规范理论与辛结构
Gauge theory and symplectic structures
中文总结 AI 辅助
该研究针对四维以上流形辛结构存在性障碍问题，在殆厄米特流形上引入椭圆方程组，证明其指标性质、解的横截性与唯一性，相关结果为高维规范理论提供了证据。
中文摘要 AI 辅助
受寻找四维以上流形上辛结构存在性障碍这一问题的驱动，我们在殆厄米特流形上引入了一个椭圆方程组，该方程组在四维时可约化为带有Taubes扰动的Seiberg-Witten方程。定义该方程需要在流形上选取一个spin^C结构。我们证明该系统对典范spin^C结构的指标为零；此外，在殆凯勒（辛）流形上，我们构造了一个典范解，并证明当扰动参数取大值时该解具有横截性，而在六维时，我们在某一假设下得到了该典范解的唯一性。我们还证明，在凯勒流形上，该方程在自然假设下可约化为涡旋方程。这些结果为可能存在的高维规范理论提供了证据，该理论能够生成殆复结构的不变量，最终为相容辛形式提供障碍。
英文摘要
Motivated by the problem of finding obstructions to the existence of symplectic structures in dimensions higher than four, we introduce an elliptic system of equations on almost Hermitian manifolds that reduces to the Seiberg--Witten equations with Taubes' perturbation in dimension four. To define the equations, one needs to choose a spin$^{\mathbb C}$ structure on the manifold. We prove that the system has index zero for the canonical spin$^{\mathbb C}$ structure. Moreover, on almost Kähler (symplectic) manifolds, we construct a canonical solution and prove its transversality with large values of the perturbation parameter, while in dimension six we obtain uniqueness of the canonical solution under a certain assumption. We also show that on Kähler manifolds the equations reduce, under a natural ansatz, to the vortex equations. These results provide evidence toward a possible higher dimensional gauge theory capable of producing invariants of almost complex structures and ultimately, obstructions to compatible symplectic forms.
