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

规范理论在低维拓扑学中仍然发挥着核心作用。特别是从拓扑学应用的角度研究了Seiberg-Witten理论。在这方面，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，由此他得到了封闭自旋4流形的不变量和10/8不等式的改进。本研究由三个部分组成。第一部分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。第三部分研究了指标定理在Guillmen和Sternberg几何量化猜想中的应用。详情如下。第一部是与南则彦合作的作品。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。与这个结果相反，我们得到了一个不消失定理。然后这个结果告诉我们Seiberg-Witten不变量不能区分太多4流形的连通和的微分结构。第二章研究了自旋4流形的Seiberg-Witten方程的模空间上的等变自旋结构。并将此结果推广到具有正第一贝蒂数的自旋4流形。这意味着我们可以用自旋结构从模空间证明10/8不等式及其推广。对于第三个问题，首席研究员报告说，可以从黎曼-罗奇数的局域化来反驳哈密顿环体作用的guillemann - sternberg猜想。

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.