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<p>For a given closed two-form, we introduce the cone Yang–Mills functional which is a Yang–Mills-type functional for a pair (<i>A</i>,&#xa0;<i>B</i>), a connection one-form <i>A</i> and a scalar <i>B</i> taking value in the adjoint representation of a Lie group. The functional arises naturally from dimensionally reducing the Yang–Mills functional over the fiber of a circle bundle with the two-form being the Euler class. We write down the Euler–Lagrange equations of the functional and present some of the properties of its critical solutions, especially in comparison with Yang–Mills solutions. We show that a special class of three-dimensional solutions satisfy a duality condition which generalizes the Bogomolny monopole equations. Moreover, we analyze the zero solutions of the cone Yang–Mills functional and give an algebraic classification characterizing principal bundles that carry such cone-flat solutions when the two-form is non-degenerate.</p>

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Mapping Cone Connections and their Yang–Mills Functional

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年6月3日

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  • Li-Sheng Tseng,
  • Jiawei Zhou

摘要

For a given closed two-form, we introduce the cone Yang–Mills functional which is a Yang–Mills-type functional for a pair (A, B), a connection one-form A and a scalar B taking value in the adjoint representation of a Lie group. The functional arises naturally from dimensionally reducing the Yang–Mills functional over the fiber of a circle bundle with the two-form being the Euler class. We write down the Euler–Lagrange equations of the functional and present some of the properties of its critical solutions, especially in comparison with Yang–Mills solutions. We show that a special class of three-dimensional solutions satisfy a duality condition which generalizes the Bogomolny monopole equations. Moreover, we analyze the zero solutions of the cone Yang–Mills functional and give an algebraic classification characterizing principal bundles that carry such cone-flat solutions when the two-form is non-degenerate.