Geometric structures on the moduli spaces in gauge theory and its applications to topology

规范理论在低维拓扑中仍然发挥着核心作用。特别是塞伯格-威滕理论已研究的观点，其应用拓扑结构。在这一领域，M。Furuta在等变同伦理论中引入了一个有限维近似来捕获方程，并由此得到了闭自旋4-流形的不变量和10/8-不等式的一个改进。在第一部分中，我们利用同伦理论研究了这个不变量。在第二部分中，我们给出了这个不变量的几何解释。第三部分研究了指数定理在Guillmen和斯滕贝格的几何量子化猜想中的应用。详细内容如下。第一个是与南纪彦的合作作品。利用Nishida的一个结果得到了4-流形连通和的精化Seiberg-Witten不变量的消失定理。与此结果相反，我们得到了一个非零定理。这个结果告诉我们，对于太多的4-流形的连通和，Seiberg-Witten不变量不能区分微分结构。第二部分研究了自旋4-流形上Seiberg-Witten方程模空间上的等变自旋结构。此外，我们将这个结果推广到第一Betti数为正的自旋4-流形.这意味着我们可以从具有自旋结构的模空间中重新证明10/8-不等式及其推广。关于第三个问题，研究小组组长得到报告说，关于Hamitonian环面作用量的Guilleman-Sternberg猜想可以从Riemann-Roch数的局部化得到重新证明。

Gauge theory has still played a central role in low dimensional topology. In particular Seiberg-Witten theory has been studied from the view point of its applications to topology. In this area 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.This research consists of three parts. In the first part we studied this invariant by using homotopy theory. In the second part we gave a geometric interpretation for this invariant. In the third part we studied the index theorem as an application to the geometric quantization conjecture of Guillmen and Sternberg. The detail is given below.The first one is a joint work with Norihiko Minami. We used a result by Nishida to get a vanishing theorem of the refined Seiberg-Witten invariants for the connected sum of 4-manifolds. In contrasts with this result, we have got a nonvanishing theorem. Then this result tells us that the Seiberg-Witten invariants cannot distinguish differential structure for the connected sum of too many 4-manifolds. In the second one we studied equivariant spin structures on the moduli spaces to Seiberg-Witten equations for a spin 4-manifold. Moreover we extended this result to a spin 4-manifold with positive first Betti number. These imply that we can reprove the 10/8-inequlatiy and its extension from the moduli space with the spin structure. As to the third one, the head investigator got a report that the Guilleman-Sternberg conjecture for Hamitonian torus actions can be reproved from the localization of the Riemann-Roch number.