SPECIAL QUANTUM-FIELD THEORIES IN 8 AND OTHER DIMENSIONS

We build nearly topological quantum field theories in various dimensio ns. We give special attention to the case of eight dimensions for whic h we first consider theories depending only on Yang-Mills fields. Two classes of gauge functions exist which correspond to the choices of tw o different holonomy groups in SO(8), namely SU(4) and Spin(7). The ch oice of SU(4) gives a quantum field theory for a Calabi-Yau fourfold. The expectation values for the observables are formally holomorphic Do naldson invariants, The choice of Spin(7) defines another eight dimens ional theory for a Joyce manifold which could be of relevance in M- an d F-theories. Relations to the eight dimensional supersymmetric Yang-M ills theory are presented. Then, by dimensional reduction, we obtain o ther theories, in particular a four dimensional one whose gauge condit ions are identical to the non-abelian Seiberg-Witten equations. The la tter are thus related to pure Yang-Mills self-duality equations in 8 d imensions as well as to the N=1, D=10 super Yang-Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensio ns.