Anomalies in Instanton Calculus

TL;DR

This paper develops a formalism for explicitly solving topological field theories with known instantons, revealing hidden quantities and complex structures, and suggests potential recursive solutions for four-dimensional theories.

Contribution

It introduces a method to solve topological field theories explicitly using known instantons, uncovering new hidden quantities and structures within these theories.

Findings

Explicit solutions for topological Yang-Mills with $k=1$ instanton.

Discovery of hidden quantities like punctures and contact terms.

Identification of link structures within topological Yang-Mills theory.

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.