Topological field theory and physics

TL;DR

This paper explores topological Yang-Mills theory, revealing its link invariants, physical relevance, and embedding into quantum field theory, providing new insights into nonperturbative effects and topological classifications in gauge theories.

Contribution

It offers a complete solution to topological Yang-Mills theory with instantons, demonstrating the physical significance of topological invariants and embedding TQFT into QFT for consistent perturbative expansions.

Findings

Link invariants persist with matter coupling

Topological Yang-Mills theory is embedded into QFT

Link numbers relate to non-abelian Aharonov-Bohm effect

Abstract

Topological Yang-Mills theory with the Belavin-Polyakov-Schwarz-Tyupkin $SU(2)$ instanton is solved completely, revealing an underlying multi-link intersection theory. Link invariants are also shown to survive the coupling to a certain kind of matter (hyperinstantons). The physical relevance of topological field theory and its invariants is discovered. By embedding topological Yang-Mills theory into pure Yang-Mills theory, it is shown that the topological version TQFT of a quantum field theory QFT allows us to formulate consistently the perturbative expansion of QFT in the topologically nontrivial sectors. In particular, TQFT classifies the set of good measures over the instanton moduli space and solves the inconsistency problems of the previous approaches. The qualitatively new physical implications are pointed out. Link numbers in QCD are related to a non abelian analogoue of the Aharonov-Bohm effect.