Massive Integrable Models and Infinite Dimensional Symmetries

点击翻译按钮获取中文摘要

In order to answer the question "What symmetries ensure the integrability or infinitely many conserved quantities of massive integrable models (quantum field theory and lattice models)?", we have been studying infinite dimensional symmetries and solvable lattice models.Corresponding to the face type and the vertex type of solvable lattice models whose Boltzmann weights are elliptic solutions of the Yang-Baxter equation, elliptic quantum groups have the face type and the vertex type. One of their differences is the counting of the energy eigenvalues; the former is homogeneous gradation and the latter is principal gradation. Free filed realizations of solvable lattice models were obtained for some face type models but not for the vertex type models direct way. To get some hint for this problem, we studied (quantum) affine Lie algebra with principal gradation. We construct a free field realization of sl^^^^_2 with principal gradation. We point out that the Lepowsky-Wilson's Z algebra, whi … More ch is obtained from sl^^^^_2 with principal gradation by splitting the Cartan part, is certain limit of the deformed Virasoro algebra. In other word the deformed Virasoro algebra can be considered as a q-deformation of Lepowsky-Wilson's Z algebra. This observation may help us to study the vertex models. For higher rank case we establish the same relationship between the sl_N version of Z algebra and the deformed W_N algebra by calculating the defining relation of the deformed W_N algebra explicitly.Calogero-Moser models are very interesting integrable models as both classical and quantum mechanics, for example, they are related to the Seiberg-Witten theory of super Yang-Mills theory, the deformed Virasoro and W algebras, etc. Recently it is pointed out that various quantities at equilibrium positions are 'quantized in integer values' even in classical theory. Motivated by this observation, we calculate the classical equilibrium position of Calogero-Moser models with rational and trigonometric potential for all finite root systems, and define Coxeter (Weyl) invariant polynomials. Coefficients of these polynomials are also integer values. Less