Anomalies in Instanton Calculus

I develop a formalism for solving topological field theories explicitly, in the case when the explicit expression of the instantons is known. I solve topological Yang-Mills theory with the k = 1 Belavin et al. instanton and topological gravity with the Eguchi-Hanson instanton. It turns out that naively empty theories are indeed nontrivial. Many unexpected interesting hidden quantities (punctures, contact terms, nonperturbative anomalies with or without gravity) are revealed. Topological Yang-Mills theory with G = SU(2) is not just Donaldson theory, but contains a certain link theory. Indeed, local and nonlocal observables have the property ofmarking cycles. Moreover, from topological gravity one learns that an object can be considered BRST exact only if it is so all over the moduli space M, boundary included. Being BRST exact in any interior point of M is not sufficient to make an amplitude vanish. Presumably, recursion relations and hierarchies can be found to solve topological field theories in four dimensions, in particular topological Yang-Mills theory with G = SU(2) on R and topological gravity with the full set of asymptotically locally Euclidean manifolds.