On Seifert fibered spaces bounding definite manifolds

We establish an inequality which gives strong restrictions on when the standard definite plumbing intersection lattice of a Seifert fibered space over $S^2$ can embed into a standard diagonal lattice, and give two applications. First, we answer a question of Neumann-Zagier on the relationship between Donaldson’s theorem and Fintushel-Stern’s $R$-invariant. We also give a short proof of the characterisation of Seifert fibered spaces which smoothly bound rational homology $S^1\times D^3$’s.