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U(1)-Gauge Theories on \(G_2\)-Manifolds

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Abstract

In this paper, we investigate two types of U(1)-gauge field theories on \(G_2\)-manifolds. One is the U(1)-Yang–Mills theory which admits the classical instanton solutions. We show that \(G_2\)-manifolds emerge from the anti-self-dual U(1) instantons, which is an analogy of Yang’s result for Calabi–Yau manifolds. The other one is the higher-order U(1)-Chern–Simons theory as a generalization of Kähler–Chern–Simons theory. We introduce the notion of higher-order U(1)-instanton, as the vacuum configurations of higher-order U(1)-Chern–Simons theory. By suitable choice of gauge and regularization technique, we calculate the partition function under semiclassical approximation. Finally, to make sure of the invariance at quantum level under the large gauge transformations, we use Deligne–Beilinson cohomology theory to give the higher-order U(1)-Chern–Simons actions (U(1)-BF-type actions) for nontrivial U(1)-principle bundles.

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Notes

  1. Some authors also call those \(G_2\)-instantons.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210014, China

    Zhi Hu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Runhong Zong

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  1. Zhi Hu
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  2. Runhong Zong
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Correspondence to Zhi Hu.

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Communicated by Christoph Kopper.

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Hu, Z., Zong, R. U(1)-Gauge Theories on \(G_2\)-Manifolds. Ann. Henri Poincaré 25, 2453–2487 (2024). https://doi.org/10.1007/s00023-023-01354-6

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