6 A ug 1 99 7 Nuclear Field Theory and Chiral Symmetry on a Calabi - Yau Manifold

The purpose of this contribution is to show how a nuclear field theory follows naturally from the structure of four–dimensional Riemannian geometry. A Yang– Mills field is introduced by constructing fibres that include all possible exchanges of spin, parity and charge such that the collective quantum numbers remain the same. In this way O(4) internal symmetry transformations are found and a connection is obtained by exponentiation of a CP–invariant operator C associated with the ground state. The metric is Calabi–Yau and Einstein. Carbon 13 is chosen as an example because it is the lightest nucleus to exhibit small spin mutations even though there is no deformation parameter in the O(4) com-mutation relations. Instead a supersymmetric transformation replaces a quantum group. Mirror symmetry is also discussed.