On orientations for gauge-theoretic moduli spaces

Abstract

Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $P\to X$ a principal $G$-bundle, and ${\mathcal{B}}_P$ the infinite-dimensional moduli space of all connections ${\nabla }_P$ on $P$ modulo gauge, as a topological stack. For each $[{\nabla }_P]\in {\mathcal{B}}_P$, we can consider the twisted elliptic operator $D^{{\nabla }_{Ad(P)}}$ on X. This is a continuous family of elliptic operators over the base ${\mathcal{B}}_P$, and so has an orientation bundle $O_P^D\to {\mathcal{B}}_P$, a principal ${\mathbb{Z}}_2$-bundle parametrizing orientations of Ker$D^{{\nabla }_{Ad(P)}}\oplus$Coker$D^{{\nabla }_{Ad(P)}}$ at each $[{\nabla }_P]$. An orientation on $({\mathcal{B}}_P,D)$ is a trivialization $O_P^D\cong {\mathcal{B}}_P\times {\mathbb{Z}}_2$. In gauge theory one studies moduli spaces $\mathcal{M}$ of connections ${\nabla }_P$ on $P$ satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds $(X,g)$. Under good conditions $\mathcal{M}$ is a smooth manifold, and orientations on $({\mathcal{B}}_P,D)$ pull back to orientations on $\mathcal{M}$ in the usual sense under the inclusion $\mathcal{M}\hookrightarrow {\mathcal{B}}_P$. This is important in areas such as Donaldson theory, where one needs an orientation on $\mathcal{M}$ to define enumerative invariants. We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on $({\mathcal{B}}_P,D)$, 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, Kapustin-Witten and Vafa-Witten equations on 4-manifolds, and the Haydys-Witten equations on 5-manifolds. Two sequels arXiv:1811.02405, arXiv:1811.09658 discuss orientations in 7 and 8 dimensions.

60 pages. (v2) sections 2.3-2.5 rewritten

References in corpus (6)

• Foundations of Topological Stacks I
• Topological and Smooth Stacks
• Gauge Theory in higher dimensions, II
• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)
• Orientability of the moduli space of Spin(7)-instantons

Cited by in corpus (10)

• Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi-Yau 4-folds
• Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections
• Canonical orientations for moduli spaces of $G_2$-instantons with gauge group SU(m) or U(m)
• The homology of moduli stacks of complexes
• Instantons on Sasakian 7-manifolds
• Batalin--Vilkovisky quantization and supersymmetric twists
• Universal structures in $\mathbb{C}$-linear enumerative invariant theories
• The Kapustin-Witten equations and nonabelian Hodge theory
• Wall-crossing for zero-dimensional sheaves and Hilbert schemes of points on Calabi-Yau 4-folds
• Localization of a $K{O}^{\ast }(\text{pt})$-valued index and the orientability of the $Pi{n}^{-}(2)$ monopole moduli space