Anomalies in Instanton Calculus

Abstract

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 {\sl 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 {\sl link} theory. Indeed, local and non-local observables have the property of {\sl marking} cycles. From topological gravity one learns that an object can be considered BRST exact only if it is so all over the moduli space ${\cal M}$, boundary included. Being BRST exact in any interior point of ${\cal M}$ 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 ${\bf R}^4$ and topological gravity on ALE manifolds.