SCALAR FIELD THEORY ON A KÄHLER MANIFOLD

The invariant one-loop counter terms for a class of non-polynomial complex scalar field theories are found using the background field geodesic expansion together with heat kernel methods. These theories have dynamics invariant under a group G with fields that transform linearly only under a subgroup H of G. The fields are coordinates on a complex Kählerian manifold corresponding to the coset space G/H. The technique is applied to the CP n model.

The spectra of operators on CP n

It is shown that the eigenvalues and degeneracies of Laplace-type operators on CP n may be obtained from those on S 2 n +1 essentially by projection along the Hopf fibration. This projection simplifies but does not completely reduce the problem, and extensive use of the SU( n +1) symmetry group of CP n and its Casimir operators is needed to complete the analysis. The entire calculation is ultimately reduced to one of group representation theory. Finally, some remarks are made about the implications of the work on the one-loop counter-terms of supersymmetric theories in CP 2 .