SOME APPLICATIONS OF COMPLEX GEOMETRY TO FIELD THEORY
POOL, ROBERT EDWIN
Doctor of Philosophy
Let be compactified complexified Minkowski space, and ('*) twistor space and dual twistor space, respectively, and ambitwistor space, a complex hypersurface in x ('*). There is a geometric correspondence between and such that the points in correspond to the null lines in . Let U be an open set in and U'' the open set in which corresponds to U under this correspondence. There exist canonical isomorphisms between the sets of solutions to generalized zero-rest-mass field equations on U and various cohomology groups on U''. There is a one-to-one correspondence between Yang-Mills fields on U and vector bundles on U'' satisfying a certain triviality condition. Further, a given Yang-Mills field is homogeneous if and only if the corresponding vector bundle on U'' can be extended to third order in x ('*), and the axial Yang-Mills current can be canonically identified with the obstruction to third order extension. The action density of the field corresponds to a certain geometric object on U''.