Sugawara's construction for Kac-Moody-Malcev algebras

We give a construction of the Virasoro algebra in terms of bilinear combinations of currents. The currents satisfy the Kac-Moody-Malcev commutation relations. The Kac-Moody-Malcev algebras are the generalization of Lie algebras of Kac-Moody type to the Malcev algebras. Thus, we give the generalization of the Sugawara construction to the case of Kac-Moody-Malcev algebras.     

A Characterization of Affine Kac-Moody Lie Algebras

We give a new characterization of the affine Kac-Moody algebras in terms of extended affine Lie algebras. We also present new realizations of the twisted affine Kac-Moody algebras. Received: 17 April 1996 / Accepted: 11 October 1996     

Kac-Moody current algebras of D = 2 massless gauge theories,their representations and applications

We give a classification of the Kac-Moody current algebras of all the possible massless fermion-gauge theories in two dimensions. It is shown that only Kac-Moody algebras based on A_N, B_N, C_N, and D_N in the Cartan classification with all possible central charge occur. The representation of local fermion fields and simply laced Kac-Moody algebras with minimal central charge in terms of free boson fields on a compactified space is discussed in detail, where stress is laid on the role played by the boundary conditions on the various collective modes. Fractional solitons and the possible soliton representation of certain nonsimply laced algebras is also analysed. We briefly discuss the relationship between the massless bound state sector of these two-dimensioned gauge theories and the critically coupled two-dimensional nonlinear sigma model, which share the same current algebra. Finally we briefly discuss the relevance of Sp(n) Kac-Moody algebras to the physics of monopole-fermion systems.     

Quivers and lie superalgebras

Indecomposable representations of quivers are in 1–1 correspondence with positive weight vectors of Kac-Moody algebras. The collection of indecomposable representations of the quiver is tame if the quiver corresponds to a Kac-Moody algebra of polynomial growth. What corresponds to positive roots of Lie algebras of polynomial growth different from Kac-Moody algebras? The classification problem for tame representations of quivers associated to Lie superalgebras is a natural step towards the answer to this question. As an aside we announce a classification of simple graded Lie superalgebras of polynomial growth.     

On the fundamental representation of borcherds algebras with one imaginary simple root

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, but the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds by hand one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory.Supported by Konrad-Adenauer-Stiftung e.V.Supported by Deutsche Forschungsgemeinschaft.     

Vertex operators for non-simply-laced algebras

A vertex operator construction is given for the level one representations of the affine Kac-Moody algebras associated with non-simply-laced finite-dimensional Lie algebras, using free boson and interacting fermion fields. The fermion fields are constructed explicitly and a detailed discussion is given of the theory of the cocycles necessary for this and other vertex operator constructions. The construction is related in detail to the folding of Dynkin diagrams and a generalisation of it yields Freudenthal's magic square.On leave from the Weizmann Institute, Israel     

Hamiltonian structures for integrable classical theories from graded Kac-Moody algebras

An infinite number of canonical representations for integrable classical field theories of non-ultralocal type are constructed from graded Kac-Moody algebras. The principal chiral field models are shown to be a particular example of this construction.     

Sugawara construction for higher genus Riemann surfaces

By the classical genus zero Sugawara construction one obtains representations of the Virasoro algebra from admissible representations of affine Lie algebras (Kac-Moody algebras of affine type). In this lecture, the classical construction is recalled first. Then, after giving a review on the global multi-point algebras of Krichever-Novikov type for compact Riemann surfaces of arbitrary genus, the higher genus Sugawara construction is introduced. Finally, the lecture reports on results obtained in a joint work with O. K. Sheinman. We were able to show that also in the higher genus, multi-point situation one obtains (from representations of the global algebras of affine type) representations of a centrally extended algebra of meromorphic vector fields on Riemann surfaces. The latter algebra is a generalization of the Virasoro algebra to higher genus.     

Kac-Moody groups,topology of the Dirac determinant bundle,and fermionization

The relation between Kac-Moody groups and algebras and the determinant line bundle of the massless Dirac operator in two dimensions is clarified. Analogous objects are studied in four space-time dimensions and a generalization of Witten's fermionization mechanism is presented in terms of the topology of the Dirac determinant bundle.     

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.     

Current algebra and Wess-Zumino model in two dimensions

We investigate quantum field theory in two dimensions invariant with respect to conformal (Virasoro) and non-abelian current (Kac-Moody) algebras. The Wess-Zumino model is related to the special case of the representations of these algebras, the conformal generators being quadratically expressed in terms of currents. The anomalous dimensions of the Wess-Zumino fields are found exactly, and the multipoint correlation functions are shown to satisfy linear differential equations. In particular, Witten's non-abelean bosonisation rules are proven.     

A generalization of the Kac-Moody algebras with a parameter on an algebraic curve and perturbations of solitons

The Lie-algebraic approach for the dynamic systems associated with a generalization of the Kac-Moody algebras on Riemann surfaces is developed. A technique of solving the inverse scattering problem of operators with spectral parameters on Riemann surfaces is presented. Some equations associated with generalized Kac-Moody algebras are presented. The connection between their hamiltonian structure and deformed Lax representation is discussed as well as its applications to some special perturbations of integrable systems.     

Modular invariance in N=2 superconformal field theories

The modular invariance properties of two-dimensional N=2 superconformal field theories are studied. It is shown that the character formulae of the central charge c<3 unitary highest weight representation for the untwisted algebras can be written in terms of the string functions and the theta functions of the affine su(2) Kac-Moody algebra. Deriving the modular transformation of the characters we construct the modular invariant partition functions on a torus. The character relation corresponding to the coset space construction of the unitary discrete series in the N=2 algebra is also obtained.     

Relating Kac-Moody,Virasoro and Krichever-Novikov algebras

We demonstrate that the Kac-Moody and Virasoro-like algebras on Riemann surfaces of arbitrary genus with two punctures introduced by Krichever and Novikov are in two ways linearly related to Kac-Moody and Virasoro algebras onS ¹. The two relations differ by a Bogoliubov transformation, and we discuss the connection with the operator formalism.     

Spontaneous symmetry breaking of affine field theory

It is possible to construct non-Abelian field theories by gauging Kac-Moody algebras. Here we discuss the spontaneous symmetry breaking of such theories via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra.     

Perturbative BPS algebras in superstring theory

This paper investigates the algebraic structure that exists on perturbative BPS states in the superstring, compactified on the product of a circle and a Calabi-Yau fourfold. This structure was defined in a recent article by Harvey and Moore. It is shown that for a toroidal compactification this algebra is related to a generalized Kac-Moody algebra. The BPS algebra itself is not a Lie algebra. However, it turns out to be possible to construct a Lie algebra with the same graded dimensions, in terms of a half-twisted model. The dimensions of these algebras are related to the elliptic genus of the transverse part of the string algebra. Finally, the construction is applied to an orbifold compactification of the superstring.     

The exceptional Jordan algebra and the superstring

Two representations of the exceptional Jordan algebra are presented, one in terms of bose vertex operators, and the other in terms of superstring vertex operators in bosonised form, including their BRST ghost contributions. It is also shown how the non-exceptional Jordan algebras may be constructed similarly.     

Symmetric Lie algebras of non-linear transformations of conformal type in quantum mechanics

The Koecher construction of simple symmetric Lie algebras is used to realize colineation and conformai Lie algebras of non-linear transformations of a pseudo-orthogonal vector space in the canonical Weyl algebras, which are used in the Schrödinger representation. The realization maps the linear sub-algebras onto symmetrized polynomials of second degree, whereas the non-linear parts are mapped onto polynomials of first and third degree. For the two examples the Meyberg Jordan algebras are explicitly given.     

Langlands Duality for Representations and Quantum Groups at a Root of Unity

We give a representation-theoretic interpretation of the Langlands character duality of [FH], and show that the “Langlands branching multiplicities” for symmetrizable Kac-Moody Lie algebras are equal to certain tensor product multiplicities. For finite type quantum groups, the connection with tensor products can be explained in terms of tilting modules.