund transformation that qenerates new solutions from a given. one can also be constructed.     

On Several Equivalent Descriptions of the Self-Dual Yang-Mills Integrable System

In this short communication we show that the new Lax pair for SDYM theory can be derivedfrom prolongation approach and some equivalent expressions for SDYM integrable theoryare given by the "gauge" transformation.     

Gauge Theory of the Generalized Symmetry on the Torus Membrane

The SDIFF(T²)_local-generalized Kac-Moody \hat G(T²) symmetry is an infinite-dimensional group on the torus membrane, whose Lie algebra is the semi-direct sum of the SDIFF(T²)_local algebra and the generalized Kac-Moody algebra \hat g(T²). In this paper, we construct the linearly realized gauge theory of the SDIFF(T²)_local-generalized Kac-Moody \hat G(T²) symmetry.     

On the topological theory of self-dual connections over noncompact Riemann surfaces

A topological action for self-dual connections over noncompact Riemann surfaces is proposed. TheJ formulation and the associated linear system are obtained. A new connection is constructed, depending on a Kac-Moody parameter such that its flatness condition is theJ-equation associated to the self-dual problem. The algebra of infinitesimal Bäcklund transformations depending on this Kac-Moody parameter is constructed.     

On the Mickelsson-Faddeev extension and unitary representations

The Mickelsson-Faddeev extension is a 3-space analogue of a Kac-Moody group, where the central charge is replaced by a space of functions of the gauge potential. This extension is a pullback of a universal extension, where the gauge potentials are replaced by operators in a Schatten ideal, as in non-commutative differential geometry. Our main result is that the universal extension cannot be faithfully represented by unitary operators on a separable Hilbert space. We also examine potential consequences of the existence of unitary representations for the Mickelsson-Faddeev extension.     

D-branes on group manifolds

Possible Dirichlet boundary states for WZW models with untwisted affine super Kac-Moody symmetry are classified for all compact simple Lie groups. They are obtained by inner- and outerautomorphisms of the group. The D-brane world-volume turns out to be a group manifold of a symmetric subgroup, so that the moduli space of D-branes is locally isomorphic to an irreducible Riemannian symmetric space. It is also clarified how these D-branes are transformed into each other under abelian T-duality of the WZW model. Our result implies, for example, that there is no D-particle on the compact simple group manifold. When the D-brane world-volume contains an S¹ factor, the D-brane moduli space becomes locally isomorphic to a hermitian symmetric space and open string world-sheet instantons are allowed.     

Gauge Theory of the Generalized Symmetry on the Torus Membrane

The SDIFF(T2)local-generalized Kac-Moody G(T2) symmetry is an infinite-dimensional group on the torus membrane, whose Lie algebra is the semi-direct sum of the SDIFF(T2)local algebra and the generalized KacMoody algebra g(T2). In this paper, we construct the linearly realized gauge theory of the SDIFF(T2)loc1al-generalized Kac-Moody G(T2) symmetry.``     

Hypergeometric solutions of Knizhnik-Zamolodchikov equations

Explicit integral formulas are presented for the solutions of Knizhnik-Zamolodchikov equations associated with an arbitrary Kac-Moody Lie algebra.     

The structure of theW ∞ algebra

A new definition of spectral data of a monopole is given for any compact Lie or Kac-Moody group. It is shown that the spectral data determines the irreducible monopole. In the case of maximal symmetry breaking the spectral data is shown to reduce to an earlier definition in terms of algebraic curves indexed by the nodes of the Dynkin diagram of the group. The structure of solutions to Nahm's equations corresponding to the monopole is discussed.Research supported in part by NSER C grant A8361 and FCAR grant EQ3518     

Supersymmetry and Kac-Moody algebras

The invariance algebra of the Majorana action contains a Kac-Moody algebra which, on shell, reduces to an Abelian algebra. In the absence of auxiliary fields in the Wess-Zumino model, supersymmetry transformations generate an infinite-dimensional Lie algebra, which is shown to be a Grassmannian extension of this Kac-Moody algebra. The corresponding Noether charges are discussed.     

Integral dimension in the string theory based on a group manifold

We study string models on a group manifold with Kac-Moody symmetry where the critical dimensiond is integer. In particular the possibility of fourdimensional models is investigated. We find that only nine group manifolds with a relevant level can have four as the critical dimension among an infinite number of compact Lee groups. They re all listed. The models with minimal conformal sectors adding to the Kac-Moody sector are investigated. In the cases with one minimal conformal sector, there are only two groups,SU (5) andSO (43), that can gived=4. Among the cases with some tensoring products of minimal conformal sectors we discuss a few special cases withk=0 andk=1. The cases based onN=1 super Kac-Moody algebra are also studied. Finally we discuss the possibility of the enlargement of gauge symmetry.