Gaudin model based on the orthosymplectic Lie superalgebra osp(1|2) is studied. The eigenvectors of the osp(1|2) invariant Gaudin hamiltonians are constructed by algebraic Bethe Ansatz. Corresponding creation operators are defined by a recurrence relation. Furthermore, explicit solution to this recurrence relation is found. The action of the creation operators on the lowest spin vector yields Bethe vectors of the model. The relation between the Bethe vectors and solutions to the Knizhnik-Zamolodchikov equation of the corresponding super-conformal field theory is established.
The problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(1|2) classical r-matrix. The eigenvectors of the trigonometric osp(1|2) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the Knizhnik-Zamolodchikov equation yields the norm of the eigenvectors. The generalized Knizhnik-Zamolodchikov system is discussed both in the rational and in the trigonometric case. 1
Schlesinger transformations are discrete monodromy preserving symmetry transformations of the classical Schlesinger system. Generalizing well-known results from the Riemann sphere we construct these transformations for isomonodromic deformations on genus one Riemann surfaces. Their action on the system's tau-function is computed and we obtain an explicit expression for the ratio of the old and the transformed tau-function.
The authors discuss the reduction of the physical degrees of freedom in the Ashtekar formulation of general relativity which leads to the Bianchi models. They also perform a canonical analysis of the Bianchi models on compact spacelike surfaces. In the case of the class B Bianchi models the constraint algebra does not close because some of the constraints are second class. This property does not exclude a consistent dynamics, but due to purely geometrical reasons, the class B models cannot exist on compact spacelike surfaces. In the case of the class A models the constraints are all first class. However, the number of independent components of the vector constraint varies from 0 (type I) to 3 (type IX). This is an agreement with the results of a similar analysis recently performed by Ashtekar and Samuel (1991) in the metric formulation. The reduced phase space is analysed through a gauge fixing procedure in the case of the type I, II and IX Biachi models.
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