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Deformations of instantons.

M F Atiyah1, N J Hitchin, I M Singer

  • 1Mathematical Institute, Oxford, OX1 3LB, England.

Proceedings of the National Academy of Sciences of the United States of America
|July 1, 1977
PubMed
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This study analyzes self-dual Yang-Mills fields in Euclidean 4-space. For SU(2) gauge theory, solutions are proven to depend on 8k - 3 parameters, where k is the Pontrijagin index.

Area of Science:

  • Theoretical Physics
  • Mathematical Physics

Background:

  • Self-dual Yang-Mills fields are crucial in understanding non-perturbative quantum field theory.
  • Euclidean 4-space provides a mathematical framework for analyzing these fields.

Purpose of the Study:

  • To rigorously determine the number of parameters characterizing solutions for SU(2) self-dual Yang-Mills fields.
  • To establish a precise relationship between solutions and the Pontrijagin index.

Main Methods:

  • The study employs rigorous mathematical analysis within the framework of Euclidean 4-space.
  • Focuses on the SU(2) gauge group to simplify and solve the field equations.

Main Results:

  • It is rigorously demonstrated that the solutions to the self-dual Yang-Mills equations for SU(2) gauge theory are parameterized by 8k - 3 variables.

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  • The parameter k, representing the Pontrijagin index, directly dictates the complexity and number of solutions.

    • Conclusions:

    • The number of independent solutions for SU(2) self-dual Yang-Mills fields is precisely determined by the Pontrijagin index.
    • This finding offers a significant advancement in the classification and understanding of these fundamental field configurations.