Quasitriangular WZW Model

Abstract

A dynamical system is canonically associated to every Drinfeld double of any affine Kac-Moody group. In particular, the choice of the affine Lu-Weinstein double gives a smooth one-parameter deformation of the standard WZW model. The deformed WZW model is exactly solvable and it admits the chiral decomposition. Its classical action is not invariant with respect to the left and right action of the loop group, however, it satisfies the weaker condition of the Poisson-Lie symmetry. The structure of the deformed WZW theory is characterized by several ordinary and dynamical r-matrices with spectral parameter. They describe the q-deformed current algebras, appear in the definition of q-primary fields and characterize the quasitriangular exchange (braiding) relations. The symplectic structure of the deformed chiral WZW theory is cocharacterized by the same elliptic dynamical r-matrix that appears in the Bernard generalization of the Knizhnik-Zamolodchikov equation, with q entering the modular parameter of the Jacobi theta functions. This reveals a remarkable connection between the classical q-deformed WZW model and the quantum standard WZW theory on elliptic curves. ContentsQuasitriangular WZW Model 681 a The classical action (1.3) makes sense even if this solution θ q exists only locally onM L . Rev. Math. Phys. 2004.16:679-808. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 04/01/15. For personal use only. Quasitriangular WZW Model 683 model is Kac-Moody Poisson-Lie symmetric. Due to this property, the quantized quasitriangular chiral WZW model will enjoy the q-Kac-Moody symmetry.Remarks. (i) The deformation of the master model (1) exists for every Drinfeld double ofG. However, the two-step symplectic reduction can be performed only ifD is the WZW double (see above). We found with satisfaction that a nontrivial WZW doubleD ofG can indeed be constructed; it is in fact nothing but the complexifi-cationD =G C of the affine Kac-Moody groupG equipped with certain invariant maximally noncompact inner product onD = Lie(D). We shall callD the "affine Lu-Weinstein double", since it turns out to be the natural affine generalization of the standard Lu-Weinstein double D 0 of the group G 0 .(ii) The Weyl alcove A + is usualy viewed as the fundamental domain of the action of the affine Weyl group on the Cartan subalgebra T of G 0 ≡ Lie(G 0 ). The affine Weyl group acts also on the Cartan subalgebraT ofG ≡ Lie(G). The alcovẽ A + is again the fundamental domain of this action. It is also important to note that the second floor chiral Hamiltoniañ

On Moment Maps Associated to a Twisted Heisenberg Double

“…We begin this section by introducing a particular example of the deformation of the WZW model which was not discussed in [9,10,11]. Then we shall perform the symplectic reduction of this u-deformed WZW model with respect to a non-anomalous quasi-adjoint action submoment map which is a sort of combination of the moment maps constructed in Secs.…”

“…It was conjectured in [9] and explained in detail in [11] that the standard WZW model [17] on a compact Lie group K is a dynamical system whose phase space can be identified with certain (decomposable) twisted Heisenberg double of a loop group LK. Moreover, the symplectic form of the WZW model is just the inverse of the fundamental Semenov-Tian-Shansky Poisson bivector (9).…”

“…Moreover, the symplectic form of the WZW model is just the inverse of the fundamental Semenov-Tian-Shansky Poisson bivector (9). The basic idea of the article [9] can be rephrased as follows: since the loop group LK may possess several different twisted Heisenberg doubles (D, κ), it makes sense to consider the dynamical system based on each of (D, κ) as a sort of generalized WZW model. The (twisted Heisenberg) double of the standard WZW model is distinguished among all other doubles of the loop group LK by the fact that the cosymmetry group B is Abelian.…”

“…Many dynamical systems can be deformed in such a way that their ordinary symmetries become Poisson-Lie. Among such systems there is also the standard WZW model [17] where the loop group symmetry gets deformed [9]. It was conjectured in [9], that the principal geometric structure underlying the deformed WZW model is the so called twisted Heisenberg double.…”

“…Among such systems there is also the standard WZW model [17] where the loop group symmetry gets deformed [9]. It was conjectured in [9], that the principal geometric structure underlying the deformed WZW model is the so called twisted Heisenberg double. We announced in [11] that, indeed, the conjecture is true, however, we could not say the full story in [11] due to the limited size of contributions to the conference proceedings.…”