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1 results

  • 8 - More Topological Saddles

  • from Part II - More on Gauge Theories
  • Piljin Yi, Korea Institute for Advanced Study
  • Book:
    Geometric Quantum Field Theories
    Published online:
    05 September 2026
    Print publication:
    03 September 2026, pp 265-294

  • Summary

    We continue exploring topological and non-perturbative saddles of path integrals for gauge field theories. Magnetic monopoles have a very close relationship to Yang-Mills instantons, although different topological quantities characterize them. We focus on the d=4 theory compactified on a circle, where (anti-)self-duality can be used for simple and explicit solutions. Monopoles obtained this way carry fractional instanton charges in addition to magnetic charges. We explain Polyakov’s confinement mechanism in d=3, which starts from magnetic monopoles as d=3 instantons, and then move on to how the same mechanism may be uplifted to d=4 compactified on a circle, where magnetic monopoles serve an important role as fractionated d=4 Yang-Mills instantons. After a brief discussion of d=2 U(1) vortices and d=5 Sp(k) theories with a discrete theta angle, we turn to the most subtle matter of cyclic bundles characterized by Stiefel-Whitney classes and cataloged by the center of the gauge algebra. The role of Lie groups as opposed to Lie algebras in these fine distinctions is mulled over.