Equivariant homotopy theory and gauge theory

在Seiberg-Witten理论中，M.Furuta引入了一个有限维近似来捕捉等变同伦理论中的方程，由此得到了封闭自旋4流形的不变量和10/8不等式的改进。在本研究中，我们通过考虑一维上同调上的四重结构改进了这个不等式。更准确地说，我们定义了闭合自旋4流形的ko -特征的变体，并得到了由该不变量决定的附加项。我们还证明，如果四重结构与四维环面结构或其副本的连通和模2一致，则可以估计我们的改进。据M.Furuta报道，该研究人员目前正在将这一结果应用于辛4流形的Seiberg-Witten不变量的研究。在完成这项工作后，我们考虑了我们的结果与方程解的模空间的几何关系。最初这是由p.k rooneheimer研究的，他考虑了低维模空间。为了将该方法推广到高维模空间，我们在模空间上引入了一类ko -特征。那么从模空间的自旋结构的对称性可以直接得到10/8不等式以及上述改进。现在我们试着把这个方法应用到其他情况下，比如Yang-Mills理论。

In Seiberg-Witten theory M.Furuta has introduced a finite dimensional approximation to capture the equation in equivariant homotopy theory, by which he has obtained a refinement of the invariant and the 10/8-inequality for closed spin 4-manifolds.In this research we improved this inequality by taking into account the quadruple structure on one dimensional cohomology. More precisely we defined a variant of KO-characteristics for closed spin 4-manifoldfs and obtained an additional term determined by this invariant. We also showed that, if the quadruple structure is congruent to the one of 4-dimesional torus or the connected sum of its copies modulo 2, our improvement can be estimated. The researcher was reported by M.Furuta that he is now applying this result to study Seiberg-Witten invariants for symplectic 4-manifolds.After finishing this work we considered how our result is related to geometry of the moduli space of solutions to the equation. Originally this was studied by P.Kroneheimer, who considered this for low-dimensional moduli spaces. To extend his method to higher dimensional moduli spaces, we introduced a sort of KO-characteristics on the moduli space. Then the 10/8-inequality, as well as the above improvement, can be directly obtained from symmetry of the moduli space with its spin structure. Now we are trying to apply this method in other situations as Yang-Mills theory.