Abstract This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed N = 2 supersymmetric Yang–Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a cohomological field theory. The well-known Donaldson–Witten theory is a twisted version of N = 2 SYM and can also be constructed using the Atiyah–Jeffrey construction (Atiyah and Jeffrey, 1990). This theory is concerned with the moduli space of anti-self-dual gauge connections, with a deformation theory controlled by an elliptic complex. More generally, supersymmetry requires considering configurations that look like either instantons or anti-instantons around fixed points, which we call flipping instantons. The flipping instantons of our 4D N = 2 theory are derived from the 5D contact instantons. The novelty is that their deformation theory is controlled by a transversally elliptic complex, which we demonstrate here. We repeat the Atiyah–Jeffrey construction in the equivariant setting and arrive at the Lagrangian (an equivariant Euler class in the relevant field space) that was also obtained from our previous work arXiv:1812.06473 . We show that the transversal ellipticity of the deformation complex is crucial for the non-degeneracy of the Lagrangian and the calculability of the theory. Our construction is valid on a large class of quasi toric 4 manifolds.

中文翻译：

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横向椭圆复形和上同调场论

摘要 这项工作是我们之前论文 arXiv:1812.06473 的延续，其中我们在具有孤立不动点的 Killing 矢量场的 4D 流形上构建了 N = 2 超对称 Yang-Mills 理论。在这项工作中，我们扩展了该理论的数学方面，特别关注其作为上同调场论的性质。著名的 Donaldson-Witten 理论是 N = 2 SYM 的扭曲版本，也可以使用 Atiyah-Jeffrey 构造（Atiyah 和 Jeffrey，1990）构建。该理论与反自双规范连接的模空间有关，具有由椭圆复形控制的变形理论。更一般地说，超对称需要考虑在固定点周围看起来像瞬时子或反瞬时子的配置，我们称之为翻转瞬时子。我们的 4D N = 2 理论的翻转瞬间子源自 5D 接触瞬间子。新颖之处在于，他们的变形理论是由横向椭圆复合体控制的，我们在这里演示。我们在等变设置中重复 Atiyah-Jeffrey 构造并得出拉格朗日（相关场空间中的等变欧拉类），这也是从我们之前的工作 arXiv:1812.06473 中获得的。我们表明变形复合体的横向椭圆度对于拉格朗日函数的非简并性和理论的可计算性至关重要。我们的构造适用于一大类拟复曲面 4 流形。我们在等变设置中重复 Atiyah-Jeffrey 构造并得出拉格朗日（相关场空间中的等变欧拉类），这也是从我们之前的工作 arXiv:1812.06473 中获得的。我们表明变形复合体的横向椭圆度对于拉格朗日函数的非简并性和理论的可计算性至关重要。我们的构造适用于一大类拟复曲面 4 流形。我们在等变设置中重复 Atiyah-Jeffrey 构造并得出拉格朗日（相关场空间中的等变欧拉类），这也是从我们之前的工作 arXiv:1812.06473 中获得的。我们表明变形复合体的横向椭圆度对于拉格朗日函数的非简并性和理论的可计算性至关重要。我们的构造适用于一大类拟复曲面 4 流形。