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Yang–Mills bar connection and holomorphic structure

  • School of Mathematical Sciences, University of Science and Technology of China; Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, Hefei 230026, China

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https://doi.org/10.52396/JUSTC-2023-0136
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  • Author Bio:

    Teng Huang is an Associate Professor at the School of Mathematical Sciences, University of Science and Technology of China (USTC). He received his Ph.D. degree in Mathematics from USTC in 2016. His research mainly focuses on mathematical physics and differential geometry

  • Corresponding author: E-mail: htmath@ustc.edu.cn
  • Received Date: 08 September 2023
  • Accepted Date: 15 February 2024
    Abstract

  • Abstract

    In this note, we study the Yang–Mills bar connection $ A $, i.e., the curvature of $ A $ obeys $ \bar{\partial}_{A}^{\ast}F_{A}^{0,2} = 0 $, on a principal $ G $-bundle $ P $ over a compact complex manifold $ X $. According to the Koszul–Malgrange criterion, any holomorphic structure on $ P $ can be seen as a solution to this equation. Suppose that $ G = SU(2) $ or $ SO(3) $ and $ X $ is a complex surface with $ H^{1}(X,\mathbb{Z}_{2}) = 0 $. We then prove that the $ (0,2) $-part curvature of an irreducible Yang–Mills bar connection vanishes, i.e., $ (P,\bar{\partial}_{A}) $ is holomorphic.

    Graphical abstract

    Yang–Mills bar connection and holomorphic structure.

    Abstract

    In this note, we study the Yang–Mills bar connection $ A $, i.e., the curvature of $ A $ obeys $ \bar{\partial}_{A}^{\ast}F_{A}^{0,2} = 0 $, on a principal $ G $-bundle $ P $ over a compact complex manifold $ X $. According to the Koszul–Malgrange criterion, any holomorphic structure on $ P $ can be seen as a solution to this equation. Suppose that $ G = SU(2) $ or $ SO(3) $ and $ X $ is a complex surface with $ H^{1}(X,\mathbb{Z}_{2}) = 0 $. We then prove that the $ (0,2) $-part curvature of an irreducible Yang–Mills bar connection vanishes, i.e., $ (P,\bar{\partial}_{A}) $ is holomorphic.

    • Public Summary

    • A connection A is called Yang–Mills bar connection if the curvature of the connection A satisfies $ \bar{\partial}_{A}^{\ast}F_{A}^{0,2} = 0 $.
    • When the structure group $ G = SU(2) $ or $ SO(3) $, we show that ${\rm {rank}}\left ( F^{0,2}_{A}+F_{A}^{2,0} \right ) \le 1$.
    • Suppose that $ H^{1}(X,\mathbb{Z}_{2}) = 0 $, following an idea from Donaldson, we prove that $ F_{A}^{0,2} = 0 $.

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    • References(13)
  • [1]
    Dai B, Guan R. Transversality for the full rank part of Vafa–Witten moduli spaces. Comm. Math. Phys., 2022, 389: 1047–1060. doi: 10.1007/s00220-021-04176-x
    [2]
    Donaldson S K. Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 1985, 50 (1): 1–26. doi: 10.1112/plms/s3-50.1.1
    [3]
    Donaldson S K, Kronheimer P B. The Geometry of Four-Manifolds. Oxford, UK: Oxford University Press, 1990 .
    [4]
    Huybrechts D. Complex Geometry: An Introduction. Berlin: Springer, 2005 .
    [5]
    Itoh M. Yang–Mills connections over a complex surface and harmonic curvature. Compositio Mathematica, 1987, 62: 95–106.
    [6]
    Le H V. Yang–Mills bar connections over compact Kähler manifolds. Archivum Mathematicum (Brno), 2010, 46: 47–69.
    [7]
    Mares B. Some analytic aspects of Vafa–Witten twisted N = 4 supersymmetric Yang–Mills theory. Thesis. Cambridge, USA: Massachusetts Institute of Technology, 2010 .
    [8]
    Koszul J L, Malgrange B. Sur certaines structures fibrées complexes. Archiv der Mathematik, 1958, 9: 102–109. doi: 10.1007/BF02287068
    [9]
    Newlander A, Nirenberg L. Complex analytic coordinates in almost complex manifolds. Ann. Math., 1957, 65 (3): 391–404. doi: 10.2307/1970051
    [10]
    Păunoiu A, Rivière T. Sobolev connections and holomorphic structures over Kähler surfaces. J. Func. Anal., 2021, 280 (12): 109003. doi: 10.1016/j.jfa.2021.109003
    [11]
    Stern M. Geometry of minimal energy Yang–Mills connections. J. Differential Geom., 2010, 86 (1): 163–188. doi: 10.4310/jdg/1299766686
    [12]
    Tanaka T. Some boundedness properties of solutions to the Vafa–Witten equations on closed 4-manifolds. The Quarterly Journal of Mathematics, 2017, 68 (4): 1203–1225. doi: 10.1093/qmath/hax015
    [13]
    Uhlenbeck K, Yau S T. On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure and Appl. Math., 1986, 39 (S1): S257–S293. doi: 10.1002/cpa.3160390714
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    [1]
    Dai B, Guan R. Transversality for the full rank part of Vafa–Witten moduli spaces. Comm. Math. Phys., 2022, 389: 1047–1060. doi: 10.1007/s00220-021-04176-x
    [2]
    Donaldson S K. Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 1985, 50 (1): 1–26. doi: 10.1112/plms/s3-50.1.1
    [3]
    Donaldson S K, Kronheimer P B. The Geometry of Four-Manifolds. Oxford, UK: Oxford University Press, 1990 .
    [4]
    Huybrechts D. Complex Geometry: An Introduction. Berlin: Springer, 2005 .
    [5]
    Itoh M. Yang–Mills connections over a complex surface and harmonic curvature. Compositio Mathematica, 1987, 62: 95–106.
    [6]
    Le H V. Yang–Mills bar connections over compact Kähler manifolds. Archivum Mathematicum (Brno), 2010, 46: 47–69.
    [7]
    Mares B. Some analytic aspects of Vafa–Witten twisted N = 4 supersymmetric Yang–Mills theory. Thesis. Cambridge, USA: Massachusetts Institute of Technology, 2010 .
    [8]
    Koszul J L, Malgrange B. Sur certaines structures fibrées complexes. Archiv der Mathematik, 1958, 9: 102–109. doi: 10.1007/BF02287068
    [9]
    Newlander A, Nirenberg L. Complex analytic coordinates in almost complex manifolds. Ann. Math., 1957, 65 (3): 391–404. doi: 10.2307/1970051
    [10]
    Păunoiu A, Rivière T. Sobolev connections and holomorphic structures over Kähler surfaces. J. Func. Anal., 2021, 280 (12): 109003. doi: 10.1016/j.jfa.2021.109003
    [11]
    Stern M. Geometry of minimal energy Yang–Mills connections. J. Differential Geom., 2010, 86 (1): 163–188. doi: 10.4310/jdg/1299766686
    [12]
    Tanaka T. Some boundedness properties of solutions to the Vafa–Witten equations on closed 4-manifolds. The Quarterly Journal of Mathematics, 2017, 68 (4): 1203–1225. doi: 10.1093/qmath/hax015
    [13]
    Uhlenbeck K, Yau S T. On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure and Appl. Math., 1986, 39 (S1): S257–S293. doi: 10.1002/cpa.3160390714
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