曲面上平坦联络的稳定模空间
The stable moduli space of flat connections over a surface
摘要：
我们计算的同伦类型的模空间的平坦，酉连接在非球面上，稳定后的秩的基础丛。在可定向曲面M^g上，我们证明了这个空间具有M^g的无穷对称积的同伦型，推广了环面上的一个著名事实。在一个非可定向曲面上，我们证明了这个空间同伦等价于两个环面的不相交并，其公共维数对应于曲面的第一（co）同调群的秩。对于曲面的乘积也有类似的计算，并与代数K理论中的奎伦-利希滕鲍姆定理有密切的相似性。这些证明利用了泰勒·劳森在变形K理论中的工作，并在很大程度上依赖于杨-米尔斯理论和规范理论。
英文摘要：
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the homotopy type of the infinite symmetric product of M^g, generalizing a well-known fact for the torus. Over a non-orientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen-Lichtenbaum conjectures in algebraic K-theory. The proofs utilize Tyler Lawson's work in deformation K-theory, and rely heavily on Yang-Mills theory and gauge theory.
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