arXiv 1811.024052018-11math.DGmath.AT

规范群为SU（m）或U（m）的$G_2$ -瞬子模空间的正则定向

Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)

Dominic Joyce, Markus Upmeier

备注：: 30 pages. Version 3: slightly revised presentation, fixed typos. Final version, to appear in the Journal of Differential Geometry

摘要：

假设$(X, g)$是一个紧的、自旋黎曼7-流形，其Dirac算子为$D$。设$G$是SU $(m)$或U $(m)$，$E\to X$是具有$G$ -结构的秩$m$复丛.将$E$上联络的无限维模空间写成${\mathcal B}_E$，模规范。在${\mathcal B}_E$中的每个$[A]$处，存在一个自然的主${\mathbb Z}_2$丛$O^D_E\to{\mathcal B}_E$，用于参数化扭曲椭圆算子$D_{{\rm Ad }A}$的det $\,D_{{\rm Ad }A}$的方向. Walpuski的一个定理表明$O^D_E$是可平凡化的。我们证明，如果我们选择一个方向为det $\,D$，并在X上的标志结构的意义乔伊斯arXiv：1610.09836，那么我们可以定义规范trivializations的$O^D_E$为所有这样的束$E\to X$，满足自然的兼容性。现在让$(X,φ,g)$是一个紧凑的$G_2$ -流形，具有d $(*φ)=0$。然后我们可以考虑$E\to X$上$G_2$ -瞬子的模空间${\mathcal M}_E^{G_2}$，它是在适当的横截性条件下的光滑流形，并且可以用${\mathcal M}_E^{G_2}\subset{\mathcal B}_E$导出一般流形。$O^D_E$对${\mathcal M}_E^{G_2}$的限制是${\mathcal M}_E^{G_2}$上的${\mathbb Z}_2$ -定向丛.因此，我们的定理在所有这样的$G_2$ -瞬子模空间${\mathcal M}_E^{G_2}$上导出了正则定向.这有助于唐纳森-西格尔计划arXiv：0902.3239，提出定义枚举不变量的$G_2$ -流形$(X,φ,g)$通过计数模空间${\mathcal M}_E^{G_2}$，与标志取决于选择的方向。本文是Joyce-Tanaka-Upmeier arXiv：1811.01096的续篇，它发展了规范论模空间上定向的一般理论，并给出了在3，4，5和6维中的应用.第三篇论文曹格罗斯乔伊斯arXiv：1811.09658研究方向的模空间在8维。

英文摘要：

Suppose $(X, g)$ is a compact, spin Riemannian 7-manifold, with Dirac operator $D$. Let $G$ be SU$(m)$ or U$(m)$, and $E\to X$ be a rank $m$ complex bundle with $G$-structure. Write ${\mathcal B}_E$ for the infinite-dimensional moduli space of connections on $E$, modulo gauge. There is a natural principal ${\mathbb Z}_2$-bundle $O^D_E\to{\mathcal B}_E$ parametrizing orientations of det$\,D_{{\rm Ad }A}$ for twisted elliptic operators $D_{{\rm Ad }A}$ at each $[A]$ in ${\mathcal B}_E$. A theorem of Walpuski shows $O^D_E$ is trivializable. We prove that if we choose an orientation for det$\,D$, and a flag structure on X in the sense of Joyce arXiv:1610.09836, then we can define canonical trivializations of $O^D_E$ for all such bundles $E\to X$, satisfying natural compatibilities. Now let $(X,φ,g)$ be a compact $G_2$-manifold, with d$(*φ)=0$. Then we can consider moduli spaces ${\mathcal M}_E^{G_2}$ of $G_2$-instantons on $E\to X$, which are smooth manifolds under suitable transversality conditions, and derived manifolds in general, with ${\mathcal M}_E^{G_2}\subset{\mathcal B}_E$. The restriction of $O^D_E$ to ${\mathcal M}_E^{G_2}$ is the ${\mathbb Z}_2$-bundle of orientations on ${\mathcal M}_E^{G_2}$. Thus, our theorem induces canonical orientations on all such $G_2$-instanton moduli spaces ${\mathcal M}_E^{G_2}$. This contributes to the Donaldson-Segal programme arXiv:0902.3239, which proposes defining enumerative invariants of $G_2$-manifolds $(X,φ,g)$ by counting moduli spaces ${\mathcal M}_E^{G_2}$, with signs depending on a choice of orientation. This paper is a sequel to Joyce-Tanaka-Upmeier arXiv:1811.01096, which develops the general theory of orientations on gauge-theoretic moduli spaces, and gives applications in dimensions 3,4,5 and 6. A third paper Cao-Gross-Joyce arXiv:1811.09658 studies orientations on moduli spaces in dimension 8.

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