Twisted Affine Lie Algebras Research Articles

Integral conjugacy classes of compact Lie groups

We show that untwisted respectively twisted conjugacy classes of a compact and simply connected Lie group which satisfy a certain integrality condition correspond naturally to irreducible highest weight representations of the corresponding affine Lie algebra. Along the way, we review the classification of twisted conjugacy classes of a simply connected compact Lie group G and give a description of their stabilizers in terms of the Dynkin diagram of the corresponding twisted affine Lie algebra.

Twisted boundary states inc= 1 coset conformal field theories

We study the mutual consistency of twisted boundary conditions in the coset conformal field theory G/H. We calculate the overlap of the twisted boundary states of G/H with the untwisted ones, and show that the twisted boundary states are consistently defined in the diagonal modular invariant. The overlap of the twisted boundary states is expressed by the branching functions of a twisted affine Lie algebra. As a check of our argument, we study the diagonal coset theory so(2n)_1 \oplus so(2n)_1/so(2n)_2, which is equivalent with the orbifold S^1/\Z_2. We construct the boundary states twisted by the automorphisms of the unextended Dynkin diagram of so(2n), and show their mutual consistency by identifying their counterpart in the orbifold. For the triality of so(8), the twisted states of the coset theory correspond to neither the Neumann nor the Dirichlet boundary states of the orbifold and yield the conformal boundary states that preserve only the Virasoro algebra.

Weyl's Character Formula for Non-connected Lie Groups and Orbital Theory for Twisted Affine Lie Algebras

We generalize I. Frenkel's orbital theory for non twisted affine Lie algebras to the case of twisted affine Lie algebras using a character formula for certain non-connected compact Lie groups.

Realization of the Space of Conformal Blocks in Lie Algebra Modules

Using integrable irreducible representations of generalized twisted affine Lie algebra modules, we give a realization of the space of conformal blocks of conformal field theory on a stable algebraic curve. Many basic properties of the conformal blocks, such as finite dimensionality of the space, invariance of the conformal blocks under suitable formal neighborhood changes, and the property of “propagation of vacua” are discussed. Finally, a relative local 1-form around a fixed point of the order two automorphism of the curve is given.

Vertex operators, semiclassical limit for soliton S-matrices and the number of bound states in affine Toda field theories

Soliton time delays and the semiclassical limit for soliton S-matrices are calculated for non-simply laced affine Toda field theories. The phase shift is written as a sum over bilinears on the soliton conserved charges. The results apply to any two solitons of any affine Toda field theory. As a by-product, a general expression for the number of bound states and the values of the coupling in which the soliton S-matrix can be diagonal are obtained. In order to arrive at these results, a vertex operator is constructed, in the principal gradation, for non-simply laced affine Lie algebras, extending the previous constructions for simply laced and twisted affine Lie algebras.

Calogero-Moser Lax pairs with spectral parameter for general Lie algebras

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite-dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of G 2).

TWISTED QUANTUM AFFINE ALGEBRAS AND SOLUTIONS TO THE YANG-BAXTER EQUATION

We construct spectral-parameter-dependent R matrices for the quantized enveloping algebras of twisted affine Lie algebras. These give new solutions to the spectral-parameter-dependent quantum Yang-Baxter equation.

Algebraic construction of higher-rank dilute A models

Starting from representations of the Birman-Wenzl-Murakami algebra underlying the critical Bn(1), Cn(1) and Dn(1) RSOS models of Jimbo, Miwa and Okado, we derive four series of solvable, critical RSOS models associated with the twisted affine Lie algebra An(2). Two of these are the critical limit of the A2n−1(2) and A2n(2) models obtained previously by Kuniba. The other two series, again one of the A2n−1(2) and one of the A2n(2) type, are new, and the latter generalizes the dilute A models to arbitrary rank n. For the two new series we present an elliptic extension which satisfies the Yang-Baxter equation, and show that for certain values of the parameters the higher-rank dilute A models break the ℤ2 symmetry of the underlying adjacency graphG, whereGis the level-l Cn(1) weight lattice.

On Some Representations of Twisted Affine Lie Algebras and Combinatorial Identities

On Some Representations of Twisted Affine Lie Algebras and Combinatorial Identities

Level one representations of the twisted affine algebrasa n (2) andd n (2)

We give an explicit description of the Kac-Peterson-Lepowsky construction of the level one representations for the twisted affine Lie algebrasan(2) anddn(2). We assign to any conjugacy class of the Weyl group ofan anddn 1) an equivalent class of maximal Heisenberg subalgebras of the corresponding twisted affine Lie algebras, and 2) multicomponent charged and neutral free fermionic fields. The boson-fermion correspondence for these fields provides us with fermionic vertex operators, whose ‘normal ordered products’ give the (twisted) vertex operators of the Kac-Peterson-Lepowsky construction.

Exact solution of solid-on-solid models for twisted affine Lie algebras A 2 n(2) and A 2 n−1 (2)

We report new exact solutions of restricted and unrestricted solid-on-solid (SOS) models associated with the twisted affine Lie algebras X n (2) = A 2 n (2) and A 2 n−1 (2). The Boltzmann weights are parametrized by elliptic theta functions and fulfill the Yang-Baxter equation. The trigonometric case of the solutions corresponds to the SOS counterpart of the solvable vertex models arising from the X n (2) quantum R-matrices. The restricted models (RSOS) are labeled by the level l as well as the rank n. We show that the level- l A 2 n−1 (2) RSOS model is equivalent to the level- n C l (1) RSOS model, which is a manifestation of the level-rank duality. One-point functions of the A 2 n−1 (2) RSOS model are obtained by using the coset pair C n (1) ⊕ A 2 n−1 (1) ⊃ C n (1). We also determine the one-point functions of the X n (2) vertex models in terms of X n (2) string functions.

Integrable representations of twisted affine Lie algebras

Integrable representations of twisted affine Lie algebras

Calculus of twisted vertex operators

Starting from an arbitrary isometry of an arbitrary even lattice, twisted and shifted vertex operators are introduced. Under commutators, these operators provide realizations of twisted affine Lie algebras. This construction, generalizing a number of known ones, is based on a self-contained "calculus."