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##### Self-Consistent-Field Method and tau-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

**Authors**: Nishiyama S, da Providencia J, Providencia C, Cordeiro F, Komatsu T

**Ref.**: SYMMETRY INTEGRABILITY AND GEOMETRY-METHODS AND APPLICATIONS **5**, 009 (2009)

**Abstract**: The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., nu (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the nu-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a nu-dependent potential with a Gamma-periodicity. A bilinear equation for the nu-HF theory has been transcribed onto the corresponding tau-function using the regular representation for the group and the Schur-polynomials. The nu-HF SCF theory on an infinite-dimensional Fock space F-infinity leads to a dynamics on an infinite-dimensional Grassmannian Gr(infinity) and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr(infinity) which is affiliated with the group manifold obtained by reducting gl(infinity) to sl(N) and su(N). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.