Yang–Mills bar connection and holomorphic structure
Authors: Huang Teng ,   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
Published: September 2024

In this note, we study the Yang–Mills bar connection <inline-formula><tex-math id="M1">\begin{document}$ A $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M1.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M1.png"/></alternatives></inline-formula>, i.e., the curvature of <inline-formula><tex-math id="M2">\begin{document}$ A $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M2.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M2.png"/></alternatives></inline-formula> obeys <inline-formula><tex-math id="M3">\begin{document}$ \bar{\partial}_{A}^{\ast}F_{A}^{0,2} = 0 $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M3.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M3.png"/></alternatives></inline-formula>, on a principal <inline-formula><tex-math id="M4">\begin{document}$ G $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M4.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M4.png"/></alternatives></inline-formula>-bundle <inline-formula><tex-math id="M5">\begin{document}$ P $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M5.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M5.png"/></alternatives></inline-formula> over a compact complex manifold <inline-formula><tex-math id="M6">\begin{document}$ X $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M6.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M6.png"/></alternatives></inline-formula>. According to the Koszul–Malgrange criterion, any holomorphic structure on <inline-formula><tex-math id="M7">\begin{document}$ P $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M7.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M7.png"/></alternatives></inline-formula> can be seen as a solution to this equation. Suppose that <inline-formula><tex-math id="M8">\begin{document}$ G = SU(2) $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M8.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M8.png"/></alternatives></inline-formula> or <inline-formula><tex-math id="M9">\begin{document}$ SO(3) $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M9.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M9.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ X $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M10.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M10.png"/></alternatives></inline-formula> is a complex surface with <inline-formula><tex-math id="M11">\begin{document}$ H^{1}(X,\mathbb{Z}_{2}) = 0 $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M11.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M11.png"/></alternatives></inline-formula>. We then prove that the <inline-formula><tex-math id="M12">\begin{document}$ (0,2) $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M12.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M12.png"/></alternatives></inline-formula>-part curvature of an irreducible Yang–Mills bar connection vanishes, i.e., <inline-formula><tex-math id="M13">\begin{document}$ (P,\bar{\partial}_{A}) $\end{document}</tex-math><alternatives><graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M13.jpg"/><graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2023-0136_M13.png"/></alternatives></inline-formula> is holomorphic.