Antiperfection of Yang-Mills Morse theory over a nonorientable surface

Anti-Perfect Morse Stratification

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincare series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface is locally equ…

Orientability in Yang-Mills Theory over Nonorientable Surfaces

In arXiv:math/0605587, the first two authors have constructed a gauge-equivariant Morse stratification on the space of connections on a principal U(n)-bundle over a connected, closed, nonorientable surface. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of this nonorientable surface. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when the nonorientable surface is not home…

不可定向曲面上的Yang-Mills联络

Yang-Mills Connections on Nonorientable Surfaces

在“The Yang-Mills equations over Riemann surfaces”中，Atiyah和Bott从莫尔斯理论的角度研究了Riemann曲面上的Yang-Mills泛函。我们将他们的研究推广到所有封闭的、紧致的、连通的、可能不可定向曲面。我们引入了曲面的基本群的“超中心扩张”的概念。它是曲面可定向时的中心延伸。我们建立了Yang-Mills联络与超中心扩张表示之间的精确对应。知道了这个确切的对应关系，我们就主要在有限维的表象变体的层次上而不是在无限维的层次上工作。

可定向和不可定向曲面上的Yang-Mills联络

Yang-Mills Connections On Orientable and Nonorientable Surfaces

在数学.SG/0605587中，我们研究了闭、连通、不可定向曲面上主G_R-丛的联络空间上的Yang-Mills泛函，其中G_R是任意紧连通李群.本文推广了Atiyah和Bott在“Riemann曲面上的Yang-Mills方程”中的讨论，并对数学进行了讨论。对于非酉典型群SO（n）和Sp（n），得到了Yang-Mills泛函在可定向曲面和不可定向曲面上Morse层的表示簇.事实证明，它与单一情况有很大不同。利用Laumon和Rapoport在“The Langlands lemma and the贝蒂数of stacks of G-bundles on a curve”中的方法，反演Atiyah-Bott递推关系，给出了全纯结构空间的半稳定层在主$SO(n,\bC)$ -丛或主$Sp(n,\bC)$ -丛上的有理等变Poincaré级数的显式表达式.