TY - JOUR

T1 - Liouville type models in the group theory framework

T2 - I. Finite-dimensional algebras

AU - Gerasimov, A.

AU - Kharchev, S.

AU - Morozov, A.

AU - Olshanetsky, M.

AU - Marshakov, A.

AU - Mironov, A.

PY - 1997/6/10

Y1 - 1997/6/10

N2 - In this series of papers we represent the "Whittaker" wave functional of the (d + 1)-dimensional Liouville model as a correlator in (d + 0)-dimensional theory of the sine-Gordon type (for d = 0 and 1). The asymptotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple Γ function factors over all positive roots of the corresponding algebras (finite-dimensional for d = 0 and affine for d = 1). This is in nice correspondence with the recent results on two-and three-point correlators in the 1 + 1 Liouville model, where emergence of peculiar double periodicity is observed. The Whittaker wave functions of (d + 1)-dimensional nonaffine ("conformal") Toda type models are given by simple averages in the (d + 0)-dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free field wave functional, which was originally a Gaussian integral over the interior of a (d + 1)-dimensional disk with given boundary conditions, as a (nonlocal) quadratic integral over the d-dimensional boundary itself. In this paper we concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions are known, and we present a survey. We also construct new "Gauss" Whittaker functions.

AB - In this series of papers we represent the "Whittaker" wave functional of the (d + 1)-dimensional Liouville model as a correlator in (d + 0)-dimensional theory of the sine-Gordon type (for d = 0 and 1). The asymptotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple Γ function factors over all positive roots of the corresponding algebras (finite-dimensional for d = 0 and affine for d = 1). This is in nice correspondence with the recent results on two-and three-point correlators in the 1 + 1 Liouville model, where emergence of peculiar double periodicity is observed. The Whittaker wave functions of (d + 1)-dimensional nonaffine ("conformal") Toda type models are given by simple averages in the (d + 0)-dimensional theories of the affine Toda type. This phenomenon is in obvious parallel with representation of the free field wave functional, which was originally a Gaussian integral over the interior of a (d + 1)-dimensional disk with given boundary conditions, as a (nonlocal) quadratic integral over the d-dimensional boundary itself. In this paper we concentrate on the finite-dimensional case. The results for finite-dimensional "Iwasawa" Whittaker functions are known, and we present a survey. We also construct new "Gauss" Whittaker functions.

UR - http://www.scopus.com/inward/record.url?scp=0000507927&partnerID=8YFLogxK

U2 - 10.1142/S0217751X97001444

DO - 10.1142/S0217751X97001444

M3 - Article

AN - SCOPUS:0000507927

VL - 12

SP - 2523

EP - 2583

JO - International Journal of Modern Physics A

JF - International Journal of Modern Physics A

SN - 0217-751X

IS - 14

ER -