On orientations for gauge-theoretic moduli spaces
On orientations for gauge-theoretic moduli spaces
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DOI：
10.1016/j.aim.2019.106957
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发表时间：
2018-11
影响因子：
1.7
通讯作者：
D. Joyce;Yuuji Tanaka;M. Upmeier
中科院分区：
文献类型：
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作者：
D. Joyce;Yuuji Tanaka;M. Upmeier
设X是紧流形，D： Γ∞(E 0)→Γ∞(E 1)是X上的实椭圆算子，G是李群，P→X是主G束，P是P模规上所有连接的无限维模空间∇P，作为拓扑堆栈。对于每个[∇P]∈B P，我们可以考虑x上的扭曲椭圆算子D∇Ad (P)： Γ∞(Ad (P)⊗E 0)→Γ∞(Ad (P)⊗E 1)。这是基底B P上的连续椭圆算子族，因此在每个[∇P]上有一个取向束O P E•→B P，一个主Z 2束参数化Ker D∇Ad (P)的取向。（B P, E•）上的取向是O P E•= B P x Z 2的简化。在规范理论中，研究了满足某些曲率条件的连接∇P on P的模空间M P ga，如黎曼4流形（X, g）上的反自对偶实例。在良好的条件下，mpga是一个光滑流形，并且（bp, E•）上的方向在包含mpga“bp”下拉回到微分几何中通常意义上的mpga上的方向，这在诸如Donaldson理论等领域是重要的，在这些领域中，人们需要mpga上的方向来定义枚举不变量。我们解释了在x上固定一些代数拓扑信息后，在（B P, E•）上证明可定向性和构造正则定向的一系列已知和新的技术。我们利用这些技术在规范理论模空间上构造正则定向，包括关于2-和3-流形、实例、4-流形上的Kapustin-Witten方程和Vafa-Witten方程的模空间的新结果，以及5-流形上的haydy - witten方程。
Let X be a compact manifold, D: Γ∞(E 0)→ Γ∞(E 1) a real elliptic operator on X, G a Lie group, P→ X a principal G-bundle, and B P the infinite-dimensional moduli space of all connections∇ P on P modulo gauge, as a topological stack. For each [∇ P]∈ B P, we can consider the twisted elliptic operator D∇ Ad (P): Γ∞(Ad (P)⊗ E 0)→ Γ∞(Ad (P)⊗ E 1) on X. This is a continuous family of elliptic operators over the base B P, and so has an orientation bundle O P E•→ B P, a principal Z 2-bundle parametrizing orientations of Ker D∇ Ad (P)⊕ Coker D∇ Ad (P) at each [∇ P]. An orientation on (B P, E•) is a trivialization O P E•≅ B P× Z 2. In gauge theory one studies moduli spaces M P ga of connections∇ P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X, g). Under good conditions M P ga is a smooth manifold, and orientations on (B P, E•) pull back to orientations on M P ga in the usual sense of differential geometry under the inclusion M P ga↪ B P. This is important in areas such as Donaldson theory, where one needs an orientation on M P ga to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (B P, E•), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2-and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.