GAUGE TRANSFORMATIONS OF TYPE (0,2) TENSOR FIELDS AND ASSOCIATED VARIATIONAL PRINCIPLES

ABSTRACT

This article is concerned with variational problems whose field variables are functions on a product manifold M x G of two manifolds M and G. These field variables are denoted by ψ_hα_j in local coordinates (h, j = 1,…,n = dim M, α = 1,…,r = dim G), and it is supposed that they behave as type (0,2) tensor fields under coordinate transformations on M. The dependance of ψ_hα_j on the coordinates of G is specified by a generalized gauge transformation that depends on a map h: M → G. The requirement that the action integral be independent of the choice of this map imposes conditions on the matrices that define the gauge transformation in a manner that gives rise to a Lie algebra, which in turn imposes a Lie group structure on the manifold G. As in the case of standard gauge theories (whose gauge potentials consist of r type (0,1) vector fields on M) certain combinations of the first derivatives of the field variations ψ_hα_j give rise to a set of r tensors on M, the so-called field strengths, that can be regarded as special representations of G. These may be used to construct appropriate connection and curvature forms of M. The Euler-Lagrange equations of the variational problem, when expressed in terms of such connections, admit a particularly simple representation and give rise a set of n conservation laws in terms of type (1,1) tensor fields.