Four-manifold geography and superconformal symmetry

A compact oriented 4-manifold is defined to be of "superconformal simple ty pe" if certain polynomials in the basic classes (constructed using the Seib erg-Witten invariants) vanish identically. We show that all known 4-manifol ds of b(2)(+) > 1 are of superconformal simple type, and that the numerical invariants of 4-manifolds of superconformal simple type satisfy a generali zation of the Noether inequality. We sketch how these phenomena are predict ed by the existence of certain four-dimensional superconformal quantum fiel d theories.