Universal Bundle, Generalized Russian Formula and Non-Abelian Anomaly in Topological Yang-Mills Theory

TL;DR

This paper explores the algebraic and geometric structures in Topological Yang-Mills theory, introducing a generalized Russian formula to identify non-Abelian anomalies that may affect Donaldson invariants.

Contribution

It provides a natural generalization of the Russian formula and descent equations using the universal bundle formalism, aiding in the analysis of non-Abelian anomalies.

Findings

Generalized Russian formula for Topological Yang-Mills

Identification of non-Abelian anomalies in the theory

Potential obstruction to defining Donaldson invariants

Abstract

We re-examine the geometry and algebraic structure of BRST's of Topological Yang-Mills theory based on the universal bundle formalism of Atiyah and Singer. This enables us to find a natural generalization of the {\it Russian formula and descent equations\/}, which can be used as algebraic method to find the non-Abelian anomalies counterparts in Topological Yang-Mills theory. We suggest that the presence of the non-Abelian anomaly obstructs the proper definition of Donaldson's invariants.