Gauge coupling of non-linear σ-model and a generalized Mazur identity
An inversionsymmetric class of non-linear σ-models is constructed. The original pure model with field values in the coset space of a classical matrix group G with respect to an isotropy subgroup under the adjoint action is generalized to a minimally gauge coupled model in which the field is a section in a bundle with group G acting on the coset space as fibre with a nontrivial connection of (for example) Yang-Mills type. It is shown that the gauge coupled models admit a natural generalisation of the identities originally constructed by Mazur for the pure nonlinear σ-models whereby the divergence of a quantity whose surface integral vanishes when suitable boundary conditions are satisfied is shown to be equal to a functional of the difference between two sets of field variables that is positive definite in many relevant situation. In such cases, which occur when the base-space metric is positive definite (so that the system is of elliptic type) and the isotropy subgroup is compact, the identities lead directly to uniqueness theorems for the solutions.
KeywordsBase Space Adjoint Action Isotropy Subgroup Coset Space Suitable Boundary Condition
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